Lecture notes on field theory in
condensed matter physics
Christopher Mudry
Condensed matter theory group, Paul Scherrer Institute, Switzerland
E-mail address: [email protected]
PREFACE
1
Preface
Reading the books from Baym, [1] Messiah, [2] and Dirac [3], while
an undergraduate student at ETHZ, taught me how enjoyable and useful it is to learn quantum mechanics from different perspectives. This
impression was reinforced as I got exposed to statistical physics and to
the diversity of approaches to it found in the books from Becker, [4]
Callen, [5] Huang, [6] and Feynman. [7]
My initiation to quantum field theory was different. In those days,
there seemed to be two separate communities doing many-body physics.
I had taken a proseminar from Prof. Klaus Hepp on the theory of
renormalization, who had told me that the only book on quantum
field theory relevant to his class was that of Itzykson and Zuber. [8]
Although I already had taken a proseminar on Fermi liquid theory
and had started reading Kittel’s Quantum theory of solid, [9] I had
not realized the close connection between the many-body physics applied to high-energy physics, statistical physics, and condensed matter
physics. At the time, the few books on quantum field theory for highenergy physics were obsessed with propagators, Feynman diagrams,
causality and positivity, and how to make sense of ultraviolet divergences. The venerable books on many-body physics in condensed matter physics were deceptive. [10]-[13] Although they claimed to do manybody physics, single-particle physics was soon enough resuscitating as
a mean-field approximation or with particles with diminutive names
(quasiparticles). Moreover, these books were full of approximations
with mysterious acronyms such as the random phase approximation,
involving some magical circular logic.
This cultural divide manifest in the books published prior to the
80’s was however in the process of disappearing. As I was moving
to UIUC to start my PhD, the standard model had established itself
as the fundamental theory for particle physics and a steady supply of
books devoted to it were being published by the late 80’s. The renormalization group had been applied to explain asymptotic freedom of
quantum chromodynamics (QCD), to solve the Kondo problem, and
had a profound impact on statistical physics. Lattice gauge theory, an
approach to solve QCD in the strong coupling problem, was turning
into a discipline of its own bridging relativistic quantum field theory to
statistical physics. Algebraic topology, an arcane discipline of mathematics to most physicists, had shown its value to classify defects in the
vacua of quantum field theories and the order parameters of symmetry
broken phases in condensed matter physics or statistical physics. Algebraic topology could explain the quantization of the quantized Hall
effect and the existence of collective excitations obeying exchange statistics evading the “spin statistics Theorem”. Integrable models from
statistical physics were used to explain the low-energy properties of
2
spin chains, impurity models, and quasi-one-dimensional metals. The
second remarkable discovery of the 80’s in condensed matter physics
was of course that of high Tc superconductivity, a class of materials
that defy a solution using perturbation theory to this date.
The first books that I am aware of aiming at overcoming the cultural
differences between high-energy physics, statistical physics, and condensed matter physics of the 60’s are those of Polyakov, [14] Parisi, [15]
and Negele and Orland [16], whose authors had made seminal contributions to the revolution brought upon by the application of the renormalization group to theoretical physics during the late 60’s and 70’s.
Another generation of authors came along in the 90’s with the intent to
explain how the machinery of quantum field theory should be applied
to condensed matter physics. [17]-[22]
Since the turn of the 21st century, concepts and techniques have
been shared from condensed matter theory to string theory. The breadth
of topics in condensed matter physics makes it impossible to cover all
applications of quantum field theory to condensed matter physics in
a single book. Correspondingly, the number of books applying quantum field theory to condensed matter physics is steadily increasing and
getting more specialized. A student has now the luxury of picking his
favorite book and taking advantage of a variety of view points.
This book is the result of teaching the class “Quantum field theory
in condensed matter physics” at ETHZ. My aim was to demystify some
of the condensed matter jargon used in seminars in condensed matter
physics for a student at the level of a master degree in physics from
ETHZ. I also wanted a student attending my class to obtain a handson experience of concepts such as spontaneous symmetry breaking,
mean-field theory, random phase approximation, screening, quantum
fluctuations, renormalization group flows, critical points, phase transitions driven by topological defects, bosonization, etc. Many books
on quantum field theory devote space to the machinery of quantum
field theory before solving problems with it. I wanted my teaching to
do the reverse, i.e., to develop the needed methodology one problem
at a time. I also did not want quantum field theory to become the
primary interest. It had to remain a tool to explain as economically as
possible fundamental principles of condensed matter physics. I am of
the opinion that the most efficient technique for this purpose is to systematically use the path integral representation of quantum mechanics.
Path integrals are thus pervasive in this book. However, I assume no
more prior knowledge than familiarity with quantum mechanics, at the
level of Baym’s book say.
PREFACE
3
The book is organized in two parts. The first part deals with bosons,
the second with fermions. In condensed matter physics, this organization principle is not as obvious as would be implied by the standard model of high-energy physics. The fundamental boson of condensed matter physics is the photon. The fundamental fermions of
condensed matter physics are the electron and the proton, the charged
constituents of the atoms from the periodic table. On the relevant energy and length scales of condensed matter physics, these elementary
constituents interact through the rules of quantum electrodynamics at
a non-vanishing density of fermionic matter in the ground state. This
is the main difference with quantum field theory aiming at explaining high-energy scattering experiments, for which the ground state
(the vacuum before and after scattering) has a vanishing density of
(fermionic) matter. This difference is of a fundamental nature. The
atomic nucleus has a much larger mass than the electrons orbiting
around it. In a material, positive charge is localized in position space on
the sites of a crystal at low temperatures. As a result, the fermionic nature of the ionic constituents becomes irrelevant. What matters greatly
however is that the normal modes of this crystal are phonons, collective excitations obeying Bose-Einstein statistics. The same can happen
with electrons. They can localize in position space, in which case the
material is called an insulator. Some localized electrons can still interact through their internal spin-1/2 degree of freedom. It is often the
case that the collective excitations resulting from the interactions between the spins of localized electrons are collective excitations obeying
Bose-Einstein statistics. They are called magnons. Electrons need not
be localized in a material, which is then called a metal. In a metal,
the mobile electrons exchange photons with each other, they interact
through the Coulomb interaction, they interact with the localized positive charge of the crystal, a one-body potential for the electrons, they
interact with the phonons, and they might interact with some of the
electrons that are localized around the crystalline sites. Solving this
many-body problem from the Schrödinger equation is and will be impossible. The Hilbert space is simply too large. Instead, effective theories motivated by phenomenology and simplicity have been the bread
and butter of theoretical condensed matter physics. In these models,
the elementary local constituents might be bosons, fermions, or more
complicated objects of which the simplest examples are quantum spin
degrees of freedom. The partition of this book into a part devoted to
bosons and a part devoted to fermions refers to the situations when
some of the low-energy collective excitations can be shown to obey
the Bose-Einstein or Fermi-Dirac statistics, respectively. Even then,
we shall show that bosons and fermions emerging from some interacting models on a one-dimensional lattice are interchangeable under the
rule of bosonization. The four chapters on bosons cover phonons (as
4
a way to introduce a quantum field theory), superfluidity, restoration
of a continuous symmetry by fluctuations at the lower critical dimension, and the Kosterlitz-Thouless phase transition, respectively. The
five chapters on fermions cover non-interacting fermions, the random
phase approximation in the jellium model, superconductivity, dissipative Josephson junction (an example of dissipative quantum mechanics), and bosonization, respectively. Each chapter ends with a section
in which material is presented as a sequence of exercises. Each chapter also comes with an appendix. Some appendices provide distracting
intermediary steps. Most appendices contain learning material.
The books of Naoto Nagaosa in Ref. [20] and Mike Stone in Ref. [21]
have been very influential when preparing my lectures. I am indebted
to these authors for these inspiring books.
Gipf-Oberfrick, July 2013
Christopher Mudry
ACKNOWLEDGMENTS
5
Acknowledgments
I must start thanking Donald E. Knuth for developing TeX. I typeset my master thesis and had I needed to do the same for a book, I
would have never written one.
I am grateful to my home institution, the Paul Scherrer Institut
(PSI), in the persons of Kurt Clausen who has been supportive of this
endeavor and Joël Mesot who has been a steady and reliable advocate of
the condensed matter theory group. Since 1999 to this date, I benefited
at PSI from a great colleague, Rudolf Morf.
I am indebted to my mentors Eduardo Fradkin (my PhD adviser),
Xiao-Gang Wen (my host at MIT), and Bertrand Halperin (my host
at Harvard) for shaping my taste in physics. I am also indebted to my
friends and long term collaborators Claudio Chamon, Akira Furusaki,
Piet Brouwer, and Shinsei Ryu who have been so influential on my
understanding of physics.
I have had the good fortune of directing the thesis of three talented
students: Andreas Schnyder, Sebastian Guerrero, and Titus Neupert.
They were all teaching assistants of my class and made important contributions to the exercises. Titus had also the kindness and patience
for converting my figures into artworks.
In the last six months Maurizio Storni has helped me polish my
lecture notes into this book. He even shares my compulsive obsession
with using TeX for baroque notation! It has been my privilege to
benefit from his dedicated and critical reading. Maurizio has been the
fairy-godmother of Cinderella for my lecture notes. I only hope there is
no midnight deadline. Of course, as convention dictates, all remaining
embarrassing mistakes are my responsibility.
Contents
Preface
Acknowledgments
Part 1.
1
4
Bosons
1
Chapter 1. The harmonic crystal
Outline
1.1. Introduction
1.2. Classical one-dimensional crystal
1.3. Quantum one-dimensional crystal
1.4. Higher-dimensional generalizations
1.5. Problems
3
3
3
3
10
17
17
Chapter 2. Bogoliubov theory of a dilute Bose gas
Outline
2.1. Introduction
2.2. Second quantization for bosons
2.3. Bose-Einstein condensation and spontaneous symmetry
breaking
2.4. Dilute Bose gas: Operator formalism at vanishing
temperature
2.5. Dilute-Bose gas: Path-integral formalism at any
temperature
2.6. Problems
25
25
25
25
Chapter 3. Non-Linear-Sigma Models
Outline
3.1. Introduction
3.2. Non-Linear-Sigma-Models (NLσM)
3.3. Fixed point theories, engineering and scaling dimensions,
irrelevant, marginal, and relevant interactions
3.4. General method of renormalization
3.5. Perturbative expansion of the two-point correlation
function up to one loop for the two-dimensional O(N )
NLσM
3.6. Callan-Symanzik equation obeyed by the spin-spin
correlator in the d = 2-dimensional O(N > 2) NLσM
3.7. Beta function in the d > 2-dimensional O(N > 2) NLσM
7
29
36
43
57
65
65
65
65
78
89
90
99
109
8
CONTENTS
3.8. Problems
119
Chapter 4. Kosterlitz-Thouless transition
157
Outline
157
4.1. Introduction
157
4.2. Classical two-dimensional XY model
157
4.3. The Coulomb-gas representation of the classical 2d–XY
model
168
4.4. Equivalence between the Coulomb gas and Sine-Gordon
model
169
4.5. Fugacity expansion of n-point functions in the Sine-Gordon
model
178
4.6. Kosterlitz renormalization-group equations
184
4.7. Problems
193
Part 2.
Fermions
Chapter 5. Non-interacting fermions
Outline
5.1. Introduction
5.2. Second quantization for fermions
5.3. The non-interacting jellium model
5.4. Time-ordered Green functions
5.5. Problems
201
203
203
203
203
207
226
240
Chapter 6. Jellium model for electrons in a solid
251
Outline
251
6.1. Introduction
251
6.2. Definition of the Coulomb gas in the Schrödinger picture 251
6.3. Path-integral representation of the Coulomb gas
255
6.4. The random-phase approximation
258
6.5. Diagrammatic interpretation of the random-phase
approximation
263
6.6. Ground-state energy in the random-phase approximation 266
6.7. Lindhard response function
267
6.8. Random-phase approximation for a short-range interaction282
6.9. Feedback effect on and by phonons
284
6.10. Problems
286
Chapter 7. Superconductivity in the mean-field and randomphase approximations
Outline
7.1. Pairing-order parameter
7.2. Scaling of electronic interactions
7.3. Time- and space-independent Landau-Ginzburg action
7.4. Mean-field theory of superconductivity
7.5. Nambu-Gork’ov representation
307
307
307
313
321
328
332
CONTENTS
7.6.
7.7.
7.8.
7.9.
9
Effective action for the pairing-order parameter
Effective theory in the vicinity of T = 0
Effective theory in the vicinity of T = Tc
Problems
334
335
354
359
Chapter 8. A single dissipative Josephson junction
Outline
8.1. Phenomenological model of a Josephson junction
8.2. DC Josephson effect
8.3. AC Josephson effect
8.4. Dissipative Josephson junction
8.5. Instantons in quantum mechanics
8.6. The quantum-dissipative Josephson junction
8.7. Duality in a dissipative Josephson junction
8.8. Renormalization-group methods
8.9. Conjectured phase diagram for a dissipative Josephson
junction
8.10. Problems
367
367
367
372
372
373
384
404
409
417
423
425
Chapter 9. Abelian bosonization in two-dimensional space and
time
459
Outline
459
9.1. Introduction
459
9.2. Abelian bosonization of the Thirring model
461
9.3. Applications
477
9.4. Problems
494
Appendix A. The harmonic-oscillator algebra and its coherent
states
A.1. The harmonic-oscillator algebra and its coherent states
A.2. Path-integral representation of the anharmonic oscillator
A.3. Higher dimensional generalizations
513
513
517
520
Appendix B. Some Gaussian integrals
B.1. Generating function
B.2. Bose-Einstein distribution and the residue theorem
521
521
522
Appendix C. Non-Linear-Sigma-Models (NLσM) on Riemannian
manifolds
525
C.1. Introduction
525
C.2. A few preliminary definitions
525
C.3. Definition of a NLσM on a Riemannian manifold
528
C.4. Classical equations of motion for NLσM:
Christoffel symbol and geodesics
529
C.5. Riemann, Ricci, and scalar curvature tensors
531
C.6. Normal coordinates and vielbeins for NLσM
538
C.7. How many couplings flow on a NLσM?
557
10
CONTENTS
Appendix D. The Villain model
Appendix E. Coherent states for fermions, Jordan-Wigner
fermions, and linear-response theory
E.1. Grassmann coherent states
E.2. Path-integral representation for fermions
E.3. Jordan-Wigner fermions
E.4. The ground state energy and the single-particle
time-ordered Green function
E.5. Linear response
559
565
565
568
569
578
583
Appendix F. Landau theory of Fermi liquids
599
Introduction
599
F.1. Adiabatic continuity
599
F.2. Quasiparticles
601
F.3. Topological stability of the Fermi surface
603
F.4. Quasiparticles in the Landau theory of Fermi liquids as
poles of the two-point Green function
612
F.5. Breakdown of Landau Fermi liquid theory
612
Appendix G. First-order phase transitions induced by thermal
fluctuations
Outline
G.1. Landau-Ginzburg theory and the mean-field theory of
continuous phase transitions
G.2. Fluctuations induced by a local gauge symmetry
G.3. Applications
615
619
626
Appendix H. Useful identities
H.1. Proof of Equation (8.75)
627
627
615
615
Appendix I. Non-Abelian bosonization
635
I.1. Introduction
635
I.2. Minkowski versus Euclidean spaces
635
I.3. Free massless Dirac fermions and the Wess-Zumino-Witten
theory
637
I.4. A quantum-mechanical example of a Wess-Zumino action 646
I.5. Wess-Zumino action in (1 + 1)–dimensional Minkowski
space and time
650
I.6. Equations of motion for the WZNW action
653
I.7. One-loop RG flow for the WZNW theory
657
I.8. The Polyakov-Wiegmann identity
659
I.9. Integration of the anomaly in QCD2
660
I.10. Bosonization of QCD2 for infinitely strong gauge coupling
673
Appendix. Bibliography
681
Part 1
Bosons
CHAPTER 1
The harmonic crystal
Outline
The classical equations of motion for a finite chain of atoms are
solved within the harmonic approximation. In the thermodynamic
limit, an approximate hydrodynamical description, i.e., a one-dimensional
classical field theory, is obtained. Quantization of the finite harmonic
chain is undertaken. In the thermodynamic limit, phonons in a onedimensional lattice are approximated by a quantum hydrodynamical
theory, i.e., a one-dimensional quantum field theory.
1.1. Introduction
To illustrate the transition from the one-body to the many-body
physics, the harmonic excitations of a crystal are derived classically
and quantum mechanically. The thematic of crystallization, i.e., of
spontaneous-symmetry breaking of translation symmetry in position
space, is addressed in section 1.5 from the point of view of an application of the Mermin-Wagner theorem.
1.2. Classical one-dimensional crystal
1.2.1. Discrete limit. For simplicity, we shall consider a onedimensional world made of N point-like objects (atoms) of mass m
and interacting through a potential V . We assume first that the potential V depends only on the coordinates ηn ∈ R, n = 1, · · · , N , of
the N atoms,
V = V (η1 , · · · , ηN ).
(1.1)
Furthermore, we assume that V has a non-degenerate minimum at
η̄n = n a,
n = 1, · · · , N,
where a is the lattice spacing. For example,
−1 a 2 N
X
2π
V (η1 , · · · , ηN ) =
κ
1 − cos
η
− ηn
2π
a n+1
n=1
N a 2
X
2π
2
+
mΩ
1 − cos
ηn
2π
a
n=1
+ boundary terms.
3
(1.2)
(1.3)
4
1. THE HARMONIC CRYSTAL
The physical interpretation of the real-valued parameters κ and Ω is
obtained as follows. For small deviations δηn about minimum (1.2), it
is natural to expand the potential energy according to
V (η̄1 + δη1 , · · · , η̄N + δηN ) = V (η̄1 , · · · , η̄N ) +
N
−1
X
n=1
1
+ m Ω2
2
N
X
2
κ
δηn+1 − δηn
2
(δηn )2 + · · · + boundary terms.
n=1
(1.4)
The dimensionful constant κ is the elastic or spring constant. It measures the strength of the linear restoring force between nearest-neighbor
atoms. The characteristic frequency Ω measures the strength of an external force that pins atoms to their equilibrium positions (1.2). To
put it differently, m Ω2 is the curvature of the potential well that pins
an atom to its equilibrium position. Terms that have been neglected in
· · · are of several kinds. Only terms of quadratic order in the nearestneighbor relative displacement δηn+1 −δηn have been accounted for, and
all interactions beyond the nearest-neighbor range have been dropped.
We have also omitted to spell out what the boundary terms are. They
are specified once boundary conditions have been imposed. In the limit
N → ∞, the choice of boundary conditions should be immaterial since
the bulk potential energy should be of order L ≡ N a, whereas the
energy contribution arising from boundary terms should be of order
L0 = 1.
To minimize boundary effects in a finite system, one often imposes
periodic boundary conditions
ηn+N = ηn ,
n = 1, · · · , N.
(1.5)
An open chain of atoms turns into a ring after imposing periodic boundary conditions. Furthermore, imposing periodic boundary conditions
endows the potential with new symmetries within the harmonic approximation defined by 1
N X
2 1
κ
2
2
Vhar (η̄1 +δη1 , · · · , η̄N +δηN ) :=
δηn+1 − δηn + m Ω (δηn ) .
2
2
n=1
(1.7)
First, changing labels according to
n → n + m,
1
n = 1, · · · , N,
∀m ∈ Z,
(1.8)
Without loss of generality, we have set the classical minimum of the potential
energy to zero,
V (η̄1 , · · · , η̄N ) = 0.
(1.6)
1.2. CLASSICAL ONE-DIMENSIONAL CRYSTAL
5
leaves Eq. (1.7) invariant. Second, translation invariance is recovered
in the absence of the pinning potential,
Eq. (1.7) with Ω = 0 =⇒
Vhar (η1 , · · · , ηN ) = Vhar (η1 + x a, · · · , ηN + x a)
(1.9)
for any real-valued x.
The kinetic energy of an open chain of atoms is simply given by
2
N N
1 X dδηn
1 X ˙ 2
T (η̄1 + δη1 , · · · , η̄N + δηN ) = m
≡ m
δηn .
2 n=1
dt
2 n=1
(1.10)
As was the case for the potential energy, the choice of boundary conditions only affects the kinetic energy by terms of order L0 . It is thus
natural to choose periodic boundary conditions if one is interested in
extensive properties of the system.
The classical Lagrangian L in the harmonic approximation and with
periodic boundary conditions is defined by subtracting from the kinetic
energy (1.10) the potential energy (1.7),
N
X
2
2
1
2
2
L :=
m δη˙ n − κ δηn+1 − δηn − m Ω (δηn ) . (1.11)
2
n=1
The classical equations of motion follow from Euler-Lagrange equations
of motion
d
∂L
∂L
,
n = 1, · · · , N.
(1.12)
=
dt ∂ δη˙ n
∂ δηn
They are
mδη¨n = κ δηn+1 + δηn−1 − 2δηn − m Ω2 δηn ,
n = 1, · · · , N,
(1.13)
with the complex-valued and traveling-wave solutions
κ
δηn (t) ∝ ei(kn−ωt) ,
ω 2 = 2 (1 − cos k) + Ω2 .
(1.14)
m
Imposing periodic boundary conditions allows to identify the normal
modes. These are countably-many traveling waves with the frequency
to wave-number relation
r
κ
2π
ωl = 2 (1 − cos kl ) + Ω2 ,
kl =
l,
l = 1, · · · , N. (1.15)
m
N
The most general real-valued solution of Euler-Lagrange equations (1.13)
obeying periodic boundary conditions is
N
X
δηn (t) =
Al e+i(kl n−ωl t) + A∗l e−i(kl n−ωl t) ,
n = 1, · · · , N.
l=1
(1.16)
6
1. THE HARMONIC CRYSTAL
Here, the complex-valued expansion coefficient Al remains arbitrary as
long as initial conditions on δηn and δη˙ n have not been specified.
To revert to the Hamilton-Jacobi formalism of classical mechanics,
one introduces the canonical momentum δπn conjugate to δηn through
δπn (t) :=
∂L
∂ δη˙ n
= − im
N
X
ωl Al e+i(kl n−ωl t) − A∗l e−i(kl n−ωl t) ,
n = 1, · · · , N,
l=1
(1.17)
and construct the Hamiltonian
N
X
2
1 (δπn )2
2
2
H=
+ κ δηn+1 − δηn + m Ω (δηn )
2
m
n=1
(1.18)
from the Lagrangian (1.11) through a Legendre transformation. HamiltonJacobi equations of motion are then
∂H
δη˙ n = +
= {δηn , H},
∂δπn
∂H
δπ˙ n = −
= {δπn , H},
∂δηn
n = 1, · · · , N,
(1.19)
2
where {·, ·} stands for the Poisson brackets.
In the long wave-number limit kl 1, the dispersion relation reduces to
κ
ωl2 = kl2 + Ω2 + O(kl4 ).
(1.21)
m
The pinning potential characterized by the curvature Ω of the potential
well has opened a gap in the spectrum of normal modes. No solutions
to Euler-Lagrange equations (1.13) can be found below the characteristic frequency Ω. By switching off the pinning potential, Ω = 0, the
dispersion relation simplifies to
κ
ωl2 = kl2 + O(kl4 ).
(1.22)
m
p
The proportionality constant κ/m between frequency and wave number is interpreted as the velocity of propagation of a sound wave in the
one-dimensional harmonic chain in units for which the lattice spacing
a has been set to unity.
2
The Poisson bracket {f, g} of two functions f and g of the canonical variables
δηn and δπn is defined by
N X
∂f ∂g
∂f ∂g
{f, g} :=
−
.
(1.20)
∂δηn ∂δπn
∂δπn ∂δηn
n=1
1.2. CLASSICAL ONE-DIMENSIONAL CRYSTAL
7
1.2.2. Thermodynamic limit. The thermodynamic limit N →
∞ emerges naturally if one is interested in the response of a onedimensional solid to external perturbations as can be induced, say,
by compressions. If the characteristic wavelength of a perturbation
applied to a solid is much larger than the atomic separation, then the
(elastic) response from this solid to this perturbation is dominated by
normal modes with arbitrarily small wave numbers k → 0. If so, it
is then much more economical not to account for the discrete nature
of this solid as is done in the Lagrangian (1.11) when computing the
(elastic) response. To this end, Eq. (1.11) is first rewritten as
"
#
N
X
δηn+1 − δηn 2 m 2
1 m ˙ 2
L=
a
δηn − κa
− Ω (δηn )2
2
a
a
a
n=1
(1.23)
N
X
=:
a Ln .
n=1
We interpret
µ :=
m
,
a
Y := κ a,
ξ :=
δηn+1 − δηn
,
a
and
Ln , (1.24)
as the mass per unit length, the Young’s modulus, 3 the elongation per
unit length, and the local Lagrangian per unit length (the Lagrangian
density) , respectively. Then, we write
" #
2
ZL
2
1
∂ϕ
∂ϕ
µ
L = dx
−Y
− µΩ2 ϕ2
2
∂t
∂x
0
(1.26)
ZL
=: dx L,
0
whereby the following substitutions
have been performed.
R
P
[1] The discrete sum n has been replaced by the integral dx/a
over the semi-open interval ]0, L].
[2] The relative displacement δηn at time t has been replaced by the
value of the real-valued function ϕ at the space-time coordinate (x, t)
obeying periodic boundary conditions in position space,
ϕ(x + L, t) = ϕ(x, t),
x ∈]0, L],
∀t ∈ R.
(1.27)
3
For an elastic rode obeying Hooke’s law, the extension ξ of the rode per unit
length is proportional to the exerted force F with the Young’s modulus Y as the
proportionality constant,
F = Y ξ.
(1.25)
8
1. THE HARMONIC CRYSTAL
[3] The time derivative of the relative displacement δηn at time
t has been replaced by the value of the time derivative (∂t ϕ) at the
space-time coordinate (x, t).
[4] The discrete difference δηn+1 − δηn at time t has been replaced
by the lattice constant times the value of the derivative (∂x ϕ) at the
space-time coordinate (x, t).
[5] The integrand L in Eq. (1.26) is called the Lagrangian density.
It is a real-valued function of space and time. From it, one obtains the
continuum limit of Euler-Lagrange equations (1.12) according to
∂t
δL(x, t)
δL(x, t)
δL(x, t)
+ ∂x
=
.
δ(∂t ϕ)(y, t)
δ(∂x ϕ)(y, t)
δϕ(y, t)
(1.28)
Here, the symbol δL(x, t) is to be interpreted as the infinitesimal functional change of L at the given space-time coordinates (x, t) induced
by the Taylor expansion
δL = L[ϕ + δϕ, (∂x ϕ) + δ(∂x ϕ), (∂t ϕ) + δ(∂t ϕ)] − L[ϕ, (∂x ϕ), (∂t ϕ)]
∂L
∂L
∂L
=
δϕ +
δ(∂x ϕ) +
δ(∂ ϕ) + · · · .
∂ϕ
∂(∂x ϕ)
∂(∂t ϕ) t
(1.29)
One must keep in mind that ϕ, (∂x ϕ), and (∂t ϕ) are independent “variables”. Moreover, one must use the rule
δϕ(x, t)
= δ(x − y) =⇒
δϕ(y, t)
ZL
dx
δϕ(x, t)
= 1,
δϕ(y, t)
y ∈]0, L],
(1.30)
0
that extends the rule
N
X
∂ηm
∂ηm
= δm,n =⇒
= 1,
∂ηn
∂ηn
m=1
n = 1, · · · , N,
(1.31)
to the continuum. Otherwise, all the usual rules of differentiation apply
to δ · /δϕ.
[6] Equations of motion (1.13) become the one-dimensional sound
wave equation
s
Y
∂t2 − v 2 ∂x2 + Ω2 ϕ = 0,
v :=
,
(1.32)
µ
after replacing the finite difference
δηn+1 + δηn−1 − 2δηn = + δηn+1 − δηn − δηn − δηn−1
(1.33)
by a2 times the value of the second-order space derivative (∂x2 ϕ) at the
space-time coordinate (x, t).
The Hamiltonian H in the continuum limit follows from Eq. (1.26)
with the help of a (functional) Legendre transform or directly from the
1.2. CLASSICAL ONE-DIMENSIONAL CRYSTAL
9
continuum limit of Eq. (1.18),
#
"
2
ZL
2
1 π
∂ϕ
H = dx
+Y
+ µΩ2 ϕ2
2 µ
∂x
0
(1.34a)
ZL
dx H,
=:
0
where the field π is the canonically conjugate to ϕ,
ZL
π(x, t) :=
dy
0
δL(y, t)
= µ(∂t ϕ)(x, t).
δ(∂t ϕ)(x, t)
(1.34b)
Probing the one-dimensional harmonic crystal on length scales much
larger than the lattice spacing a blurs our vision to the point where the
crystal appears as an elastic continuum. Viewed without an atomic
microscope, the relative displacements δηn , n = 1, · · · , N , become a
field ϕ(x, t) where x can be any real-valued number provided N is sufficiently large. 4
The mathematics that justifies this blurring or coarse graining is
that, for functions f that vary slowly on the lattice scale,
Z
X
dx
f (n a) −→
f (x).
(1.35)
a
n
In particular,
Z
X δm,n
X
a
δm,n f (n a) =
f (n a) −→ f (x) = dy δ(x−y)f (y),
f (m a) =
a
n
n
(1.36)
justifies the identification
δm,n
−→ δ(x − y).
a
(1.37)
Equation (1.37) tells us that the divergent quantity δ(x = 0) in position
space should be thought of as the inverse, 1/a, of the lattice spacing a.
In turn, the number 1/a can be interpreted as the spacing of normal
modes in reciprocal space per unit volume 2π/N in wave-number space
by the following argument,
k − kl
1
1
= l+1
×
,
a
a
2π/N
4
kl :=
2π
l.
N
(1.38)
In mathematics, a (real-valued scalar) field ϕ is a mapping ϕ : Rd+1 →
R, (r, t) 7→ ϕ(r, t). In physics, a field is often abbreviated by the value ϕ(r, t) it
takes at the point (r, t) in (d + 1)-dimensional (position) space and time.
10
1. THE HARMONIC CRYSTAL
How does one go from a discrete Fourier sum to a Fourier integral?
Start from an even number N of sites for which
N
X
2π
eikl (m−n) = N δm,n ,
kl :=
l.
(1.39)
N
l=1
Multiply both sides of this equation by the inverse of the system size
L = N a,
N
1 X ikl (m−n) δm,n
.
(1.40)
e
=
L l=1
a
Since the right-hand side should be identified with δ(x − y) in the
thermodynamic limit N → ∞, the left-hand side should be identified
with
2π/a
Z
Z+∞
N
1 X i kl (m−n) a
dk ik(x−y)
dk ik(x−y)
e a
−→
e
≈
e
,
(1.41)
L l=1
2π
2π
−∞
0
whereby
kl
−→ k,
(m − n) a −→ x − y.
(1.42)
a
To see this, recall first that the periodic boundary conditions tell us that
l = 1, · · · , N could have equally well be chosen to run between −N/2+1
and +N/2. Hence, it is permissible to adopt the more symmetrical rule
N
1X ˜
f (kl ) −→
L l=1
+π/a
Z
dk ˜
f (k)
2π
(1.43)
−π/a
to convert a finite summation over wave numbers into an integral over
the first Brillouin zone (reciprocal space) ]−π/a, +π/a] as the thermodynamic limit N = L/a → ∞ is taken. Now, if f (x) is a slowly varying
function on the lattice scale a, its Fourier transform f˜(k) will be essentially vanishing for |k| 1/a. In this case, the limits ±π/a can safely
be replaced by the limits ±∞ on the right-hand side of Eq. (1.43). We
then arrive at the desired integral representation of the delta function
in position space,
Z+∞
dk ik(x−y)
δ(x − y) =
e
.
(1.44)
2π
−∞
Observe that factors of 2π appear in an asymmetrical way in integrals over position (x) and reciprocal (k) spaces. Although this is
purely a matter of convention when defining the Fourier transform,
there is a physical reasoning behind this choice. Indeed, Eq. (1.43) implies that dk/(2π) has the physical meaning of the number of normal
modes in reciprocal space with wave number between k and k + dk
1.3. QUANTUM ONE-DIMENSIONAL CRYSTAL
11
per unit volume L in position space. Correspondingly, the divergent
quantity 2π δ(k = 0) in reciprocal space has the physical meaning of
being the divergent volume L → ∞ of the system as is inferred from
Z+∞
Z+∞
dk ikx
δ(x) =
e ⇐⇒ 2πδ(k) =
dx e−ikx .
(1.45)
2π
−∞
−∞
1.3. Quantum one-dimensional crystal
1.3.1. Reminiscences about the harmonic oscillator. We now
turn to the task of giving a quantum-mechanical description for a nondissipative one-dimensional harmonic crystal. One possible route consists in the construction of a Hilbert space with the operators acting
on it and whose expectation values can be related to measurable properties of the crystal. 5 In this setting, the time evolution of physical
quantities can be calculated either in the Schrödinger or in the Heisenberg picture. We will begin by reviewing these two approaches in the
context of a single harmonic oscillator. The extension to the harmonic
crystal will then follow in a very natural way.
The classical Hamiltonian that describes a single particle of unit
mass m = 1 confined to a quadratic well with curvature ω 2 is
1 2
H :=
p + ω 2 x2 .
(1.46)
2
Hamilton-Jacobi equations of motion are
∂H
dx(t)
= {x, H} = +
= p(t),
dt
∂p
(1.47)
dp(t)
∂H
2
= {p, H} = −
= −ω x(t).
dt
∂x
Solutions to these classical equations of motion are
x(t) = A cos(ωt) + B sin(ωt),
(1.48)
p(t) = ω [−A sin(ωt) + B cos(ωt)] .
The energy E of the particle is a constant of the motion that depends
on the choice of initial conditions through the two real-valued constants
A and B,
1 2
A + B 2 ω2.
(1.49)
E=
2
In the Schrödinger picture of quantum mechanics, the position x of
the particle and its canonical conjugate p become operators x̂ and p̂ that
(i) act on the Hilbert space of twice-differentiable and square-integrable
functions Ψ : R → C and (ii) obey the canonical commutation relation
[x̂, p̂] := x̂ p̂ − p̂ x̂ = i~.
5
(1.50)
Another route to quantization is by means of the path-integral representation
of quantum mechanics as is shown in appendix A.
12
1. THE HARMONIC CRYSTAL
The time evolution (or dynamics in short) of the system is encoded by
Schrödinger equation
i~ ∂t Ψ(x, t) = Ĥ Ψ(x, t),
(1.51a)
where the quantum Hamiltonian Ĥ is given by
1 2
(1.51b)
Ĥ =
p̂ + ω 2 x̂2 .
2
The time evolution of the wave function Ψ(x, t) is unique once initial
conditions Ψ(x, t = 0) are given.
Solving the time-independent eigenvalue problem
Ĥ ψn (x) = εn ψn (x)
(1.52a)
is tantamount to solving the time-dependent Schrödinger equation through
the Ansatz
X
Ψ(x, t) =
cn ψn (x) e−iεn t/~ .
(1.52b)
n
The expansion coefficients cn ∈ C are time independent and uniquely
determined by the initial condition, say Ψ(x, t = 0).
As is well known, the energy eigenvalues εn are given by
1
εn = n +
~ ω,
n = 0, 1, 2, · · · .
(1.53)
2
The energy eigenfunctions ψn (x) are Hermite polynomials multiplying
a Gaussian,
ω 1/4 1 ω 2
ψ0 (x) =
e− 2 ~ x ,
π~
1/4
1ω 2
4 ω 3
xe− 2 ~ x ,
ψ1 (x) =
π ~
1ω 2
ω 1/4 ω
(1.54)
2 x2 − 1 e − 2 ~ x ,
ψ2 (x) =
4π~
~
..
.
n 1/2 n
1ω 2
~
ω 1/4 ω
d
1
x−
e− 2 ~ x .
ψn (x) = n
2 n! ω
π~
~
dx
The Heisenberg picture of quantum mechanics is better suited than
the Schrödinger picture to a generalization to quantum field theory. In
the Heisenberg picture, and contrary to the Schrödinger picture, operators are explicitly time dependent. For any operator Ô, the solution
to the equation of motion 6
i~
6
dÔ(t)
= [Ô(t), Ĥ]
dt
(1.55a)
The assumption that the system is non-dissipative has been used here in that
Ĥ does not depend explicitly on time, ∂t Ĥ = 0.
1.3. QUANTUM ONE-DIMENSIONAL CRYSTAL
13
that replaces Schrödinger equation is
Ô(t) = e+i
Ĥt
~
Ô(t = 0) e−i
Ĥt
~
.
(1.55b)
By definition, the algebra obeyed by operators in the Schrödinger picture holds true in the Heisenberg picture provided operators are taken
at equal time. For example,
x̂(t) := e+i
Ĥt
~
x̂(t = 0) e−i
Ĥt
~
,
p̂(t) := e+i
Ĥt
~
p̂(t = 0) e−i
Ĥt
~
, (1.56)
obey by construction the equal-time commutator
∀t ∈ R.
[x̂(t), p̂(t)] = i~,
(1.57)
Finding the commutator of x̂(t) and p̂(t0 ) at unequal times t 6= t0 requires solving the dynamics of the system, i.e., Eq. (1.55a) with Ô
substituted for x̂ and p̂, respectively,
dx̂(t)
dp̂(t)
= +p̂(t),
= −ω 2 x̂(t).
(1.58)
dt
dt
In other words, the Heisenberg operators x̂(t) and p̂(t) satisfy the same
equations of motion as the classical variables they replace,
dx̂(t)
d2 x̂(t)
2
+
ω
x̂(t)
=
0,
p̂(t)
=
.
(1.59)
dt2
dt
The solution (1.48) can thus be borrowed with the caveat that A and
B should be replaced by time-independent operators  and B̂.
At this stage, it is more productive to depart from following a strategy dictated by the real-valued classical solution (1.48). The key observation is that the quantum Hamiltonian for the harmonic oscillator
takes the quadratic form 7
1
†
Ĥ = ~ ω â (t) â(t) +
,
(1.60a)
2
if the pair of canonically conjugate Hermitean operators x̂(t) and p̂(t)
is traded for the pair ↠(t) and â(t) of operators defined by
r
r
~ ~ †
x̂(t) =:
â(t) + â (t) ,
p̂(t) =:
−iωâ(t) + iω↠(t) .
2ω
2ω
(1.60b)
†
Once the equal-time commutator [â(t), â (t)] is known, the Heisenberg
equations of motion are easily derived from
d↠(t)
= [↠(t), Ĥ],
dt
With the help of
r ω
p̂(t)
†
â (t) =
x̂(t) − i
,
2~
ω
i~
7
i~
dâ(t)
= [â(t), Ĥ].
dt
r
â(t) =
(1.61)
ω
p̂(t)
x̂(t) + i
,
2~
ω
(1.62a)
We are anticipating that Ĥ does not depend explicitly on time.
14
1. THE HARMONIC CRYSTAL
one verifies that
[x̂(t), p̂(t)] = i~,
[x̂(t), x̂(t)] = [p̂(t), p̂(t)] = 0 ⇐⇒
[â(t), ↠(t)] = 1,
[â(t), â(t)] = [↠(t), ↠(t)] = 0.
(1.62b)
The change of Hermitean operator-valued variables to non-Hermitean
operator-valued variables is advantageous in that the equations of motion for ↠(t) and â(t) decouple according to
d↠(t)
= +iω ↠(t),
dt
dâ(t)
= −iω â(t),
dt
↠(t) = ↠(t = 0) e+iωt ,
(1.63)
â(t) = â(t = 0) e
−iωt
,
respectively. Below, we will write â for â(t = 0) and similarly for ↠.
The time evolution of x̂(t), p̂(t), and Ĥ is now explicitly given by
r
~
â e−iωt + ↠e+iωt ,
2ω
r
~ω
p̂(t) = −i
â e−iωt − ↠e+iωt ,
2
1
†
.
Ĥ = ~ ω â â +
2
x̂(t) =
(1.64a)
(1.64b)
(1.64c)
As must be by the absence of dissipation, Ĥ is explicitly time independent, ∂t Ĥ = 0.
The Hilbert space can now be constructed explicitly with purely
algebraic methods. The Hilbert space is defined by all possible linear
combinations of the eigenstates
n
â†
|ni := √ |0i,
n!
Ĥ|ni = εn |ni,
n = 0, 1, 2, · · · .
(1.65)
Here, the ground state or vacuum |0i is defined by the condition
â|0i = 0.
(1.66)
One verifies that ψ0 (x) in Eq. (1.54) uniquely (up to a phase) satisfies
Eq. (1.66) by using the position-space representation of the operator â.
1.3.2. Discrete limit. In the spirit of the Heisenberg picture for
the harmonic oscillator and guided by the Fourier expansions in Eqs. (1.16)
1.3. QUANTUM ONE-DIMENSIONAL CRYSTAL
15
and (1.17), we begin by defining the operators
s
N
i
1 X
~ h +i(kl n−ωl t)
η̂n (t) := √
âl e
+ â†l e−i(kl n−ωl t) , n = 1, · · · , N,
N l=1 2ωl
r
N
i
1 X ~ ωl h +i(kl n−ωl t)
† −i(kl n−ωl t)
π̂n (t) := −i √
âl e
− âl e
, n = 1, · · · , N,
2
N l=1
(1.67a)
where the frequency ωl and the integer label l are related by Eq. (1.15),
i.e., (remember that we have chosen units in which the mass is given
by m = 1)
q
2π
ωl = 2κ (1 − cos kl ) + Ω2 ,
kl :=
l,
l = 1, · · · , N, (1.67b)
N
and the operator-valued expansion coefficients â†l and âl obey the harmonic oscillator algebra
[âl , â†l0 ] = δl,l0 ,
[âl , âl0 ] = [â†l , â†l0 ] = 0,
l, l0 = 1, · · · , N. (1.67c)
√
The normalization factor 1/ N is needed to cancel the factor of N
present in the Fourier series
N
X
eikl (m−n) = N δm,n
(1.68)
l=1
that shows up when one verifies that the equal-time commutators
m, n = 1, · · · , N,
(1.69)
hold for all times. We are now ready to define in a consistent way
the Hamiltonian Ĥ for the quantum one-dimensional harmonic crystal
[compare with Eq. (1.18)]
[η̂m (t), π̂n (t)] = i~ δm,n ,
[η̂m (t), η̂n (t)] = [π̂m (t), π̂n (t)] = 0,
N
o
X
2
1n
Ĥ :=
[π̂n (t)]2 + κ η̂n+1 (t) − η̂n (t) + Ω2 [η̂n (t)]2 .
2
n=1
(1.70)
With the help of the algebra (1.67c), one verifies that Ĥ is explicitly
time independent and given by
N
X
1
†
Ĥ =
~ ωl âl âl +
.
(1.71)
2
l=1
The next task is to construct the Hilbert space for the one-dimensional
quantum crystal by algebraic methods. Assume that there exists a
unique (up to a phase) normalized state |0i, the ground state or vacuum, defined by
h0|0i = 1,
âl |0i = 0,
l = 1, · · · , N.
(1.72)
16
1. THE HARMONIC CRYSTAL
If so, the state
|n1 , n2 , · · · , nN i :=
N
Y
l=1
1 † n l
p
âl |0i,
nl !
n1 , n2 , · · · , nN = 0, 1, 2, · · · ,
(1.73)
is normalized to one and is an eigenstate of Ĥ with the energy eigenvalue
N
X
1
εn1 ,··· ,nN :=
.
(1.74)
~ ωl n l +
2
l=1
The ground-state energy is of order N and given by
N
ε0,··· ,0
1X
:=
~ ωl .
2 l=1
(1.75)
Excited states have at least one nl > 0. They are called phonons. The
eigenstate |n1 , n2 , · · · , nN i is said to have n1 phonons in the first mode,
n2 phonons in the second mode, and so on. Phonons can be thought
of as identical elementary particles since they possess a definite energy
and momentum. Because the phonon occupation number
nl = hn1 , · · · , nl , · · · , nN |â†l âl |n1 , · · · , nl , · · · , nN i
(1.76)
is an arbitrary positive integer, phonons obey Bose-Einstein statistics.
Upon switching on a suitable interaction [say by including cubic and
quartic terms in the expansion (1.4)], phonons scatter off one other just
as other “-ons” (mesons, photons, gluons, and so on) known to physics
do. Although we are en route towards constructing the quantum field
η̂(x, t) out of η̂n (t), we have encountered particles. The duality between
quantum fields and particles is the essence of quantum field theory.
The vector space spanned by the states labeled by the phonon occupation numbers (n1 , · · · , nN ) ∈ {0, 1, 2, · · · }N in Eq. (1.73) is the
Hilbert space of the one-dimensional quantum crystal. The mathematical structure of this Hilbert space is a symmetric tensor product of N
copies of the Hilbert space for the harmonic oscillator. In physics, this
symmetric tensor product is called a Fock space when the emphasis is
to think of the phonon as an “elementary particle”.
1.3.3. Thermodynamic limit. Taking the thermodynamic limit
N → ∞ is a direct application to section 1.3.2 of the rules established
in the context of the classical description of sections 1.2.2 Hence, with
1.3. QUANTUM ONE-DIMENSIONAL CRYSTAL
the identifications
N
X
17
8
Z
a −→
dx,
n=1
N
1 X
−→
N a l=1
Z
dk
,
2π
r
√
v2
ωl −→ ω(k) = 2 2 [1 − cos(k a)] + Ω2 ≈ v 2 k 2 + Ω2 , if |k a| 1,
a
kl n −→ kx,
1
âl −→ √
â(k),
Na
√
η̂n (t) −→ a η̂(x, t),
√
π̂n (t) −→ a π̂(x, t),
(1.77)
the canonically conjugate pairs of operators η̂n (t) and π̂n (t) are replaced
by the quantum fields
s
Z
dk
~ η̂(x, t) :=
â(k) e+i[kx−ω(k)t] + ↠(k) e−i[kx−ω(k)t] ,
2π 2ω(k)
r
Z
dk ~ ω(k) â(k) e+i[kx−ω(k)t] − ↠(k) e−i[kx−ω(k)t] ,
π̂(x, t) := −i
2π
2
(1.78)
respectively. 9 Their equal-time commutators follow from the harmonic
oscillator algebra
[â(k), ↠(k 0 )] = 2πδ(k − k 0 ),
[â(k), â(k 0 )] = [↠(k), ↠(k 0 )] = 0.
(1.79)
They are
[η̂(x, t), π̂(y, t)] = i~ δ(x−y),
8
[η̂(x, t), η̂(y, t)] = [π̂(x, t), π̂(y, t)] = 0.
(1.80)
Limits of integrations in position and reciprocal spaces are left unspecified
at this stage as we want to remain free to choose how the thermodynamic limit
N → ∞ is to be taken. For example, we could keep a finite in which case the
thermodynamic limit N → ∞ implies L → ∞. Alternatively, we could keep L
finite in which case the thermodynamic limit N → ∞ implies a → 0.
9 The substitution rules â −→ √ 1 â(k), η̂ (t) −→ √a η̂(x, t), and π̂ (t) −→
n
n
l
Na
R dk
PN
√
a π̂(x, t), are needed to cancel the volume factor N a in l=1 −→ N a 2π
.
18
1. THE HARMONIC CRYSTAL
The Hamiltonian is
Z
1
Ĥ = dx
[π̂(x, t)]2 + v 2 [∂x η̂(x, t)]2 + Ω2 [η̂(x, t)]2
2
Z
dk 1
=
~ ω(k) ↠(k)â(k) + â(k)↠(k) .
2π 2
(1.81)
The excitation spectrum is obtained by making use of the commutator between ↠(k) and â(k). It is given by
Z
dk
~ ω(k) ↠(k) â(k)
(1.82)
Ĥ − E0 :=
2π
and is observed to vanish for the vacuum |0i. The operation of subtracting from the Hamiltonian the ground state energy E0 is called
normal ordering. It amounts to placing all annihilation operators â(k)
to the right of the creation operators ↠(k). The ground state energy
E0 := h0|Ĥ|0i
Z
dk 1
=
~ ω(k) × 2πδ(k = 0)
2π 2
Z
= (Volume in position space) ×
=
dk 1
~ ω(k)
2π 2
(1.83)
X 1
~ω
2 modes
modes
can be ill-defined for two distinct reasons. First, if N → ∞ with a
held fixed, there exists an upper cut-off to the integral over reciprocal
space at the Brillouin zone boundaries ±π/a and E0 is only infrared
divergent due to the fact that 2πδ(k = 0) is the diverging volume
L = N a in position space. Second, even if L = N a is kept finite
while both the infrared N → ∞ and ultraviolet a → 0 limits are taken,
the absence of an upper cut-off in the k integral can cause the zeropoint energy density E0 /L to diverge as well. Divergences of E0 or
E0 /L are only of practical relevance
if one can control experimentally
P
ω(k) or the density of states modes and thereby measure changes in
E0 or E0 /L. For example, this can be achieved in a resonant cavity
whose size is variable. If so, changes of E0 with the cavity size can be
measured. These changes in the zero point energy are known as the
Casimir energy. Sensitivity to E0 with measurable consequences also
occurs when, upon tuning of some internal parameters entering the microscopic Hamiltonian, the vacuum state |0i becomes unstable, i.e., is
not the true ground state anymore. The system then undergoes a quantum phase transition. Finally, divergences of E0 /L matter greatly if the
energy-momentum tensors of “matter fields” are dynamical variables
as is the case in cosmological models.
1.5. PROBLEMS
19
1.4. Higher-dimensional generalizations
Generalizations to higher dimensions are straightforward. The coordinates x ∈ R1 and k ∈ R1 in position and reciprocal one-dimensional
spaces need only be replaced by the vectors r ∈ Rd and k ∈ Rd , in
position and reciprocal d-dimensional spaces, respectively.
1.5. Problems
1.5.1. Absence of crystalline order in one and two dimensions.
Introduction. We are going to prove the Mermin-Wagner theorem
for the case of crystalline order in two (and one) dimensions of position
space. [23] The Mermin-Wagner theorem states that classical particles
in a box, i.e., particles that are subject to hard-wall boundary conditions, cannot exhibit crystalline order in one and two dimensions,
provided that the pair potential Φ(r) through which they interact satisfies certain conditions [see Eq. (1.109)].
Before we start with the derivation, let us set up some notation.
Given the pair potential Φ(r), the internal energy of a configuration
of N particles with coordinates r 1 , · · · , r N in d-dimensional position
space is given by
N
1 X
U (r 1 , · · · , r N ) =
Φ(r i − r j ).
2 i6=j
(1.84)
Using this, we can define the (classical) ensemble average of a realvalued function f of the coordinates r 1 , · · · , r N by
1
hf i :=
Z
Z
B
N
Y
!
dd r i
e−β U (r1 ,··· ,rN ) f (r 1 , · · · , r N )
(1.85a)
i=1
and
Z
N
Y
B
i=1
Z :=
!
dd r i
e−β U (r1 ,··· ,rN ) .
(1.85b)
Here, β is the inverse temperature after the Boltzmann constant kB
has been set to unity and B denotes the box over which the integration
is taken.
20
1. THE HARMONIC CRYSTAL
Step 1: Proof of Bogoliubov’s inequality. The proof of the MerminWagner theorem will be crucially based on an inequality due to Bogoliubov, which for our purposes can be formulated as
2
N
P
+
* N
hϕ(r i )∇ψ(r i )i
2
X
i=1
+,
ψ(r i ) ≥ *
N
N
P
P
2
i=1
β
|∇ϕ(r )|2
∆Φ(r − r ) ϕ(r ) − ϕ(r ) +
2
i
j
i
i
j
i=1
i,j=1
(1.86)
for a real-valued function ϕ that is continuous and differentiable and
vanishes on the boundary ∂B of B, while ψ is complex valued and
sufficiently smooth. Our first task is to prove Eq. (1.86).
Exercise 1.1: Convince yourself that the bilinear map
h·, ·i : L × L → R,
(ϕ, ψ) 7→ hϕ, ψi := hϕ∗ ψi ,
(1.87)
for two complex-valued functions ϕ and ψ belonging to the set L of
continuous differentiable functions from B to R with the standard definition of a product of two functions, is a scalar product. We then have
the Schwarz inequality
2 2
|f1 |
|f2 | ≥ |hf1 f2 i|2
(1.88)
for any pair of functions f1 and f2 from L at our disposal.
Exercise 1.2: Use the Schwarz inequality (1.88) with the choice
f1 (r 1 , · · · , r N ) :=
N
X
ψ(r i ),
(1.89a)
i=1
N
X
1
,r N )
f 2 (r 1 , · · · , r N ) := − eβ U (r1 ,··· ,rN )
∇i ϕ(r i ) e−β U (r1 ,···(1.89b)
,
β
i=1
to prove Eq. (1.86). Hint: Use partial integration.
Step 2: Densities on the lattice. We now want to use the Bogoliubov inequality (1.86) to probe the tendency towards crystalline order
(we specialize to d = 2 for simplicity, but without loss of generality).
Suppose that the crystalline order has the Bravais lattice vectors a1
and a2 and consists of N1 × N2 sites so that
B = r ∈ Rd |r = x1 a1 N1 + x2 a2 N2 ,
0 ≤ x1 , x2 < 1 . (1.90)
The reciprocal lattice vectors K are given by
K := n1 b1 + n2 b2 ,
n1 , n2 ∈ Z,
and a general wave vector k is given by
n
n
k := 1 b1 + 2 b2 ,
N1
N2
bi · aj = 2π δij ,
n1 , n2 ∈ Z.
i, j = 1, 2,
(1.91)
(1.92)
1.5. PROBLEMS
21
To probe whether particles form a crystal, we have to compute their
density at the reciprocal lattice vectors. In position space, the density
of a configuration of N particles is
ρ(r) :=
N
X
δ(r − r i ).
(1.93)
i=1
Its Fourier component at momentum k is given by
Z
2
−ik·r
d re
ρk :=
ρ(r) =
N
X
e−ik·ri .
(1.94)
i=1
B
This allows us to sharpen a criterion for crystalline order as follows. A
crystal has formed, if
1
hρk i = 0, if k is not a reciprocal lattice vector,
N1 ,N2 →∞ N
1
lim
hρK i 6= 0, for at least one reciprocal lattice vector K.
N1 ,N2 →∞ N
(1.95)
lim
The thermodynamic limit is taken in such a way that the filling of the
system with particles n := N/(N1 N2 ) is held constant.
Exercise 2.1: Define the following momenta. Let k be an arbitrary
wave vector from the first Brillouin zone as given by Eq. (1.92). Its
components with respect to the basis b1 and b2 of the reciprocal lattice
are
ki = ni bi /Ni
(1.96)
for i = 1, 2. Let K be a reciprocal lattice vector, for which Eq. (1.95)
is claimed not to vanish.
(a) Show that the functions defined by
ψ(r) := e−i(k+K)·r ,
ϕ(r) := sin(k1 · r) sin(k2 · r),
(1.97)
are such that ϕ vanishes on the boundary ∂B of the box B.
(b) Show that the Bogoliubov inequality (1.86) with these functions yields
Nk,K
ρ+k+K ρ−k−K ≥
,
Dk,K
Nk,K
(1.98a)
where the numerator is
E2
(k + K)2 D
:=
ρK + ρK+2k − ρK+2k1 − ρK+2k2 , (1.98b)
16 β
22
1. THE HARMONIC CRYSTAL
while the denominator is (∆ is Laplace operator in two-dimensional
position space)
Dk,K :=
N
2 E
1 XD
∆Φ(r i − r j ) sin(k1 · r i ) sin(k2 · r i ) − sin(k1 · r j ) sin(k2 · r j )
2
i,j=1
+
N
E
1 XD
|k2 sin(k1 · r i ) cos(k2 · r i ) + (1 ↔ 2)|2 .
β
i=1
(1.98c)
Exercise 2.2:
(a) Show that there exists an A > 0 that depends only on b1 and
b2 such that the estimates
A (k1 + k2 )2 ≥ (ν1 k1 + ν2 k2 )2 , (1.99)
A (k1 + k2 )2 ≥ k21 + k22 ,
for any pair ν1 and ν2 of real numbers of magnitudes less or
equal to one, |ν1 | ≤ 1, |ν2 | ≤ 1, hold. We are now going to
make use of these inequalities.
(b) Establish upper bounds on the trigonometric functions in the
denominator (1.98c) to infer that
1 1
ρ+k+K ρ−k−K ≥ 2
N
N
β Nk,K
A k2 1 +
β
N
!.
N P
∆Φ(r − r ) (r − r )2
i
j
i
j
i,j=1
(1.100)
It will be the quadratic k-dependence in the denominator on which
the argument crucially relies (it would break down for a k-linear or
constant term). From here on, the task is to find suitable estimates for
the remaining factors.
To refine the estimate of the denominator on the right-hand side of
Eq. (1.100), we have to impose conditions on the asymptotic behavior
of the pair potential Φ(r) at small and large r. To that end, we consider
a family of pair potentials labeled by a real number λ > 0
Φλ (r) := Φ(r) − λ r 2 |∆Φ(r)|.
(1.101)
We define the free energy F to be the functional from the space of pair
potentials to the real-valued numbers that assigns to any pair potential
Φ̃ the value
!
!
Z Y
N
N
X
β
− β F [Φ̃] := ln
dd r i exp −
Φ̃(r i − r j ) . (1.102)
2 i6=j
i=1
B
Exercice 2.3: Show that
N
F [Φ0 ] − F [Φλ ]
1 X
≥
(r i − r j )2 |∆Φ(r i − r j )| ≥ 0.
Nλ
2N i,j=1
(1.103)
1.5. PROBLEMS
23
Hint: Use the representation
Zλ
F [Φ0 ] − F [Φλ ] = −
dλ0
∂F [Φλ0 ]
∂λ0
(1.104)
0
and the fact that (prove!)
∂ 2 F [Φλ ]
∂F [Φλ ]
= hDiλ ,
−
= β (D − hDiλ )2 λ , (1.105)
2
∂λ
∂λ
where h· · · iλ denotes the ensemble average using the potential Φλ and
−
N
1X
D(r 1 , · · · , r N ) :=
(r − r j )2 |∆Φ(r i − r j )|.
2 i,j=1 i
(1.106)
The inequality that we are seeking applies to a restricted class of
two-body potentials {Φ}. This restriction comes about because we need
to insure that the thermodynamic limit N1 , N2 → ∞ is well defined.
More precisely, we need the existence of the free energy per particle
f0 :=
lim
N1 ,N2 →∞
F [Φ0 ]
< ∞,
N
(1.107)
where we made use of the definition (1.101) for Φ0 ≡ Φ. Furthermore,
we need the existence of at least one λ > 0 for which
F [Φλ ]
fλ := lim
< ∞,
(1.108)
N1 ,N2 →∞
N
i.e., fλ is intensive. 10 For any λ > 0 that satisfies Eq. (1.108), we
can then use the fraction (f0 − fλ )/λ to estimate the right-hand side
of Eq. (1.100) by writing
β Nk,K
1 1
.
ρ+k+K ρ−k−K ≥ 2
2
N
N A k 1 + 2λβ (f0 − fλ )
(1.110)
Exercice 2.4: By assumption (1.95), the averages hρK+2k i, hρK+2k1 i,
and hρK+2k2 i vanish in the thermodynamic limit if 2k, 2k1 , and 2k2 are
not reciprocal lattice vectors, respectively. Starting from Eq. (1.110),
show that
1 X
1 K 20 g(|K| + |K 0 |/2) hρK i2 1 X
1
g(|q|) ρq ρ−q ≥
,
2
β
2
VN q
64 A 1 + λ (f0 − fλ )
N V
k2
|k|<|K
|/2
0
|
{z
}
(1.111)
10
It can be shown (see Ref. [23]) that a sufficient condition on Φλ for
Eq. (1.108) to hold is that
Φλ (r)
|r|→∞
∼
|r|−(2+) ,
|r|→0
0
Φλ (r) > const × |r|−(2+ ) ,
0
where const is a positive number and , are two positive numbers.
(1.109)
24
1. THE HARMONIC CRYSTAL
where K 0 is the reciprocal lattice vector with smallest magnitude and
the positive function g : R → R+ , k → g(k) > 0 is a Gaussian centered
at the origin.
The strategy to complete the proof will now be as follows. By inspection of the right-hand side of Eq. (1.111), we anticipate that the
factor that is underbraced is non-vanishing but finite in the thermodynamic limit. In contrast, the sum over 1/k2 , once turned into an
integral, diverges logarithmically near the origin. (What happens if
d = 1 or d > 2?) If the left-hand side of Eq. (1.111) turns out to
have a finite upper bound in the thermodynamic limit, the logarithmic
divergence forces
hρK i2
N2
N1 ,N2 →∞
−→
0.
(1.112)
This is not compatible with our criterion for crystalline order.
To make this line of arguments work, it thus remains to show that
the left-hand side of Eq. (1.111) is bounded from above in the thermodynamic limit. To that end, we define the function
Z
d2 q
δΦ(r) :=
g(|q|) eiq·r .
(1.113)
(2π)2
Exercice 2.5: Show that, in the thermodynamic limit,
δΦ(0) + 2
F [Φ] − F [Φ − δΦ]
N
N
1 X
≥ δΦ(0) +
δΦ(r i − r j )
N i6=j
1 X
g(|q|) ρ+q ρ−q ,
=
VN q
(1.114)
where F [Φ−δΦ] is the free energy for the system with the pair potential
Φ − δΦ as defined in Eq. (1.102).
The left-hand side of Eq. (1.114) contains the difference in free
energy per particle for the pair potential Φ and Φ − δΦ. We require
(and used above already) that F [Φ]/N is finite in the thermodynamic
limit. If a Gaussian is added to the pair potential, this behavior is
unaffected, as the additive contribution δΦ(r) is well behaved both for
small and large r. It follows that also F [Φ − δΦ]/N is finite in the
thermodynamic limit.
The estimate (1.114) thus allows us to recast the inequality in (1.111)
for sufficiently large N in the form
hρK i2 1
c>
N2 V
X
|k|<|K 0
1
,
k2
|/2
(1.115)
1.5. PROBLEMS
25
where c < ∞ is a constant independent of N . For large but finite N ,
the k-sum diverges as
1 X
1
∼ ln N.
(1.116)
V
k2
|k|<|K |/2
0
Hence, the density at every given reciprocal lattice vector goes to zero
as
hρK i
1
∼√
.
(1.117)
N
ln N
In this sense, the divergence that leads to the conclusion that there
is no crystalline order in two dimensions is very weak. (What about
one- or three-dimensional position space?) In turn, weak violations
of the assumptions that lead us to this conclusion may already cause
some crystalline ordering phenomenon in two dimensions. One example, where this happens, is graphene. There, a weak “buckling” of the
plane in which atoms arrange themselves, i.e., slight deviations from
the strictly two-dimensional geometry, suffices to allow for a crystalline
ordering.
CHAPTER 2
Bogoliubov theory of a dilute Bose gas
Outline
Second quantization for bosons is reviewed. Bose-Einstein condensation for non-interacting bosons is interpreted as an example of
spontaneous-symmetry breaking. The spectrum of a dilute Bose gas
with hardcore repulsion is calculated within Bogoliubov mean-field
theory using the operator formalism. It is shown that a Goldstone
mode, an acoustic phonon, emerges in association with spontaneoussymmetry breaking. Landau criterion for superfluidity is presented.
Bose-Einstein condensation as well as superfluidity at non-vanishing
temperatures are treated using the path integral formalism.
2.1. Introduction
This chapter is devoted to the study of a dilute Bose gas with a
repulsive contact interaction. We shall see that the phenomenon of superfluidity takes place at sufficiently low temperatures. Superfluidity
is an example of the spontaneous breaking of a continuous symmetry. The continuous symmetry is the global U (1) gauge symmetry that
is responsible for conservation of total particle number. We shall also
carefully distinguish Bose-Einstein condensation from superfluidity. Interactions are necessary for superfluidity to take place. Interactions are
not needed for Bose-Einstein condensation.
We begin this chapter with the formalism of second quantization for
bosons. We then interpret Bose-Einstein condensation at zero temperature as an example of the spontaneous breaking of a continuous symmetry through an explicit construction of a ground state that breaks
the global U (1) gauge symmetry. The emphasis is here on how the
global U (1) gauge symmetry organizes the Hilbert space spanned by
eigenstates of the Hamiltonian. In this construction, the thermodynamic limit plays an essential role.
Next, we treat a repulsive contact interaction through a mean-field
approximation first proposed by Bogoliubov.
We revisit this approximation using path-integral techniques to
show that it is nothing but a saddle-point approximation. We also
present two effective field theories with different physical contents. The
first one deals with single-particle excitations. The second one deals
with collective excitations.
27
28
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
2.2. Second quantization for bosons
The terminology “second quantization” is rather unfortunate in
that it might be perceived as implying concepts more difficult to grasp
than the passage from classical to quantum mechanics. Quite to the
contrary the relation between “second” and “first” quantization 1 is
nothing but a matter of convenience. Going from first to second quantization is like going from a real-space representation of Schrödinger
equation to a momentum-space representation when the Hamiltonian
has translation symmetry.
Second quantization is a formalism that aims at describing a system made of identical “particles”, bosons or fermions, in which creation and annihilation of particles is easily and naturally accounted
for. Hence, the quantum “particle number” need not be sharp in this
representation, very much in the same way as position is not a sharp
quantum number for a momentum eigenstate. Another analogy for the
relationship between first quantization, in which the quantum “particle number” is a sharp quantum number, and second quantization, in
which it need not be, is that between the canonical and grand-canonical
ensembles of statistical mechanics. In the canonical ensemble, particle number is given. In the grand-canonical ensemble, particle number
fluctuates statistically as it has been traded for a fixed chemical potential.
The formalism of second quantization can already be introduced
at the level of a single harmonic oscillator, but it is for interacting
many-body systems that it becomes very powerful. It is nevertheless
instructive to develop the formalism already at the level of a singleparticle Hamiltonian since, to a large extent, many-body physics is
glorified perturbative physics about some non-interacting limit.
We shall now generalize the construction of a second-quantized formalism in terms of creation and annihilation operators for the onedimensional harmonic oscillator that we presented in chapter 1. We
shall thus consider a finite volume V of d-dimensional space on which
the single-particle Hilbert space H(1) of square-integrable and twicedifferentiable functions is defined. In turn, the single-particle Hamiltonian is represented by (~ = 1 and ∆ is Laplace’s operator in ddimensional space)
H=−
1
∆
+ U (r),
2m
By first quantization is meant Schrödinger equation.
(2.1a)
2.2. SECOND QUANTIZATION FOR BOSONS
29
and possesses the complete, orthogonal, and normalized basis of eigenfunctions
Z
X
ϕ∗n (r) ϕn (r 0 ) = δ(r−r 0 ).
H ϕn (r) = εn ϕn (r),
dd r ϕ∗m (r) ϕn (r) = δm,n ,
n
V
(2.1b)
The index n belongs to a countable set after appropriate boundary
conditions, say periodic, have been imposed at the boundaries of the
finite volume V . We assume that the single-particle potential U (r)
is bounded from below, i.e., there exists a single-particle and nondegenerate ground-state energy, say ε0 . Hence the energy eigenvalue
index runs over the positive integers, n = 0, 1, 2, · · · . The time evolution of any solution of Schrödinger equation
i∂t Ψ(r, t) = H Ψ(r, t),
can be written as
X
Ψ(r, t) =
An ϕn (r) e−iεn t ,
Ψ(r, t = 0) given,
Z
An =
n
(2.2a)
dd r ϕ∗n (r) Ψ(r, t = 0).
V
(2.2b)
The formalism of second quantization starts with the following two
postulates.
(1) There exists a set of pairs of adjoint operators â†n (creation
operator) and ân (annihilation operator) labeled by the energy
eigenvalue index n and obeying the bosonic algebra 2
[âm , â†n ] = δm,n ,
[âm , ân ] = [â†m , â†n ] = 0,
m, n = 0, 1, 2, · · · .
(2.3)
(2) There exists a non-degenerate vacuum state |0i that is annihilated by all annihilation operators,
ân |0i = 0,
n = 0, 1, 2, · · · .
(2.4)
With these postulates in hand, we define the Heisenberg representation
for the operator-valued field (in short, quantum field),
X
ϕ̂† (r, t) :=
â†n ϕ∗n (r) e+iεn t
(2.5a)
n
together with its adjoint
ϕ̂(r, t) :=
X
ân ϕn (r) e−iεn t .
(2.5b)
n
2 The conventions for the commutator and anticommutator of any two “objects”
A and B are [A, B] := AB − BA and {A, B} := AB + BA, respectively.
30
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
The bosonic algebra (2.3) endows the quantum fields ϕ̂† (r, t) and ϕ̂(r, t)
with the equal-time algebra 3
[ϕ̂(r, t), ϕ̂† (r 0 , t)] = δ(r−r 0 ),
[ϕ̂(r, t), ϕ̂(r 0 , t)] = [ϕ̂† (r, t), ϕ̂† (r 0 , t)] = 0.
(2.9)
The quantum fields ϕ̂† (r, t) and ϕ̂(r, t) act on the “big” many-particle
space


N
∞


M O
(1)
H
.
(2.10a)
F :=


sym
N =0
Here, each
N
N
sym
H(1) is spanned by states of the form
m i
†
Y âi
p
|m0 , · · · , mi−1 , mi , mi+1 , · · · i :=
|0i,
mi !
i
mi = 0, 1, 2, · · · ,
(2.10b)
with the condition on the non-negative integers mi that
X
mi = N.
(2.10c)
i
N
(1)
The algebra obeyed by the â’s and their adjoints ensures that N
sym H
is the N -th symmetric power of H(1) , i.e., that the state |m0 , · · · , mi−1 , mi , mi+1 , · · · i
made of N identical particles of which mi have energy εi is left unchanged by any permutation of the N particles. Hence, the “big” manyparticle Hilbert space (2.10a) is the sum over the subspaces spanned by
wave functions for N identical particles that are symmetric under any
permutation of the particles labels. This “big” many-particle Hilbert
space is called the bosonic Fock space in physics.
The rule to change the representation of operators from the Schrödinger
picture to the second quantized language is best illustrated by the following examples.
3
Alternatively, if we start from the classical Lagrangian density
L := (ϕ∗ i∂t ϕ)(r, t) −
1
|∇ϕ|2 (r, t) − |ϕ∗ |2 (r, t) U (r),
2m
(2.6)
we can elevate the field ϕ(r, t) and its momentum conjugate
π(r, t) :=
δL
= iϕ∗ (r, t)
δ(∂t ϕ)(r, t)
(2.7)
to the status of quantum fields ϕ̂(r, t) and π̂(r, t) = iϕ̂† (r, t) obeying the equal-time
bosonic algebra
[ϕ̂(r, t), π̂(r 0 , t)] = iδ(r − r 0 ),
[ϕ̂(r, t), ϕ̂(r 0 , t)] = [π̂(r, t), π̂(r 0 , t)] = 0.
(2.8)
2.2. SECOND QUANTIZATION FOR BOSONS
31
Example 1: The second-quantized representation Ĥ of the singleparticle Hamiltonian (2.1a) is
Z
Ĥ := dd r ϕ̂† (r, t) H ϕ̂(r, t)
(2.11)
V
=
X
εn â†n
ân .
n
As it should be, it is explicitly time independent.
Example 2: The second-quantized total particle-number operator
Q̂ is
Z
Q̂ := dd r ϕ̂† (r, t) 1 ϕ̂(r, t)
(2.12)
V
=
X
â†n
ân .
n
It is explicitly time independent as follows from the continuity equation
0 = (∂t ρ)(r, t) + (∇ · J )(r, t),
ρ(r, t) := |Ψ(r, t)|2 ,
(2.13a)
1
∗
∗
J (r, t) :=
[Ψ (r, t) (∇Ψ) (r, t) − (∇Ψ ) (r, t)Ψ(r, t)] ,
2mi
obeyed by Schrödinger equation (2.2a). The number operator Q̂ is the
infinitesimal generator of global gauge transformations by which all
states in the bosonic Fock space are multiplied by the same operatorvalued phase factor. Thus, for any q ∈ R, a global gauge transformation
on the Fock space is implemented by the operation
|m0 , · · · , mi−1 , mi , mi+1 , · · · i → e+iq Q̂ |m0 , · · · , mi−1 , mi , mi+1 , · · · i
(2.14)
on states, or, equivalently, 4
ân → e+iq Q̂ ân e−iq Q̂ = e−iq ân ,
(2.16a)
and
â†n → e+iq Q̂ â†n e−iq Q̂ = e+iq â†n ,
(2.16b)
for all pairs of creation and annihilation operators, respectively. Equation (2.16b) teaches us that any creation operator carries the particle
number +1. Equation (2.16a) teaches us that any annihilation operator
carries the particle number −1.
4
we made use of
†
[â â, â] = ↠ââ−â↠â = ↠ââ−↠ââ+↠ââ−â↠â = ↠[â, â]+[↠, â]â = −â, (2.15a)
and, similarly,
[↠â, ↠] = +↠.
(2.15b)
32
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
Example 3: The second-quantized local particle-number density
operator ρ̂ and the particle-number current density operator Ĵ are
ρ̂(r, t) := ϕ̂† (r, t) 1 ϕ̂(r, t),
(2.17a)
and
1 †
ϕ̂ (r, t) (∇ϕ̂) (r, t) − ∇ϕ̂† (r, t)ϕ̂(r, t) ,
2mi
respectively. The continuity equation
0 = (∂t ρ̂)(r, t) + ∇ · Ĵ (r, t)
Ĵ (r, t) :=
(2.17b)
(2.17c)
that follows from evaluating the commutator between ρ̂ and Ĥ is
obeyed as an operator equation.
The operators Ĥ, Q̂, ρ̂, and Ĵ all act on the Fock space F. They
are thus distinct from their single-particle counterparts H, Q, ρ, and J
(1)
whose actions are restricted to the Hilbert space
By construction,
N1 H . (1)
the action of Ĥ, Q̂, ρ̂ and Ĵ on the subspace sym H of F coincides
with the action of H, Q, ρ, and J on H(1) , respectively.
2.3. Bose-Einstein condensation and spontaneous symmetry
breaking
Given a many-body system made of identical bosons, say atoms
carrying an integer-valued total angular momentum, how does one construct the ground state? The simplest answer to this question occurs
when bosons are non-interacting. In this case, the ground state is simply obtained by putting all bosons in the lowest energy single-particle
state. If the number of bosons is taken to be N , then the ground state
is |N, 0, · · · i with energy N ε0 . This straightforward observation underlies the phenomenon of Bose-Einstein condensation. A non-vanishing
fraction of bosons occupies the single-particle energy level ε0 below the
Bose-Einstein transition temperature TBE in the thermodynamic limit
of infinite volume V but non-vanishing particle density.
From a conceptual point of view, it is more fruitful to associate
Bose-Einstein condensation with the phenomenon of the spontaneous
breaking of a continuous symmetry than with macroscopic occupation
of a single-particle level. The continuous symmetry in question is the
freedom in the choice of the global phase of the many-particle wave
functions. This symmetry is responsible for total particle-number conservation. In mathematical terms, the vanishing commutator
[Ĥ, Q̂] = 0
(2.18)
between the total number operator Q̂ and the single-particle Hamiltonian Ĥ implies a global U (1) gauge symmetry.
The concept of spontaneous symmetry breaking is subtle. For one
thing it can never take place when the normalized ground state |Φ0 i of
2.3. BOSE-EINSTEIN CONDENSATION AND SPONTANEOUS SYMMETRY BREAKING
33
the many-particle Hamiltonian (possibly interacting) is non-degenerate,
i.e., unique up to a phase factor. Indeed, the transformation law of
the ground state |Φ0 i under any symmetry of the Hamiltonian must
then be multiplication by a phase factor. Correspondingly, the ground
state |Φ0 i must transform according to the trivial representation of the
symmetry group, i.e., |Φ0 i transforms as a singlet. In this case there
is no room for the phenomenon of spontaneous symmetry breaking by
which the ground state transforms non-trivially under some symmetry
group of the Hamiltonian.
Now, the Perron-Frobenius theorem for finite dimensional matrices with positive entries, see Refs. [24] and [25], or its extension, see
Ref. [26], to single-particle Hamiltonians of the form (2.1a) guarantees that the ground state is non-degenerate forNa non-interacting N (1)
body Hamiltonian defined on the Hilbert space N
sym H . When the
ground
state of an interacting Hamiltonian defined on the Hilbert space
NN
(1)
H
is non-degenerate, then spontaneous symmetry breaking is
sym
ruled out for this interacting Hamiltonian.
Before evading this “no-go theorem” by taking advantage of the
thermodynamic limit of infinite volume V but non-vanishing particle
density, we want to investigate more closely the consequences of having a non-degenerate ground state. We consider the cases of both
non-interacting many-body Hamiltonians such as Ĥ in Eq. (2.11) and
interacting many-body Hamiltonians 5 that commute with Q̂. The
Hilbert space will be the bosonic Fock space F in Eq. (2.10a) on which
the quantum field operator ϕ̂(r, t) in Eq. (2.5b) is defined. We shall
see that the expectation value of ϕ̂(r, t) in the ground state |Φ0 i of
the many-body system can be used as a signature of the spontaneous
breaking of the U (1) symmetry. More generally, we shall interpret the
quantum statistical average of ϕ̂(r, t) as a temperature dependent order
parameter.
As follows from Eq. (2.16a), the quantum field ϕ̂(r, t) transforms
according to
e+iq Q̂ ϕ̂(r, t) e−iq Q̂ = e−iq ϕ̂(r, t),
∀r, t,
(2.19)
under any global gauge transformation labeled by the real-valued number q. The quantum field ϕ̂(r, t)Pcarries U (1) charge −1 as it lowers
the bosonic occupation numbers i mi by one on any state (2.10b) of
the bosonic Fock space F. By hypothesis, the ground state |Φ0 i of Ĥ
is non-degenerate. Thus, it transforms like a singlet under U (1),
e−iq Q̂ |Φ0 i = e−iq Q0 |Φ0 i,
hΦ0 | e+iq Q̂ = hΦ0 | e+iq Q0 .
(2.20)
What then follows for the expectation value hΦ0 |ϕ̂(r, t)|Φ0 i?
∃ Q0 ∈ R,
5
Interactions are easily introduced through polynomials in creation and annihilation operators of degree larger than 2.
34
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
It must vanish. Indeed,
hΦ0 | e+iq Q̂ ϕ̂(r, t)e−iq Q̂ |Φ0 i = e−iq hΦ0 |ϕ̂(r, t)|Φ0 i,
∀r, t, (2.21)
by Eq. (2.19) and
hΦ0 |e+iq Q̂ ϕ̂(r, t) e−iq Q̂ |Φ0 i = hΦ0 |ϕ̂(r, t)|Φ0 i,
∀r, t, (2.22)
by Eq. (2.20) hold simultaneously for any q ∈ R. The vanishing of
hΦ0 |ϕ̂(r, t)|Φ0 i, in view of the fact that ϕ̂(r, t) carries U (1) charge −1
and thus transforms non-trivially under U (1), can be traced to the
assumption that the ground state |Φ0 i is unique, i.e., that |Φ0 i is an
eigenstate of Q̂. In more intuitive terms, the action of ϕ̂(r, t) on an
eigenstate of Q̂ such as |Φ0 i is to lower the total number of particles
by one, thereby producing a state orthogonal to |Φ0 i. Conversely, a
non-vanishing expectation value of ϕ̂(r, t) in some state |φi ∈ F is only
possible if |φi ∈ F is not an eigenstate of Q̂. 6
Evading the “no-go theorem” for spontaneous symmetry breaking
thus requires quantum degeneracy of the ground state with orthogonal
ground states that are related by the action of the U (1) symmetry
group. In turn, this can be achieved by constructing a ground state
|φi ∈ F that is an eigenstate of ϕ̂(r, t) and thus cannot be an eigenstate
of Q̂.
A prerequisite to evade the “no-go theorem” for spontaneous symmetry breaking is that the thermodynamic limit of infinite volume V
but non-vanishing particle density be taken. This idealized mathematical limit is often an excellent approximation in condensed-matter
physics or in cold-atom physics. When the thermodynamic limit N, V →
∞ with N/V held fixed is well defined, there is no difference between
approaching this limit by working at N
fixed volume and at fixed parN
(1)
ticle number with the Hilbert space
sym H or approaching the
thermodynamic limit by working at fixed external
P∞pressure
NN and(1)at fixed
chemical potential with the Fock space F = N =0 sym H . The
first approach to the thermodynamic limit defines the so-called canonical ensemble of quantum statistical mechanics. The second approach
to the thermodynamic limit defines the so-called grand-canonical ensemble of quantum statistical mechanics. The thermodynamic limit is
also needed to recover spontaneous symmetry breaking even when the
Hilbert space of finitely-many degrees of freedom is endowed with the
structure of a Fock space. 7
NN
It is impossible for ϕ̂(r, t) to acquire an expectation value on sym H(1) .
7 This occurs when the bosons of the many-body system are collective excitations, say phonons in a solid, spin waves in an antiferromagnet, or excitons in a
semiconductor, i.e., when the finitely-many degrees of freedom are ions, spins, or
band electrons, respectively.
6
2.3. BOSE-EINSTEIN CONDENSATION AND SPONTANEOUS SYMMETRY BREAKING
35
To underscore the role played by the thermodynamic limit to evade
the “no-go theorem” for spontaneous symmetry breaking, we now restrict ourself to the many-body and non-interacting Hamiltonian
Ĥµ := Ĥ − µ Q̂
(2.23a)
with
∆
(2.23b)
2m
in Eq. (2.1a) so that translation invariance holds at the single-particle
level. The real-valued parameter µ is called the chemical potential.
Since Ĥ commutes with Q̂ by hypothesis, an eigenstate of Ĥ is also an
eigenstate of Ĥµ and conversely. Eigenenergies of Ĥ and Ĥµ may differ,
however. For example, the single-particle eigenfunctions ϕn (r) of H in
Eq. (2.1a) are also single-particle eigenfunctions of Ĥµ on H(1) but with
the rigidly shifted spectrum of energy eigenvalues εn − µ. Furthermore,
the dimensionalities of the eigenspaces of Ĥ can change dramatically
by the addition of −µQ̂. To see this, observe that the choice µ = ε0
insures that the single-particle ground-state energy of Ĥµ vanishes and
√
that the corresponding normalized eigenfunction ϕ(r) = 1/ V . 8 This
choice also guarantees that all states
H=−
(↠)m0
|0i,
|m0 , 0, · · · i = p0
m0 !
m0 = 0, 1, 2, · · · ,
(2.25)
are orthogonal eigenstates of Ĥµ in F with the same vanishing energy. 9
The choice µ = ε0 guarantees that Ĥµ has countably-many orthogonal
ground states provided the volume V is finite.
Any linear combination of states of the form (2.25) is a ground state
of Ĥµ with µ = ε0 . Of all these possible linear combinations, consider
the continuous family of normalized 10 ground states labeled by the
8
A time-dependent gauge transformation plays the same role as the chemical
potential if one chooses to work in the canonical instead of the grand-canonical
statistical ensemble. For example, setting ε0 to 0 in the single-particle Hilbert
space H(1) is achieved with the help of the time-dependent gauge transformation
Ψ(r, t) → eiε0 t Ψ(r, t)
(2.24)
on the single-particle Schrödinger equation (2.2a).
9 The same states are also eigenstates of Ĥ in F but with distinct energy
eigenvalues m0 ε0 .
10 Observe that the operator
Y
†
∗
D(φ1 , φ2 , · · · ) :=
e(φn ân −φn ân ) ,
φ1 , φ2 , · · · ∈ C,
(2.26)
n
is unitary.
36
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
complex-valued parameter φ,
V
2
=e
− V2
|φ|2
=e
− V2
|φ|2
|φigs := e− 2 |φ|
â0 |0i = 0
√
=e
√
∞
X
m0 =0
√
V φ â†0
e
m0
Vφ
p
|m0 , 0, · · · i
m0 !
|0i
√
+ V φ â†0
e
φ â†0 −φ∗ â0
V(
e
√
− V φ∗ â0
(2.27)
|0i
) |0i
√
=: D̂( V φ, 0, · · · )|0i.
To reach the penultimate line, we made use of [[A, B], A] = √
[[A, B], B] =
A B
[A,B]/2 A+B
0 =⇒ e e = e
e
. Here, the unitary operator D̂( V φ, 0, · · · )
rotates the vacuum into the bosonic coherent state (see appendix A)
√
√
†
(2.28)
| V φ, 0, · · · ics := e V φ â0 |0i,
up to the proportionality constant exp(− V2 |φ|2 ). Bosonic coherent
states form an overcomplete set of the Fock space (see appendix A).
The overlap between any two coherent states is always non-vanishing
(see appendix A),
Y ∗
eαn βn ,
αn , βn ∈ C,
cs hα0 , α1 , · · · |β0 , β1 , · · · ics =
n
cs hα0 , α1 , · · · |
:= h0|
Y
α∗n ân
e
αn ∈ C,
,
(2.29)
n
|β0 , β1 , · · · ics :=
Y
†
eβn ân |0i,
βn ∈ C.
n
The same is true of the overlaps (see appendix A)
gs hφ|0i
V
2
= e− 2 |φ| ,
0
−V
gs hφ|φ igs = e
|φ−φ0 |2
2
(2.30)
.
The rational
for having scaled the arguments of the unitary opera√
tor D̂( V φ, 0, · · · ) by the square root of the volume V of the system
in Eq. (2.27) is to guarantee that all the rotated vacua in Eq. (2.27)
become orthogonal in the thermodynamic limit. The thermodynamic
limit is thus essential in providing an escape to the absence of spontaneous symmetry breaking in systems of finite sizes. InNthe therN
(1)
modynamic limit, we need not distinguish Ĥ defined on
sym H
from Ĥµ defined on F. It is only in the thermodynamic limit that
the ground-state manifold ∼
= C of Ĥµ , µ = ε0 , in Eq. (2.27) becomes
the ground-state manifold ∼
= C of Ĥ. Where does this degeneracy
of
Ĥ
comes
from?
When
V
and N are finite and Ĥ is restricted to
NN
(1)
sym H the ground-state energy is N ε0 . The ground-state energy
2.3. BOSE-EINSTEIN CONDENSATION AND SPONTANEOUS SYMMETRY BREAKING
37
N ±1 (1)
NN
(1)
of Ĥ in N
sym H differs from that in sym H by a term of order
N 0 namely ±ε0 . In the Fock space F, the energy difference per particle
between hN, 0, · · · |Ĥ|N, 0, · · · i and hN ± δN, 0, · · · |Ĥ|N ± δN, 0, · · · i
scales like 1/N as the thermodynamic limit N → ∞, δN/N → 0, and
N/V non-vanishing is taken. Hence, more and more states have an
energy of order N 0 above the ground-state energy N ε0 as the system
size is increased. The surprising result is that it is not a mere countable
infinity of states that become degenerate with the ground state in the
thermodynamic limit but an uncountable infinity.
It remains to verify that each ground state |φigs in Eq. (2.27) is an
eigenstate of the quantum fields ϕ̂(r, t), 11 but is not an eigenstate of
Q̂,
ϕ̂(r, t) |φigs = φ |φigs ,
(2.31)
e−iαQ̂ |φigs = |e−iα φigs .
The U (1) “multiplet” structure of the manifold of ground states ∼
= C in
Eq. (2.27) is displayed by Eq. (2.31). Circles in the complex plane φ ∈
C correspond to U (1) “multiplets”.
Normalization of the single-particle
√
eigenfunction ϕ0 (r) = 1/ V and the property that coherent states are
eigenstates of annihilation operators guaranty that the quantum field
ϕ̂(r, t) acquires the expectation value φ ∈ C with the particle density
|φ|2 in the ground-state manifold (2.27),
gs hφ|ϕ̂(r, t)|φigs
gs hφ|ϕ̂
†
= φ,
(r, t)ϕ̂(r, t)|φigs = |φ|2 .
(2.32)
In an interacting system the non-interacting trick relying on fine
tuning of the chemical potential µ → ε0 to construct explicitly the
many-body ground state breaks down. The chemical potential is chosen
instead by demanding that the particle density,
N
hΦ0 |ϕ̂† (r, t)ϕ̂(r, t)|Φ0 i = ,
(2.33)
V
at zero temperature, 12 be held fixed to the value N/V as the thermodynamic limit is taken. At non-vanishing temperature the right-hand
side is unchanged whereas the left-hand side becomes a statistical average in the grand-canonical ensemble. A degenerate manifold of ground
states satisfying Eqs. (2.32) is not anymore parametrized by φ ∈ C but
by arg(φ) ∈ [0, 2π[, since the modulus |φ|2 = N/V is now given. The
11
Remember
that the single-particle ground-state wave function ϕ0 (r) is the
√
constant
1/
V
.
Make
use of the expansion (2.5b) applied to (2.27) whereby
√
√then √
√1 â | V φi = √1 ( V φ)| V φi
cs
cs must be used.
V 0
V
12As before, |Φ i denotes the many-body ground state which, in practice, can0
not be constructed exactly when interactions are present. We are implicitly assuming translation invariance. This is the reason why the right-hand side does not
depend on r.
38
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
U (1) symmetry group parametrized by exp(iα Q̂), α ∈ [0, 2π[ is said
to act transitively on the ground-state manifold. Construction of the
ground-state manifold relies on approximate schemes such as meanfield theory. These approximations are non-perturbative in the sense
that they yield variational wave functions that cannot be derived from
the non-interacting limit to any finite order of the perturbation theory
in the interaction strength.
Spontaneous symmetry breaking is said to occur when the ground
state |Φ0 i of a many-body system is no longer a singlet under the action of a symmetry group of the system. A quantity like hΦ0 |ϕ̂(r, t)|Φ0 i
that must vanish when the ground state is a singlet, but becomes nonvanishing in a phase with spontaneous symmetry breaking is called
an order parameter. An order parameter is a probe to detect spontaneous symmetry breaking. In condensed-matter physics, some order
parameters can be directly observed in static measurements. For example, elastic-neutron scattering can show Bragg peaks corresponding
to crystalline or magnetic order. An order parameter can also be indirectly observed in a dynamical measurement. For instance, inelasticneutron scattering can show a gapless branch of excitations, Goldstone
modes, corresponding to phonons or spin waves. Some consequences of
symmetries such as selections rules and degeneracies of the excitation
spectrum no longer hold in their simplest forms when the phenomenon
of spontaneous symmetry breaking occurs. The mass distributions of
mesons, hadrons, photon, W and Z bosons are interpreted as a manifestation of spontaneous symmetry breaking leading to the standard
model of strong, weak, and electromagnetic interactions.
How does one go about detecting spontaneous symmetry breaking
in the canonical ensemble? This question is of relevance to numerical
simulations where the dimensionality of the Hilbert space is necessarily
finite. A probe for spontaneous symmetry breaking is off-diagonal longrange order. Let |ΦN i be the ground state of the many-body system
NN
(1)
in the Hilbert space
sym H . We denote with ϕ̂(r) the quantum
field ϕ̂(r, t = 0) in the Schrödinger picture. Here, the Schrödinger picture can be implemented numerically through exact diagonalization of
matrices say. We assume translation invariance, i.e., the single-particle
potential U (r) = 0 in Eq. (2.1a). Define the one-particle density matrix
by
R(r 0 , r) :=
1
hΦ |ϕ̂† (r 0 ) ϕ̂(r)|ΦN i.
V N
(2.34)
By translation invariance
R(r 0 , r) = R(r 0 − r),
(2.35)
2.3. BOSE-EINSTEIN CONDENSATION AND SPONTANEOUS SYMMETRY BREAKING
39
which we use to deduce the dependence on r 0 − r. First, we insert into
Eq. (2.34) the Fourier expansions (2.5)
Z
Z d 0
dd k
d k i(k·r−k0 ·r0 )
0
(2.36)
e
hΦN |â†k0 âk |ΦN i.
R(r , r) =
d
d
(2π)
(2π)
Second, we take advantage of R(r 0 , r) = R(r 0 − r) to do
Z
1
0
R(r , r) =
dd yR(r 0 + y, r + y).
(2.37)
V
Third, we combine Eqs. (2.36) and (2.37) into
Z
Z
Z d 0
dd k
d k i[k·(r+y)−k0 ·(r0 +y)]
1
d
0
d y
e
hΦN |â†k0 âk |ΦN i
R(r , r) =
d
d
V
(2π)
(2π)
Z
Z d 0
d
d k
d k
1
0 0
(2π)d δ(k − k0 )ei(k·r−k ·r ) hΦN |â†k0 âk |ΦN i.
=
d
d
V
(2π)
(2π)
(2.38)
Finally, the integration over the momentum k0 yields
Z
1
dd k ik·(r−r0 )
0
R(r , r) =
e
hΦN |â†k âk |ΦN i
V
(2π)d
(2.39)
Z
dd k ik·(r−r0 )
e
nk .
=:
(2π)d
The ground-state expectation value nk is the number of particles per
unit volume with momentum k. When r 0 − r = 0, the one-particle
density matrix R(r 0 − r) is just the total number of particles per unit
volume n0 = N/V . Bose-Einstein condensation means that
nk = n0 (2π)d δ(k) + f (k),
(2.40a)
with f (k) some smooth function that satisfies
Z
dd k
f (k) = 0.
(2.40b)
(2π)d
In position space, Bose-Einstein condensation thus amounts to
Z
dd k −ik·r
0
0
e
f (k),
lim F (r) = 0.
R(r , r) = n0 +F (r −r),
F (r) :=
|r|→∞
(2π)d
(2.41)
The non-vanishing of lim|r|→∞ R(r, 0) is another signature of spontaneous symmetry breaking associated to Bose-Einstein condensation.
We conclude this section with some field-theoretical terminology.
States |Θi for which
lim
|r 1 −r 2 |→∞
hΘ|Ô1 (r 1 )Ô2 (r 2 )|Θi = hΘ|Ô1 (r 1 )|ΘihΘ|Ô2 (r 2 )|Θi
(2.42)
holds for any pair of operators Ô1 (r) and Ô2 (r) defined on the Fock
space F are said to satisfy the cluster decomposition property or to be
clustering. The ground state |ΦN i in Eq. (2.41) does not satisfy the
40
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
clustering property. 13 The manifold of states |φ ∈ Cigs in Eq. (2.27)
does satisfy the clustering property by Eq. (2.32).
2.4. Dilute Bose gas: Operator formalism at vanishing
temperature
2.4.1. Operator formalism. Bogoliubov introduced in 1947 an
interacting model for superfluid 4 He. [27] This model turns out not
to be a very good one for superfluid 4 He in that the assumption of
pairwise interactions made by Bogoliubov fails. However, this model
has been conceptually very important. Moreover, this is a realistic
model in the field of cold atoms that came into maturity in 1995 with
the experimental realization of Bose-Einstein condensation. [28]
The model for weakly interacting bosons proposed by Bogoliubov, a
dilute Bose gas in short, is defined by the second-quantized Hamiltonian
Z
Ĥµ,λ =
∆
λ † 2
d r ϕ̂ (r, t) −
− µ ϕ̂(r, t) +
ϕ̂ ϕ̂ (r, t) .
2m
2
d
†
V
(2.43a)
The chemical potential µ determines the number N (µ) of particles in
the interacting ground state |Φgs i from
*
+
Z
d
†
N (µ) = Φgs d r ϕ̂ ϕ̂ (r, t) Φgs .
(2.43b)
V
Conversely, fixing the total particle number to N determines µ(N ). The
interaction is a two-body, short-range, and repulsive density-density
interaction. In the limit in which the range of this interaction is much
smaller than the average particle separation, this interaction is well
approximated by a delta function repulsion (this is the justification for
the adjective dilute),
Z
Z
λ
d
Ĥλ :=
d r dd r 0 ρ̂(r, t) δ(r−r 0 ) ρ̂(r 0 , t),
ρ̂(r, t) := ϕ̂† ϕ̂ (r, t).
2
V
V
(2.43c)
The real-valued parameter λ ≥ 0 measures the strength of the repulsive
interaction and carries the units of (energy×volume). Bosons are said
to have a hardcore.
When periodic boundary conditions are imposed in the volume V ,
it is natural to expand the pair of canonical conjugate quantum fields
13
Choose Ô1 = ϕ̂† and Ô2 = ϕ̂. The left-hand side of Eq. (2.42) is nonvanishing. On the other hand, since the ground state has a well-defined number N
of particle, the right-hand side must vanish.
2.4. DILUTE BOSE GAS: OPERATOR FORMALISM AT VANISHING TEMPERATURE
41
ϕ̂(r, t) and iϕ̂† (r, t) in the basis of plane waves,
1 X † −i(k·r−εk t)
ϕ̂† (r, t) = √
,
âk e
V k
1 X
ϕ̂(r, t) = √
âk e+i(k·r−εk t) .
V k
(2.44a)
Here, the summation over reciprocal space is infinite but countable,
k=
2π
l,
L
l ∈ Zd ,
Ld ≡ V,
(2.44b)
and we have introduced the single-particle dispersion
k2
εk =
.
2m
(2.44c)
We observe that the single-particle plane wave with the lowest energy is
1
ϕ0 (r) = √ ,
V
ε0 = 0.
(2.45)
The representation of the Hamiltonian in terms of creation and
annihilation operators â†k and âk , respectively, is
Ĥµ,λ =
X
k
λ
λ X
εk − µ + δ(r = 0) â†k âk +
δk1 +k2 ,k3 +k4 â†k1 â†k2 âk3 âk4 .
2
2V k ,k ,k ,k
1
2
3
4
(2.46)
Normal ordering has resulted in the (divergent) shift in the chemical
potential − λ2 δ(r = 0).
The strategy that we shall use to study the energy spectrum of the
dilute Bose gas is to try a variational Ansatz for the ground state. This
variational state is taken to be the ground state in the non-interacting
limit. Define the Bose-condensate wave function |Φ0 i to be state (2.27)
with
√
p
V φ = N0 .
(2.47)
The variational Ansatz |Φ0 i is the ground state of Eq. (2.43a) with µ =
λ = 0. It depends on a single variational parameter, the expectation
value N0 of the number operator â†0 â0 in the state |Φ0 i. The presence
of repulsive interactions results in the possibility that N0 is smaller
than N , i.e., causes a depletion of the Bose condensate in the noninteracting limit. By construction, N0 /V remains non-vanishing in the
thermodynamic limit whereas the expectation value of â†k âk in the state
|Φ0 i vanishes for all k 6= 0.
In view of the very special role played by the reciprocal vector
k = 0, all contributions to the Hamiltonian that depend on k = 0 are
42
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
singled out,
Ĥµ,λ
λ
λ † †
=
δ(r = 0) − µ â†0 â0 +
â â â â
2
2V 0 0 0 0
X
λ
εk − µ + δ(r = 0) â†k âk
+
2
k6=0
λ X † †
+
4âk â0 âk â0 + â†+k â†−k â0 â0 + â†0 â†0 â+k â−k
2V
k6=0
λ X † †
+
â0 âk+k0 âk âk0 + â†k+k0 â†0 âk âk0 + â†k â†k0 â0 âk+k0 + â†k â†k0 âk+k0 â0
2V
0
k,k 6=0
+
λ
2V
X
δk1 +k2 ,k3 +k4 â†k1 â†k2 âk3 âk4 .
k1 ,k2 ,k3 ,k4 6=0
(2.48)
Interaction terms have been arranged by decreasing number of â†0 or
â0 . Momentum conservation prevents terms linear (cubic) in â†0 or
â0 arising from the kinetic energy (interaction). Only the first line
contributes to the expectation value of Ĥ in the variational state |Φ0 i.
The new ground-state energy, to first order in λ/V , is thus
p 2
λ
λ p 4
δ(r = 0) − µ
N0 +
N0 .
(2.49)
2
2V
p
It is permissible to replace any â†0 or â0 by N0 on the subspace
spanned by acting with the creation operators â†k , k 6= 0, on the variational Ansatz |Φ0 i. Hence, on this subspace,
Ĥµ,λ
λ
λ 2
→
δ(r = 0) − µ N0 +
N
2
2V 0
X
X †
λ
λ
+
εk − µ + δ(r = 0) â†k âk +
N0
4âk âk + â†+k â†−k + â+k â−k
2
2V
k6=0
k6=0
X †
λ p
+2
N0
âk+k0 âk âk0 + â†k â†k0 âk+k0
2V
0
k,k 6=0
λ
+
2V
X
δk1 +k2 ,k3 +k4 â†k â†k âk âk .
1
2
3
4
k1 ,k2 ,k3 ,k4 6=0
(2.50)
After absorbing the divergent C-number − λ2 δ(r = 0) into a redefinition
λ
(2.51)
µren := µ − δ(r = 0)
2
of the chemical potential µ, Eq. (2.50) suggests the approximation by
which the p
right-hand side is truncated to the first two leading terms in
powers of N0 , i.e., the first two lines, provided the full Fock space F is
2.4. DILUTE BOSE GAS: OPERATOR FORMALISM AT VANISHING TEMPERATURE
43
restricted to the subspace spanned by the tower of states obtained from
acting on |Φ0 i with â†k6=0 . Hence, the task of solving for the spectrum
of Ĥ in the Fock space F has been replaced by the simpler problem of
solving for the spectrum of Ĥmf in the Fock space Fmf ,
Ĥµ,λ → Ĥmf :=
X
εk − µren â†k âk − µren N0
k6=0
F → Fmf
!
X †
λ
,
+
N N0 +
4âk âk + â†+k â†−k + â+k â−k
2V 0
k6=0
(
)
Y † mk
:= span
âk
|Φ0 i, mk = 0, 1, 2, · · · .
k6=0
(2.52)
This approximation is called a mean-field approximation. It is useful
because it can be solved exactly, for Ĥmf is quadratic in creation and
annihilation operators. It should be a good approximation if N0 is very
close to N . The self-consistency of this approximation is verified once
the variational parameter N0 has been expressed in terms of the total
number of bosons, or, equivalently, in terms of the chemical potential.
We note the presence of the additive C-number
λ 2
N − µren N0
2V 0
(2.53)
in the mean-field Hamiltonian (2.52). A first estimate of the variational
parameter N0 follows from minimization of this C-number,
N0
µ
= ren .
V
λ
(2.54)
Insertion of N0 = µren V /λ into the mean-field Hamiltonian then yields
Ĥmf =
X
k6=0
V
µ X † †
εk + µren â†k âk + ren
â+k â−k + â+k â−k − µ2ren .
2 k6=0
2λ
(2.55)
We will discard the last C-number, since we are only interested in the
dependence on k of the excitation spectrum of Ĥmf and in the change
in the variational wave function |Φ0 i induced by the interactions within
the mean-field approximation.
Diagonalization of Eq. (2.55) on the Fock space Fmf is performed
with the help of a canonical transformation (also called a Bogoliubov
44
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
transformation in this context)
14
â†+k = sinh(θ+k ) b̂−k + cosh(θ+k ) b̂†+k ,
(2.57a)
â+k = cosh(θ+k ) b̂+k + sinh(θ+k ) b̂†−k ,
where (see chapter 2 of [9] or section 35 in chapter 10 of [12])
εk + µren
cosh(2θk ) = q
,
2
2
(εk + µren ) − µren
sinh(2θk ) = q
−µren
.
)2
µ2ren
(εk + µren −
(2.57b)
This transformation preserves the bosonic algebra (hence the terminology canonical),
[âk , â†k0 ] = δk,k0 , [âk , âk0 ] = [â†k , â†k0 ] = 0 ⇐⇒
[b̂k , b̂†k0 ] = δk,k0 , [b̂k , b̂k0 ] = [b̂†k , b̂†k0 ] = 0.
(2.58)
Correspondingly, there exists a unitary transformation Û on the meanfield Fock space such that b̂k = Û âk Û −1 ,
!
X † †
.
(2.59)
θk âk â−k − â−k âk
Û = exp +
k6=0
Up to the C-number
E0 := −
i
X 1 h
V 2
µren −
εk + µren − ξk ,
2λ
2
k6=0
the mean-field Hamiltonian has become
q
X
†
ξk := (εk + µren )2 − µ2ren .
ξk b̂k b̂k ,
Ĥmf =
(2.60a)
(2.60b)
k6=0
This is the Hamiltonian of a gas of free bosons with dispersion ξk . For
small |k|,
s
r
µren
λ N0
ξk ≈
|k| =
|k| ≡ v0 |k|.
(2.61a)
m
mV
This is the dispersion relation of sound waves in a fluid that propagate
with the speed
r
λ N0
v0 :=
.
(2.61b)
mV
14
In matrix form the Bogoliubov transformation reads
! !
! â+k
b̂+k
b̂+k
cosh θk sinh θk
cosh θk
=
⇐⇒ †
=
â†−k
sinh θk cosh θk
b̂†−k
b̂−k
− sinh θk
where θk = θ−k .
− sinh θk
cosh θk
â+k
â†−k
(2.56)
!
2.4. DILUTE BOSE GAS: OPERATOR FORMALISM AT VANISHING TEMPERATURE
45
For large |k|, the dispersion crosses over to the usual free-particle expression
k2
ξk ≈
.
(2.61c)
2m
Having found the mean-field excitation spectrum, we must evaluate the change on the unperturbed ground state |Φ0 i induced by the
Bogoliubov transformation Û . The “rotated” ground state is the one
annihilated by all b̂k , k 6= 0. The state annihilated by all b̂k is
|Φmf i := Û |Φ0 i.
(2.62)
With the mean-field ground state at hand, and recalling that the
total number of particle N (µ) is the expectation value of the total
particle-number operator Q̂ in the ground state, we find the relation
*
!
+
X †
†
N (µ) ≈
Φmf â0 â0 +
âk âk Φmf
k6=0
+
!
*
X
b̂k b̂†k sinh2 θ−k Φmf
=
Φmf N0 +
k6=0
*
+
!
1
X †
b̂k b̂k + 1
=
Φmf N0 +
cosh(2θk ) − 1 Φmf
2
k6=0


X
1
 q εk + µren
Eq. (2.57b)
= N0 +
(2.63)
− 1 .
2 k6=0
2
2
(εk + µren ) − µren
For comparison, had we estimated N (µ) using the variational state
hΦ0 i, we would have found
* ! +
X †
†
N (µ) ≈ Φ0 â0 â0 +
âk âk Φ0
(2.64)
k6=0
= N0 .
The number N (µ) of particles present in |Φmf i exceeds the number N0
present in the single-particle condensate |Φ0 i by the sum over momenta
on the right-hand side of Eq. (2.63). Conversely, had we fixed the number of bosons to be N instead of fixing the chemical potential µ, then
the number N0 of weakly interacting bosons that form a Bose-Einstein
condensate in the mean-field ground state |Φmf i is smaller than N by
an amount that depends on the dimensionality of space, the density
N /V , and the coupling strength λ. The mean-field approximation is
self-consistent if this amount is small, i.e., if and only if the sum over
momenta on the right-hand side of Eq. (2.63) can be shown to be small.
It is shown with Eq. (2.114) that this is the case in three-dimensional
space for either a dilute hardcore Bose gas or for small λ.
46
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
Had we not chosen N0 by minimization, we would have found that
would not vanish anymore in the limit k → 0. Hence, for any other
value of N0 than the one in Eq. (2.54), we could lower the trial energy
by either removing or adding particles in the condensate, i.e., varying
the parameter N0 of the trial wave function |Φ0 i.
We close the discussion of this mean-field theory with a word of
caution. The main prediction of this mean-field analysis is the existence of a mean-field gapless spectrum. Is this prediction robust?
This prediction is predicated on the minimization (2.54). As such, it
would be robust if and only if this local minimum is the global one,
as shall become clear when we derive the mean-field approximation
from the path-integral formalism. In practice, such a proof can not be
achieved and the “validity” of a mean-field approximation rests on two
verifications, namely that it is self-consistent and that it agrees with
experiments.
ξk2
2.4.2. Landau criterion for superfluidity. We shall assume
that the mean-field spectrum that was derived for the dilute Bose gas
is exact. We shall also assume that the excitations, phonons, described
by the pair b̂†k and b̂k of annihilation and creation operators are the
only ones. Although neither assumptions are realistic, the point made
by Landau is that they are sufficient to understand the phenomenon of
superfluidity.
Consider a body of large mass M moving in the dilute Bose gas
(the fluid from now on) at velocity V . By hypothesis, the only way for
the body to experience a retarding force or drag is for it to emit some
phonons. In doing so, energy
1
1
δε := M V 2 − M (V − δV )2 = M V · δV + O[(δV )2 ]
2
2
(2.65a)
and momentum
δk := M V − M (V − δV ) = M δV
(2.65b)
are lost to the phonons with momenta ki and energies εki , i.e.,
X
δε = +
εk i ,
(2.65c)
i
δk = +
X
ki .
(2.65d)
i
By hypothesis phonons in the model have a non-vanishing minimum
phase velocity
εk
v0 = inf k
> 0.
(2.66)
|k|
2.5. DILUTE-BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE
47
The chain of inequalities
|δε| =
X
εki ≥ v0
X
i
i
X |ki | ≥ v0 ki = v0 |δk|
(2.67)
i
then follows. To leading order in M −1 , we have established that
|δk||V | ≥ |δk · V | = |δε| + O(M −1 ) ≥ v0 |δk| + O(M −1 )
(2.68)
for any permitted δk. Such a δk can only exist if
|V | ≥ v0 ,
(2.69)
i.e., the body must exceed a minimum velocity before experiencing any
drag. By moving sufficiently slowly, a heavy body suffers no loss of energy and momentum from the medium. This property of the medium is
called superfluidity. It originates here from the fact that the mean-field
excitation spectrum is bounded from below by a linear dispersion. In
turn, this is a consequence of the interactions conspiring together with
spontaneous symmetry breaking in the existence of Goldstone modes,
acoustic phonons. Interactions are essential to superfluidity. The excitation spectrum remains quadratic in the non-interacting limit and
the velocity threshold below which a moving body does not suffer drag
is v0 = 0. Bose-Einstein condensation alone (i.e., without Goldstone
modes) is not sufficient for superfluidity to occur.
2.5. Dilute-Bose gas: Path-integral formalism at any
temperature
The partition function for the dilute Bose gas at inverse temperature β and chemical potential µ is
Z(β, µ) := Tr e−β Ĥµ,λ ,
Z
λ † 2
∆
d
†
− µ ϕ̂(r) +
ϕ̂ ϕ̂ (r) .
Ĥµ,λ = d r ϕ̂ (r) −
2m
2
(2.70a)
V
The total number of bosons N (β, µ) at inverse temperature β and
chemical potential µ is obtained from
*Z
N (β, µ) :=
+
dd r ϕ̂† ϕ̂ (r)
V
≡ β −1 ∂µ ln Z(β, µ).
Z(β,µ)
(2.70b)
48
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
We have seen in appendix A that the path-integral representation
Z
Z(β, µ) = D[ϕ∗ , ϕ] exp (−SE )
 β

Z
Z
Z
(2.71a)
∗
d


=
D[ϕ , ϕ] exp − dτ d rLE ,
0
V
where
∆
λ
λ
LE = ϕ (r, τ ) ∂τ −
− µ + δ(r = 0) ϕ(r, τ )+ [ϕ∗ (r, τ )]2 [ϕ(r, τ )]2 ,
2m
2
2
(2.71b)
of this partition function exists. Integration variables are the real and
imaginary parts of the complex-valued function ϕ(r, τ ) or, equivalently,
its complex conjugate ϕ∗ (r, τ ). They obey periodic boundary conditions in imaginary time τ ,
∗
ϕ∗ (r, τ ) = ϕ∗ (r, τ + β),
ϕ(r, τ ) = ϕ(r, τ + β).
(2.71c)
Boundary conditions in space, say periodic ones, are also present. The
total number of bosons N (β, µ) is now represented by
*Zβ
+
Z
N (β, µ) = β −1
dd r (ϕ∗ ϕ) (r, τ )
dτ
0
V
(2.71d)
Z(β,µ)
= β −1 ∂µ ln Z(β, µ).
The choice of periodic boundary conditions in space and time suggests to change integration variable in the path-integral representation
of the partition function by performing the Fourier transforms
1 XX ∗
ϕ∗ (r, τ ) = √
ak,$l e−i(kr−$l τ ) ,
βV k l
a∗k,$l
1
=√
βV
Zβ
Z
dτ
0
(2.72a)
d
∗
d r ϕ (r, τ ) e
+i(kr−$l τ )
,
V
on the one hand, and
1 XX
ϕ(r, τ ) = √
ak,$l e+i(kr−$l τ ) ,
βV k l
ak,$l
1
=√
βV
Zβ
Z
dτ
0
(2.72b)
d
d r ϕ(r, τ ) e
−i(kr−$l τ )
,
V
on the other hand. Here,
2π
$l =
l,
l ∈ Z,
β
k=
2π
l,
L
l ∈ Zd .
(2.72c)
2.5. DILUTE-BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE
49
This change of integration variable turns the path-integral representation of the partition function into
Z
Z(β, µ) = D[a∗ , a] exp (−SE )
(2.73a)
with the Euclidean action
XX
k2
λ
∗
SE =
ak,$l −i$l +
− µ + δ(r = 0) ak,$l
2m
2
l
k
λ 1
+
2 βV
l1 ,l2 ,l3 ,l4
X
δl1 +l2 ,l3 +l4 δk1 +k2 ,k3 +k4 a∗k1 ,$l a∗k2 ,$l ak3 ,$l ak4 ,$l .
1
k1 ,k2 ,k3 ,k4
2
3
4
(2.73b)
The total number of bosons N (β, µ) is represented by
*
+
X X
N (β, µ) =β −1
a∗k,$l ak,$l
l
k
Z(β,µ)
(2.73c)
= β −1 ∂µ ln Z(β, µ).
2.5.1. Non-Interacting limit λ = 0. In the non-interacting limit
λ = 0, we need to solve the quadratic problem
Z
Z(β, µ) = D[a∗ , a] exp (−SE ) ,
(2.74a)
XX
k2
∗
− µ ak,$l .
SE =
ak,$l −i$l +
2m
l
k
The path integral is a multi-dimensional Gaussian integral, one Gaussian integral of the form
Z
Z
d(x − iy) d(x + iy) −(x−iy) K (x+iy)
dz ∗ dz −z∗ Kz
e
≡
e
2πi
2πi
Z+∞ Z+∞
1
2
2
=
dx
dy e−K (x +y )
π
−∞
=
1
(2π)
π
−∞
Z+∞
dr r e−K r
2
0
1
1 −K r2 0
=
(2π)
e
π
2K
+∞
1
=
,
K ∈ R+ ,
K
(2.75)
50
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
for each pair (a∗k,$l , ak,$l ), provided the counterpart −i$l +
K has a positive real part, i.e.,
k2
> µ.
2m
k2
2m
− µ to
(2.76)
This is symbolically written as an inverse determinant (see appendix
B.1)
Z(β, µ) =
≡
1
Det ∂τ −
YY
l
k
∆
2m
−µ
1
−i$l +
k2
2m
(2.77)
−µ
,
µ < 0.
The total number of bosons N (β, µ) is thus given by the expression
N (β, µ) = β −1 ∂µ ln Z(β, µ)
XX
1
= β −1
,
k2
l
k −i$l + 2m − µ
µ < 0,
(2.78)
in the non-interacting limit. It is shown in appendix B.2 that the
imaginary-time summation can be written as a contour (Γ) integral in
the complex z-plane for any given k,
Z
X
1
fBE (z)
dz
= +β
k2
2πi −z + k2 − µ
l −i$l + 2m − µ
2m
Γ
Z
dz
fBE (z)
= −β
2πi z − k2 + µ
2m
Γ
Z
dz
fBE (z)
≡ −β
,
µ < 0,
(2.79a)
2πi z − εk + µ
Γ
where fBE (z) is the Bose-Einstein distribution function
fBE (z) :=
eβz
1
,
−1
(2.79b)
and εk is the single-particle dispersion,
k2
εk :=
.
2m
(2.79c)
A second application of the residue theorem (see appendix B.2) turns
the z-integral over the counterclockwise Γ contour into
Z
dz
fBE (z)
−β
= (−)2 βfBE (εk − µ),
µ < 0.
(2.80)
2πi z − εk + µ
Γ
2.5. DILUTE-BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE
51
We conclude that the total number of bosons is given by
X
N (β, µ) =
fBE (εk − µ)
k
=
X
k
−1
2
k
−µ
−1
,
exp β
2m
(2.81)
µ < 0.
When β is fixed, N (β, µ) is a monotonically increasing function of
µ. When µ is fixed, N (β, µ) is a monotonically decreasing function of
β. If the temperature dependence of µ is determined by fixing the lefthand side of Eq. (2.81) to be some constant number, say the average
total particle number N in the grand-canonical ensemble, then µ(β) is
a monotonically increasing function of β. However, µ(β) is necessarily
bounded from above by
µc := inf εk = 0,
k
(2.82)
for, if it was not, there would be an inverse critical temperature βc
above which µ(β) > µc and the integral over the so-called zero mode 15
(a∗k=0,$l =0 , ak=0,$l =0 ),
Z
da∗0,0 da0,0
exp +|µ(β)|a∗0,0 a0,0 ,
(2.85)
2πi
would diverge in contradiction with the assumption that there exists
a well defined vacuum |0i. The alternative scenario by which µ(β) is
pinned to µc above βc is actually what transpires from a numerical
solution of Eq. (2.81) which now reads
P

fBE (εk − µ),
if β < βc .


k
(2.86)
N=
P


−1
2

fBE (εk − µc ), if β ≥ βc .
β |a0,0 | +
k6=0
Equation (2.86) determines the chemical potential as a function of the
inverse temperature. It also determines the macroscopic number of
bosons β −1 |a0,0 |2 (β) that occupy the lowest single-particle energy ε0
above the critical inverse temperature βc . In the limit of vanishing
temperature, limβ→∞ fBE (εk − µc ) = 0, k 6= 0, and all N bosons occupy the single-particle ground-state energy ε0 . Before tackling the
15
A zero mode is a configuration ϕ(r, τ ) that does not depend on space or
time:
ϕ(r, τ ) = ϕ0 .
The only non-vanishing Fourier component of ϕ0 is
p
ak=0,$l =0 = βV ϕ0 .
(2.83)
(2.84)
52
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
interacting case, it is important to realize that Bose-Einstein condensation did not alter the single-particle dispersion εk . According to the
Landau criterion, superfluidity cannot take place in the non-interacting
limit. We shall see below how the excitation spectrum becomes linear
at long wavelengths due to a conspiracy between spontaneous symmetry breaking and interactions, thus enabling superfluidity.
Before leaving the non-interacting limit, we introduce the singleparticle Green function
Gk,$l := −
1
−i$l +
k2
2m
−µ
.
(2.87)
The sign is convention. The Green function (2.87) is, up to a sign,
the inverse of the Kernel in Eq. (2.74a). Furthermore, because of the
identities (2.75) and
Z
Z
dz ∗ dz ∗
∂2
dz ∗ dz −z∗ Kz+J ∗ z+Jz∗ −z ∗ Kz
(z z) e
=
e
∗
2πi
∂J ∗ ∂J
2πi
J =J=0
Z
∗
2
∗
dz dz −(z− KJ ) K (z− KJ )+J ∗ K1 J ∂
e
=
∗
∂J ∗ ∂J
2πi
J =J=0
1
J
2 +J ∗ K
∂ e
Eq. (2.75)
= (1/K)
∂J ∗ ∂J ∗
J =J=0
(1/K)
=
,
K
K ∈ R+ ,
(2.88)
the Green function (2.87) is the covariance or two-point function
D
E
∗
Gk,$l := − ak,$l ak,$l
Z
R
(2.89)
∗
−SE
D[a , a] e
a∗k,$l ak,$l
≡− R
.
D[a∗ , a] e−SE
2.5.2. Random-phase approximation. The first change relative to the analysis of the non-interacting limit that is brought by
switching a repulsive contact interaction, λ > 0, is the breakdown of
the stability argument that leads to the pinning of the chemical potential. To see this, consider as in Eq. (2.85) the action of the zero
mode
λ
(0) ∗
4
2
ϕ(r, τ ) := ϕ0 , ∀r, τ =⇒ SE [ϕ0 , ϕ0 ] := βV −µren |ϕ0 | + |ϕ0 | ,
2
(2.90)
where now λ > 0 implies that the renormalized chemical potential
λ
µren := µ − δ(r = 0)
2
(2.91)
2.5. DILUTE-BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE
53
can become arbitrarily large as a function of inverse temperature without endangering the convergence of the contribution
Z
dϕ∗0 dϕ0 −SE(0) [ϕ∗0 ,ϕ0 ]
e
(2.92)
Z0 :=
2πi
from the zero modes to the partition function.
An estimate of Eq. (2.92) that becomes exact in the limit of β, V →
∞ is obtained from the saddle-point approximation. In the saddlepoint approximation, the modulus |ϕ0 (µren )|2 is given by the solution
[compare with Eq. (2.54)]


if µren < 0,
0,
2
|ϕ0 (µren )| =
(2.93)

 µren , if µ ≥ 0,
ren
λ
to the classical equation of motion
(0)
δSE
,
0=
δ|ϕ0 |2
(2.94)
and
Z0 ≈


1,
if µren < 0.
(2.95)

exp +βV
µ2ren
2λ
, if µren ≥ 0.
In turn, the dependence on β of the renormalized chemical potential is
determined by demanding that
+
*Zβ
Z
N = V |ϕ0 (µren )|2 + β −1
dd r (ϕ
e∗ ϕ)
e (r, τ )
dτ
0
V
.
(2.96)
Z/Z0
The tilde over ϕ
e∗ (r, τ ) and ϕ(r,
e τ ) as well as the subscript Z/Z0 are
reminders that zero modes should be removed from the path integral
in the second term on the right-hand side [compare with Eq. (2.71d)],
as they would be counted twice when µren ≥ 0 otherwise.
The strategy that we shall pursue to go beyond the zero-mode approximation consists in the following steps.
Step 1: We assume that Eq. (2.93) holds with some µren > 0.
Step 2: We choose a convenient parametrization of the fluctuations
ϕ
e∗ (r, τ ) and ϕ(r,
e τ ) about the zero modes.
Step 3: We construct an effective theory in ϕ
e∗ (r, τ ) and ϕ(r,
e τ ) to the
desired accuracy.
Step 4: We solve Eq. (2.96) with the effective theory of step 3 and verify
the self-consistency of step 1 within the accuracy of the approximation
made in step 3.
This approximate scheme is called the random-phase approximation
(RPA) when the effective theory in step 3 is non-interacting. It is
54
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
nothing but an expansion of the action up to quadratic order in the
fluctuations about the saddle-point or mean-field solution.
The zero-mode approximation in Eq. (2.93) leaves the choice of the
phase of the zero mode ϕ0 arbitrary. This is the classical implementation of the spontaneous breaking of the U (1) symmetry associated
with total particle-number conservation. Without loss of generality,
the (linear ) parametrization that we choose is
r
r
µ
µren
ren
ϕ∗ (r, τ ) =
+ϕ
e∗ (r, τ ),
ϕ(r, τ ) =
+ ϕ(r,
e τ ). (2.97)
λ
λ
Here, we are also assuming that
0 = hϕ
e∗ (r, τ )iZ/Z0 = hϕ(r,
e τ )iZ/Z0 .
(2.98)
This parametrization is the natural one if the approximation to the
action is meant to linearize equations of motion. Correspondingly, the
action is expanded up to second order in the deviations ϕ
e∗ (r, τ ) and
ϕ(r,
e τ ) from the saddle-point or mean-field Ansatz (2.93)
p
p
(0)
SE [ϕ∗ , ϕ] ≈ SE [ µren /λ, µren /λ]
Zβ
+
µren ∗
∆
2
∗
(ϕ
e + ϕ)
e (r, τ )
d r ϕ
e ∂τ −
ϕ
e+
2m
2
Z
d
dτ
0
V
+ ··· .
(2.99)
If partial integrations are performed and all space or time total derivatives are dropped owing to the periodic boundary conditions in space
and time, we find the approximation
p
p
(0)
SE [ϕ∗ , ϕ] ≈ SE [ µren /λ, µren /λ]
Zβ
+
Z
dτ
V
Zβ
Z
dτ
V
Zβ
Z
dτ
0
d
d r
0
+
∆
(Re ϕ)
e −
+ 2µren (Re ϕ)
e (r, τ )
2m
∆
(Im ϕ)
e −
2m
d r
0
+
d
(Im ϕ)
e (r, τ )
dd r [(Re ϕ)i∂
e τ (Im ϕ)
e − (Im ϕ)i∂
e τ (Re ϕ)]
e (r, τ )
V
+ ··· .
(2.100)
A quite remarkable phenomenon is displayed in Eq. (2.100). A
purely imaginary term has appeared in the Euclidean effective action.
2.5. DILUTE-BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE
55
Hence, this effective action cannot be interpreted as some classical action. The purely imaginary term is an example of a Berry phase. Upon
be
be
canonical quantization, the commutator between Re ϕ(r)
and Im ϕ(r)
is the same as that between the position and momentum operators,
respectively, in quantum mechanics (see chapter 1).
If we ignore for one instant the Berry phase term, i.e., ignore quantum mechanics, we can interpret Re ϕ(r,
e τ ) as a massive mode and
Im ϕ(r,
e τ ) as a massless mode. The massive mode Re ϕ(r,
e τ ) originates from the radial motion of a classical particle which, at rest, is
sitting somewhere at the bottom of the circular potential well
(0)
U0 [ϕ∗0 , ϕ0 ]
S [ϕ∗ , ϕ ]
λ
:= E 0 0 = −µren |ϕ0 |2 + |ϕ0 |4 .
βV
2
(2.101)
The massless mode Im ϕ(r,
e τ ) originates from the angular motion of
this particle along the bottom of the circular potential well U0 [ϕ∗0 , ϕ0 ].
At the classical level, the dispersions above the gap thresholds 2µren
and 0 of Re ϕ(r,
e τ ) and Im ϕ(r,
e τ ), respectively, are both quadratic in
the momentum k. Including quantum fluctuations through the Berry
phase dramatically alters this picture. Indeed, these “two classical
modes” are not independent, since they interact through the Berry
phase terms. As we now show, including the Berry phase couplings
allows us to interpret ϕ
e as a mode with a linear (quadratic) dispersion
relation at long (short) wavelengths.
The explicit dispersion can be obtained from Fourier transforming
Eq. (2.100) into
(0)
SE ≈ SE
T k2
X X (Re ϕ)
e −k,−$l
+ 2µren
2m
+
(Im
ϕ)
e
−$l
−k,−$l
Lk
l∈Z
(0)
= SE
2π
+$l
∈Zd
† k2
X X (Re ϕ)
e −k,−$l
+ 2µren
(0)
2m
= SE +
(Im ϕ)
e −k,−$l
+$l
Lk
l∈Z
2π
∈Zd
(Re ϕ)
e +k,+$l
(Im ϕ)
e +k,+$l
k2
2m
∈Zd
2π
† k2
X X (Re ϕ)
e k,$l
+ 2µren
2m
+
(Im
ϕ)
e
−$l
k,$l
Lk
l∈Z
+$l
k2
2m
−$l
k2
2m
(Re ϕ)
e k,$l
(Im ϕ)
e k,$l
(Re ϕ)
e −k,−$l
.
(Im ϕ)
e −k,−$l
(2.102)
To reach the second line, we have used the fact that the real and
imaginary parts of ϕ(r, τ ) are real-valued functions. To reach the third
line, we made the relabeling k → −k and $l → −$l , under which the
$l -dependence of the kernel is odd while the k-dependence of the kernel
is even.
56
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
Define the 2×2 matrix-valued Green function by its matrix elements
in the k-$l basis,
*
+
†
(Re ϕ)
e −k,−$l
(Re ϕ)
e −k,−$l (Im ϕ)
e −k,−$l
G−k,−$l := −
.
(Im ϕ)
e −k,−$l
Z/Z0
(2.103)
The − sign is convention. Evaluation of the Green function (2.103)
is now an exercise in Gaussian integration over independent complexvalued integration variables that is summarized in appendix B.1. Hence,
for any 0 6= Lk
∈ Zd and l ∈ Z,
2π
−1
k2
1 2m
+ 2µren −$l
G−k,−$l = −
k2
2
+$l
2m
k2
1
1
+$
l
2m
=−
.
k2
2 k2 k2 + 2µ
−$
+
2µ
2
ren
l
2m
ren + ($l )
2m 2m
(2.104)
The factor 1/2 comes from the fact that only half of 0 6= Lk
∈ Zd are
2π
to be counted as independent labels, for (Re ϕ)(r,
e
τ ) and (Im ϕ)(r,
e
τ)
are real valued.
The Green function (2.104) has first-order poles whenever
s
k2 k2
i$l = ±
+ 2µren
2m 2m
q
|k|
=±
k2 + 4 m µren
(2.105)
2m
q
|k|
≡±
k2 + (k0 )2
2m
≡ ± ξk .
We have recovered with ξk the dispersion in Eq. (2.60b). For long
wavelengths, the dispersion is linear
" #
2
k0
|k|
ξk =
|k| + O
,
(2.106a)
2m
k0
with the speed of sound
k
v0 ≡ 0 :=
2m
r
µren
.
m
(2.106b)
For short wavelengths, the dispersion relation of a free particle emerges,
" #
2
k2
k0
ξk =
+O
.
(2.107)
2m
|k|
2.5. DILUTE-BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE
57
It is time to verify the selfconsistency of the assumptions encoded
by Eqs. (2.93), (2.97), and (2.100). This we do by solving Eq. (2.96) in
the Gaussian approximation. Fourier transform of Eq. (2.96) yields
N = V (ϕ0 )2 + δN,
(2.108a)
where
δN := β −1
*Zβ
Z
dτ
0
= β −1
+
dd r (ϕ
e∗ ϕ)
e (r, τ )
V
Z/Z
X X D
l∈Z
+ iβ −1
0
E
Re ϕ
e+k,+$l Re ϕ
e−k,−$l + (Re → Im)
Lk
∈Zd
2π
X X D
l∈Z
Re ϕ
e+k,+$l
Z/Z0
E
Im ϕ
e−k,−$l − (Re ↔ Im)
Lk
∈Zd
2π
.
Z/Z0
(2.108b)
The Gaussian approximation gives the estimate
X X ei0+ $l k2
−1
δN ≈ β
+ µren + i$l ,
2
2
$
+
ξ
2m
l
k
d
l∈Z Lk
2π
(2.109)
∈Z
as can be read from the Green function (2.104). A convergence factor
exp(i0+ $l ) was introduced to regulate the poles of the Green function (2.104). One verifies that the summand with k and l fixed, while
µ = 0 in Eq. (2.78) is recovered in the non-interacting limit µren = 0.
At zero temperature, the summation over l turns into an integral
X
l∈Z
Z+∞
→β
d$
.
2π
(2.110)
−∞
The integrand is nothing but nk . As a function of $ ∈ C, it has
two first-order
poles along the imaginary axis at ±iξk with residues
2
k
+ µren ∓ ξk . The convergence factor exp(i0+ $) allows one
± i2ξ1 2m
k
to close the real-line integral by a very large circle in the upper complex
plane $ ∈ C. Application of the residue theorems then yields
2
k
1
nk ≈
+ µren − ξk
2ξk 2m


(2.111)
2
2
1  k + (k0 ) /2

q
=
−1 .
2 |k| k2 + (k )2
0
58
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
In the thermodynamic limit,
V
−1
X
d
nk ≈ γd (k0 ) ,
k6=0
Ωd
γd :=
2(2π)d
Z∞
d−2
dxx
x2 + 1/2
√
−x .
x2 + 1
0
(2.112)
Here,
√ Ωd is the area of the unit sphere in d dimensions. Since k0 =
2 mλϕ0 , we conclude that ϕ0 is determined by
N
V
= (ϕ0 )2 + 2d γd (mλ)d/2 (ϕ0 )d
= (ϕ0 )2 1 + 2d γd (mλ)d/2 (ϕ0 )d−2 .
n :=
(2.113)
The Gaussian approximation is selfconsistent if the quantum correction 2d γd (mλ)d/2 (ϕ0 )d is smaller than the semi-classical result (ϕ0 )2 ,
i.e., if
√
ϕ0 ∼ n
(2.114)
−1/(d−2)
2d γd (mλ)d/2
.
The constant γd is finite if and only if 1 < d < 4. For d = 1, γ1 has
an infrared logarithmic divergence. For d = 4, γ4 has an ultraviolet
logarithmic divergence. When Eq. (2.114) is satisfied,
(ϕ0 )2 ≈ n − 2d γd (mλ)d/2 nd/2 .
(2.115)
The RPA (Gaussian approximation) in d = 3 is thus appropriate in
the dilute limit or when the interacting coupling constant λ is small.
For d = 2, Eq. (2.115) indicates that quantum corrections to the semiclassical result scale in the same way as a function of the particle
density n, but with the opposite sign. This is an indication that fluctuations are very important in two-dimensional space and that the RPA
might then break down.
The case of two-dimensional space is indeed very special. To see this
one can estimate the size of the fluctuations about the semi-classical
value of the order parameter by calculating the root-mean-square deviation
s
h|ϕ0 |2 − ϕ∗ (r, τ )ϕ(r, τ )iZ
|ϕ0 |2
(2.116)
within the RPA. The root-mean-square deviation should be smaller
than one in the thermodynamic limit if there is true long-range order, i.e., below the transition temperature. It should diverge upon
approaching the transition temperature from below. To evaluate
hϕ∗ (r, τ )ϕ(r, τ )iZ
(2.117)
2.5. DILUTE-BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE
59
at inverse temperature β one must perform a Fourier integral over the
entries of the Green function (2.104). These integrals are dominated
at long wavelengths by the contribution
Z d
d k
−1
β
(2.118)
k2
coming from the acoustic mode. This contribution is logarithmically
(linearly) divergent in d = 2 (d = 1) whenever β < ∞. Hence, the
root-mean-square deviation diverges in the thermodynamic limit and
within the RPA, signaling the breakdown of spontaneous symmetry
breaking and off-diagonal long-range order at any non-vanishing temperature when d ≤ 2. It is said that d = 2 is the lower-critical dimension at which and below which the U (1) continuous symmetry cannot
be spontaneously broken at any non-vanishing temperature within the
RPA. Absence of spontaneous symmetry breaking of the U (1) symmetry in the dilute Bose gas within the RPA is an example of the
Hohenberg-Mermin-Wagner-Coleman theorem. It can be shown that
thermal fluctuations due to acoustic modes downgrade the long-range
order of the ground state to quasi-long-range order within the RPA.
Quasi-long-range order is the property that the one-particle density
matrix in Eq. (2.34) decays algebraically fast with |r 0 − r| at long separations. Quasi-long-range order cannot maintain itself at arbitrary
high temperatures. The mechanism by which quasi-long-range order
is traded for exponentially fast decaying spatial correlations is called
the Kosterlitz-Thouless transition. The Kosterlitz-Thouless transition
cannot be accounted for within the RPA, since it is intrinsically a nonlinear phenomenon. Chapter 4 is devoted to the Kosterlitz-Thouless
transition.
2.5.3. Beyond the random-phase approximation. We close
the discussion of a repulsive dilute Bose gas by sketching how one can
go beyond the RPA defined by Eq. (2.100). The key to capturing
physics beyond the RPA (2.100) is to choose the parametrization
p
p
ϕ(r, τ ) = ρ(r, τ ) e+iθ(r,τ ) , (2.119)
ϕ∗ (r, τ ) = ρ(r, τ ) e−iθ(r,τ ) ,
of the fields entering the path integral (2.71). This parametrization is
non-linear. It reduces to the linear parametrization (2.97) if one works
to linear order in θ and makes the identifications
µ
ρ(r, τ ) → ren ,
iρ(r, τ )θ(r, τ ) → ϕ(r,
e τ ).
(2.120)
λ
If we were able to solve the repulsive dilute Bose gas model exactly, the
choice of parametrization of the fields in the path integral (2.71) would
not matter. However, performing approximations, say linearization of
the equations of motion, can lead to very different physics depending
on the initial choice of parametrization of the fields in the path integral.
For example, the RPA on the linear parametrization (2.97) breaks down
60
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
in d = 1, 2 due to the dominant role played by infrared fluctuations of
the Goldstone mode. On the other hand, we shall argue that the nonlinear parametrization (2.119) can account for these strong fluctuations.
The physical content of Eq. (2.119) is to parametrize the fields in the
path integral of Eq. (2.71) in terms of the density
ρ(r, τ ) := ϕ∗ (r, τ ) ϕ(r, τ )
(2.121)
1
[ϕ∗ (r, τ ) (∇ϕ) (r, τ ) − (∇ϕ∗ ) (r, τ )ϕ(r, τ )]
2mi
1
= ρ(r, τ )(∇θ)(r, τ )
m
(2.122)
and the currents
J (r, τ ) :=
associated to the global U (1) gauge invariance of the theory.
We begin by inserting Eq. (2.119) into the Lagrangian in Eq. (2.71)
1
1
λ
2
2
LE = iρ∂τ θ +
(∇ρ) + ρ(∇θ) − µren ρ + ρ2
2m 4ρ
2
(2.123)
1 2
1
+ (∂τ ρ) − (∇ ρ) − i∇ [ρ(∇θ)] .
2
2
The second line does not contribute to the action, since fields obey
periodic boundary conditions in imaginary time and in space.
Next, we expand the first line to quadratic order in powers of δρ,
to zero-th order in powers of (∇δρ)2 , and to zero-th order in powers of
(δρ)(∇θ)2 , where
µ
ρ0 := ren ,
λ
ρ(r, τ ) = ρ0 + δρ(r, τ ),
Zβ
Z
dτ
0
dd r δρ(r, τ ) = 0.
V
(2.124)
This expansion is a good one at low temperatures when the renormalized chemical potential is strictly positive. In particular, note that, to
the contrary of the RPA (2.100), we are not assuming that θ is small
(only ∇θ is taken small). We find the quadratic action
(0)
SE = SE
Zβ
+
Z
dτ
0
(0)
= SE
V
X X δρ+k,+$ † λ
2$
l
+
l
θ
−
+k,+$l
2
Lk
l
−
ρ0
λ
2
2
d r i(δρ)∂τ θ +
(∇θ) + (δρ) − (δµ)(δρ) + · · ·
2m
2
d
2π
X X
l
∈Zd
$l δρ+k,+$l
2
ρ0 2
θ+k,+$l
k
2m
+
δµ−k,−$l δρ+k,+$l + · · · ,
Lk
∈Zd
2π
(2.125)
2.5. DILUTE-BOSE GAS: PATH-INTEGRAL FORMALISM AT ANY TEMPERATURE
61
and the currents
ρ0
(∇θ)(r, τ ) + · · · .
(2.126)
m
Here, we have also substituted µren by µren + δµ(r, τ ). The source term
Rβ R
δµ(r, τ ), dτ dd r (δµ)(r, τ ) = 0, is a mathematical device to probe
J (r, τ ) =
0
V
the response to an external “scalar” potential.
The first term on the right-hand side of the first line of Eq. (2.125)
is the Berry phase that converts the radial (δρ) and angular (θ) semiclassical modes into a quantum harmonic oscillator. It implies that δρ
and θ are coupled through the classical equations of motion
0 = +i∂τ θ + λδρ − δµ ≡ +i∂τ θ − δµeff ,
(2.127)
ρ
0 = −i∂τ δρ − 0 ∆θ ≡ −i∂τ δρ + ∇ · J ,
m
in imaginary time. These equations are called the Josephson equations.
Chapter 8 is devoted to their study in the context of superconductivity
and quantum decoherence.
The second term on the right-hand side of the first line of Eq. (2.125)
is only present below the U (1) symmetry-breaking transition temperature. It endows the angular degree of freedom θ with a rigidity since,
ρ
classically, 2m0 (∇θ)2 is the penalty in elastic energy paid by a gradient
of the phase. Alternatively, this term corresponds to the kinetic energy
of a “point-like particle” of mass m/ρ0 .
The third term on the right-hand side of the first line of Eq. (2.125)
represents the potential-energy cost induced by the curvature of the
semi-classical potential well (2.101) if the “point-like particle” moves
by the amount δρ away from the bottom of the well.
Poles in the counterpart
ρ 2
$ 0
1/2
k − 2l
col
2m
Gk,$l = − λρ 2
(2.128)
$
λ
$l 2
1 0
+ 2l
k
+
2
4 m
2
for δρ and θ to the Green function (2.104) give the linear dispersion
r
r
λρ0
µren
col
ξk =
|k| =
|k|.
(2.129)
m
m
There is a one-to-one correspondence between the existence of a linear
dispersion relation and the existence of a rigidity ∝ ρ0 in our effective
model. If we interpret the rigidity of the phase θ as superfluidity, we
have establish the one-to-one correspondence between superfluidity and
the existence of a Goldstone mode associated with spontaneous symmetry breaking. In turn, this Goldstone mode can only exist when λ > 0,
i.e., Bose-Einstein condensation at λ = 0 cannot produce superfluidity.
The Green functions (2.104) and (2.128), although very similar,
have a very different physical content. In the former case, the Green
62
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
function describes single-particle properties. In the latter case, the
Green function describes collective excitations (δρ and θ). For example,
the equal-time density-density correlation function Sk is given by
V 1 X
Sk :=
δρ+l,+k δρ−l,−k
Nβ l
ρ0
|k|2
V 1X
m
N β l (ξkcol )2 + ($l )2
=
ρ
=
ρ0 ≈
N
V
≈
V 2m0 |k|2
+ O(β −1 )
N ξkcol
|k|2
+ O(β −1 ).
2 m ξkcol
(2.130)
The so-called Feynman relation
ξkcol ≈
|k|2
,
2mSk
(2.131)
which is valid at zero temperature, implies that the long-range correlation
Sk ∝ |k|
(2.132)
is equivalent to an acoustic wave dispersion for density fluctuations.
Feynman relation can be used to establish the existence of superfluidity in d = 2 when the criterion built on the RPA (2.100) fails. However,
effective theory (2.125) is still too crude for a description of superfluidity in (d = 2)-dimensional space.
2.6. Problems
2.6.1. Magnons in quantum ferromagnets and antiferromagnets as emergent bosons.
Introduction. We are going to study spin-wave excitations (magnons)
on top of a ferromagnetically and antiferromagnetically ordered state
of quantum spins arranged on a lattice. We will treat this problem using the so-called Holstein-Primakoff transformation,[29] which allows
to rewrite the spin operators at each lattice site in terms of creation
and annihilation operators of a boson. The virtue of this transformation is that the algebra of bosons is much simpler than that of spins,
for example perturbation theory simplifies greatly by the availability of
Wick theorem for bosons. However, the price to pay is that the bosonic
operators are acting on a larger local Hilbert space at every lattice site
than the spin operators. In this way, one might end up with solutions
of the bosonic problem, that do not map back to physical states in the
spin variables. This problem is avoided in the approximation by which
spins only deviate slightly from the ferromagnetic or antiferromagnetic
orientation, as we shall see below.
2.6. PROBLEMS
63
Our goal is to derive the dispersion relations of spin waves and to
understand their connections to the symmetries of the problem. For
small momenta k, the dispersions will be quadratic in k in the ferromagnet and linear in k in the antiferromagnet.
We consider a Bravais lattice Λ that is spanned by the orthonormal
vectors a1 , · · · , ad with integer-valued coefficients and made of N sites
labeled by r. Each site r ∈ Λ is assigned a spin-S degree of freedom.
The three components Ŝrα with α = x, y, z of the spin operator Ŝ r =
ex Ŝrx +ey Ŝry +ez Ŝrz act on the (2S +1)-dimensional local Hilbert space
Hrs = span | − Sir , | − S + 1ir , · · · , |S − 1ir , |Sir ,
(2.133a)
where we have chosen to represent Hrs with the eigenbasis of Ŝrz (~ = 1),
Ŝrz |mir = m |mir ,
m = −S, −S + 1, · · · , S − 1, S.
(2.133b)
The components of the spin-operator at each site r ∈ Λ obey the
algebra
[Ŝrα , Ŝrβ ] = i αβγ Ŝrγ ,
α, β, γ = x, y, z,
(2.134)
and commute between different sites. Equivalently, the operators
Ŝr± := Ŝrx ± i Ŝry
(2.135a)
obey the algebra
[Ŝr+ , Ŝr− ] = 2Ŝrz ,
[Ŝr± , Ŝrz ] = ∓Ŝr± ,
(2.135b)
at each site r ∈ Λ and commute between different sites.
Ferromagnetic spin waves. The Heisenberg Hamiltonian for a ferromagnet in the uniform magnetic field B ez , that only includes the
nearest-neighbor Heisenberg exchange coupling J > 0 is given by
X
X
ĤF := −J
Ŝ r · Ŝ r0 − B
Ŝrz ,
(2.136)
hr,r 0 i
r∈Λ
where hr, r 0 i indicates that the sum is only taken over directed nearestneighbor lattice sites. Periodic boundary conditions are imposed.
If B > 0, all the magnetic moments align along the positive ez
direction in internal spin space and the ground state is given by
O
|0i =
|Sir .
(2.137)
r∈Λ
Consider introducing a bosonic degree of freedom at every lattice
site r ∈ Λ with creation and annihilation operators â†r and âr , respectively, that act on the local infinite-dimensional Hilbert space
(â†r )n
b
Hr = span |n)r := √ |0)r n = 0, 1, 2, · · · ,
âr |0)r = 0 ,
n!
(2.138)
64
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
and obey
[âr , â†r0 ] = δr,r0 ,
(2.139)
with all other commutators vanishing. The Holstein-Primakoff transformation is defined by the following substitutes for the spin operators
Ŝr+
q
â†r
2S − âr âr ,
q
−
†
Ŝr := âr 2S − â†r âr ,
(2.140b)
Ŝrz := S − â†r âr .
(2.140c)
:=
(2.140a)
Exercise 1.1: Show that the operators Ŝr+ , Ŝr− , and Ŝrz defined
in Eq. (2.140) on Hrb obey the same algebra (2.135b) as the operators
Ŝr+ , Ŝr− and Ŝrz on Hrs .
Exercise 1.2: As we are going to study wave-like excitations, it
will be convenient to express the theory in the Fourier components of
the bosonic operators, that is
1 X +i k·r
1 X −i k·r †
ĉ†k := √
e
âr ,
ĉk := √
e
âr ,
(2.141)
N r∈Λ
N r∈Λ
where the wave number k belongs to the first Brillouin zone. Show
that ĉ†k and ĉk obey the algebra
[ĉk , ĉ†k0 ] = δk,k0 ,
(2.142)
with all other commutators vanishing.
Now, the central assumption that will allow us to simplify the theory when written in the bosonic variables is that the fraction of reversed
spins above the ferromagnetic ground state is small
† âr âr
1
(2.143)
S
for all r ∈ Λ. Within the range of validity of this assumption, it is
justified to expand the square-roots that enter Eq. (2.140) in â†r âr /2S.
Exercise 1.3: Using the spin-wave representation (2.140) of the
spin operators, show that the Hamiltonian (2.136) becomes
X γk + γ−k
X †
0
2
− 1 ĉ†k ĉk +B
ĉk ĉk ,
ĤF = −J N z S −B N S−2 J z S
2
k
k
(2.144)
when expanded to second order in the spin-wave variables ĉ†k and ĉk and
all higher-order terms are neglected. Here, z is half the coordination
number of the Bravais lattice and the form factor
1 X +ik·δ
γk :=
e
(2.145)
z δ
2.6. PROBLEMS
65
includes a sum over the z vectors δ that
P are the directed connections
to nearest-neighbor sites. (Verify that k γk = 0.)
∗
, it follows that the non-constant part of
With the help of γk = γ−k
the Hamiltonian (2.144) reads
X
ĤF00 =
(2.146)
[2J z S(1 − Re γk ) + B] ĉ†k ĉk .
k
This allows to directly read off the dispersion relation of ferromagnetic
magnons on Bravais lattices
ωk := 2 J z S(1 − Re γk ) + B.
(2.147)
Exercise 1.4: Show that for a hypercubic lattice with lattice constant a, the magnon dispersion becomes
ωk ≈ B + J S a2 |k|2
(2.148)
in the limit a |k| 1. This is equivalent to the dispersion relation of
a free massive particle of mass m∗ := 1/(2 J S a2 ).
Exercise 1.5: Suppose we had considered Hamiltonian (2.136)
with B < 0. Then, the naive expectation for the ground state is the
one with all the magnetic moments aligned along the negative ez -axis
in internal spin space,
O
|0i =
| − Sir .
(2.149)
r∈Λ
What is the counterpart to the operators (2.140) for this situation?
Antiferromagnetic spin waves. To study spin waves above an antiferromagnetic ground state, our starting point has to be modified in
three respects. First, we have to assume that the lattice Λ is bipartite,
i.e., it can be divided into two interpenetrating sublattices ΛA and ΛB
such that all nearest neighbors of any lattice site in ΛA are lattice sites
in ΛB and vice versa. Hence,
Λ = ΛA ∪ ΛB ,
ΛA ∩ ΛB = ∅.
(2.150)
For example, in two dimensions, the square lattice is bipartite, but the
triangular lattice is not (the triangular lattice is tripartite). Second,
for an antiferromagnetic coupling, the sign of the interaction parameter J that enters the Hamiltonian (2.136) has to be changed (we will
implement this sign change explicitly below, keeping J > 0 as before).
Third, the ferromagnetic source field B ez must be replaced by a staggered source field ±B ez , where one sign is assigned to sublattice ΛA
and the other sign is assigned to sublattice ΛB .
We will thus consider the Hamiltonian
X
X
X
ĤAF := J
Ŝ r · Ŝ r0 − B
Ŝrz + B
Ŝrz ,
(2.151)
hr,r 0 i
r∈ΛA
r∈ΛB
66
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
where J > 0 and B > 0 and the sums in the second-to-last and last
term on the right-hand side only run over lattice sites in sublattices ΛA
and ΛB , respectively. Periodic boundary conditions are imposed.
This time, the Holstein-Primakoff transformation differs on sublattice ΛA from that on sublattice ΛB (recall exercise 1.5). It is given by
Ŝr+
q
:=
2S −
â†r
âr
Ŝr−
âr ,
:=
â†r
q
2S − â†r âr ,
Ŝrz := S−â†r âr ,
(2.152a)
for all sites on sublattice ΛA and
q
q
†
†
−
+
†
Ŝr :=
2S − b̂r b̂r b̂r ,
Ŝr := b̂r 2S − b̂r b̂r ,
Ŝrz := b̂†r b̂r −S,
(2.152b)
for all sites on sublattice ΛB .
Exercise 2.1: We define the magnon variables on the two sublattices as
1 X −i k·r †
1 X +i k·r
ĉ†k := p
e
âr ,
e
(2.153a)
âr ,
ĉk := p
NA r∈Λ
NA r∈Λ
A
A
X
X
1
1
dˆ†k := p
e+i k·r b̂†r ,
e−i k·r(2.153b)
b̂r ,
dˆk := p
NB r∈Λ
NB r∈Λ
B
B
where NA = NB = N/2 are the numbers of sites on sublattice ΛA and
ΛB , which we take to be equal. What is the first Brillouin zone for
k, how does it compare in size to that of the case of a ferromagnet
with the same lattice Λ? Assuming the limit in which the spin-wave
approximation (2.143) is valid, expand the operators Ŝr± and Ŝrz from
Eq. (2.152) on both sublattices ΛA and ΛB to second order in the
bosonic variables â†r , âr , and b̂†r , b̂r . Then, express the result in terms
of the magnon variables ĉ†k , ĉk , dˆ†k , and dˆk defined in Eq. (2.153).
Exercise 2.2: Show that the Hamiltonian (2.151), when expanded
to quadratic order in the magnon variables, becomes
0
HAF
= − J N z S2 − B N S
X
X †
† ˆ†
† ˆ
ˆ
ˆ
+ 2J z S
Re (γk ) ĉk dk + ĉk dk + (B + 2J z S)
ĉk ĉk + dk dk .
k
k
(2.154)
The remaining task is to diagonalize the Hamiltonian (2.154), which
is already quadratic in the bosonic variables. To that end, we make
the Ansatz
α̂ := u ĉ − v dˆ† ,
β̂ := u dˆ − v ĉ† ,
(2.155a)
k
k k
k
k
k
k
k
k k
for some real-valued functions uk and vk with
u2k − vk2 = 1.
(2.155b)
2.6. PROBLEMS
67
Condition (2.155b) ensures the bosonic commutation relations
[α̂k , α̂k† ] = [β̂k , β̂k† ] = 1.
(2.156)
Exercise 2.3: Show that for an appropriate choice of uk and vk ,
the Hamiltonian becomes
X †
1
0
= −J N z S(S +1)− B N (2S +1)+
HAF
ωk α̂k α̂k + β̂k† β̂k + 1 ,
2
k
(2.157a)
with the magnon dispersion
q
ωk := + (2 J z S + B)2 − (2 J z S γk )2 .
(2.157b)
Why did we choose the positive square root and not the negative one?
Exercise 2.4: Show that in the limit of small momenta a |k| 1
and vanishing staggered source field ±B ez = 0, the magnon dispersion
on a simple cubic lattice with lattice constant a is given by the two
degenerate solutions
√
(2.158)
ωk = 2 3 J S a |k|.
When a ferromagnet and an antiferromagnet share the same lattice Λ
with the cardinality N very large but finite, is there a difference in the
total number of their magnons?
Comparison of antiferromagnetic and ferromagnetic cases. Having
worked out the Holstein-Primakoff treatment of spin wave for both
ferromagnetic and antiferromagnetic order, we will now comment on
the reasons for the differences between the two cases. Both cases can
be understood as an instance of spontaneous symmetry breaking, if
the magnitude |B| of the source field is taken to zero at the end of the
calculation. Then, the operator that multiplies the source field, i.e.,
X
X
X
M̂F :=
Ŝrz ,
M̂AF :=
Ŝrz −
Ŝrz ,
(2.159)
r∈Λ
r∈ΛA
r∈ΛB
is the order parameter for either case.
Exercise 3.1: Convince yourself that
[Ĥ0 , M̂F ] = 0
(2.160a)
[Ĥ0 , M̂AF ] 6= 0,
(2.160b)
while
where Ĥ0 is
Ĥ0 := −J
X
Ŝ r · Ŝ r0 .
(2.161)
hr,r 0 i
We thus observe the following fundamental difference. The Hamiltonian Ĥ0 commutes with the symmetry-breaking field for the case of
ferromagnetism and it does not commute with the symmetry-breaking
field in the case of antiferromagnetism.
68
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
We can expand on this observation by considering the symmetries
of Hamiltonians (2.136) and (2.151) in the limit B = 0, i.e., Hamiltonian (2.161) with J > 0 and J < 0, respectively, when Λ is bipartite.
The Hamiltonian Ĥ0 , if the lattice Λ has the point group PΛ as symmetry group, has the symmetry group
GΛ = SO(3) × Z2 × PΛ .
(2.162)
Here, the factor groups are the symmetry group of proper rotations
in spin space SO(3) ∼
= SU (2)/Z2 , the symmetry group Z2 that represents time reversal, which acts like inversion in spin space, and the
point group PΛ . For the antiferromagnet, we anticipate the breaking
of the sublattice symmetry by factoring the symmetry group Z2 that
interchanges the two sublattices from the point group PΛ of the lattice
Λ, so that
PΛ = Z2 × PΛA .
(2.163)
Here, the factor group Z2 is the group generated by the interchange of
sublattices ΛA and ΛB . The factor group PΛA is made of the pointgroup transformations of sublattice ΛA . Note that reversal of time is
represented by an antiunitary operator, while all other symmetries are
unitary.
For the ferromagnet, the order parameter M̂F is one of the generators of the continuous global symmetry group SO(3) ∼
= SU (2)/Z2 .
As a corollary, M̂F commutes with Ĥ0 , as we have verified explicitly.
Thus, M̂F and with it the ferromagnetic ground state both break the
global symmetry group SO(3) down to the subgroup SO(2). At the
same time, M̂F breaks the time-reversal symmetry, i.e., the inversion
symmetry in spin space. The symmetry-breaking pattern of the ferromagnet with the Hamiltonian Ĥ0 obtained by taking the limit N → ∞
before taking the limit B → 0 in Hamiltonian (2.136) is thus
GΛ = SO(3) × Z2 × PΛ −→ HFΛ = SO(2) × PΛ .
(2.164)
For the antiferromagnet, the order parameter M̂AF also breaks the
rotation group SO(3) down to the subgroup SO(2) and breaks timereversal symmetry. However, in contrast to the ferromagnet, M̂AF is
unchanged under a composition of time-reversal and exchange of sublattices ΛA and ΛB . The symmetry breaking pattern of the antiferromagnet with the Hamiltonian Ĥ0 obtained by taking the limit N → ∞
before taking the limit B → 0 in Hamiltonian (2.151) is thus
Λ
GΛ = SO(3) × Z2 × Z2 × PΛA −→ HAF
= SO(2) × Z2 × PΛA . (2.165)
A fundamental difference between the broken symmetry groups HFΛ
Λ
and HAF
is that all symmetries in HFΛ are represented by unitary operΛ
ators, while the Z2 symmetry in HAF
is represented by an antiunitary
operator, i.e., a composition of sublattice exchange and time-reversal.
2.6. PROBLEMS
69
In that sense, it is seen that the antiferromagnet preserves an effective
or emergent time-reversal symmetry.
The fact that M̂F commutes with Ĥ0 makes the classical [eigenstate
of Ĥ defined by Eq. (2.136) in the limit B/J → ∞ taken after the
thermodynamic limit N → ∞] and the quantum mechanical [eigenstate
of Ĥ defined by Eq. (2.136) in the limit B/J → 0 taken after the
thermodynamic limit N → ∞] ground states coincide.
Furthermore, the ferromagnet allows for an exact treatment of the
one-magnon excitations above the ground state, as we shall now explore. For simplicity, we consider a one-dimensional lattice Λ with a
spin-1/2 degree of freedom on every lattice site r ∈ Λ. We impose
periodic boundary conditions. The state
|0i = | ↑↑ · · · ↑↑i
(2.166)
is a ground state of Hamiltonian (2.136) for ferromagnetic J > 0. We
first try the states
|ri := Ŝr− |0i,
r ∈ Λ,
(2.167)
as candidates for excited states. However, these are not eigenstates of
the Hamiltonian (2.136). Instead, consider the superposition
X
|Ψα i :=
αr |ri,
(2.168)
r∈Λ
with some coefficients αr ∈ C.
Exercise 3.2: Assuming a one-dimensional lattice
√ Λ with the latik r
tice spacing a, and with the Ansatz αr = e / N , show that the
magnon dispersion
Ek = E0 + J 1 − cos(k a)
(2.169)
follows from Ĥ0 . Here, E0 = −J N/4 is the ground state energy. Which
values of k are allowed by the periodic boundary conditions?
The exact one-magnon dispersion (2.169) coincides with the dispersion (2.147) when Λ is a linear chain with the lattice spacing a.
However, in the derivation of Eq. (2.147), we had made the approximation of small fractional spin reversal (2.143), not knowing that we
would nevertheless obtain an exact result.
For the antiferromagnet, such an exact treatment cannot be carried out. The reason is that, unlike with the ferromagnet, the classical
[eigenstate of Ĥ defined by Eq. (2.151) in the limit B/J → ∞ taken
after the thermodynamic limit N → ∞] and the quantum mechanical
[eigenstate of Ĥ defined by Eq. (2.151) in the limit B/J → 0] taken
after the thermodynamic limit N → ∞] ground states of the antiferromagnet do not coincide due to the lack of commutativity between Ĥ0
and the antiferromagnetic order parameter (the staggered magnetization) [see Eq. (2.160)]. While in the ferromagnetic ground state at zero
temperature, all spins are fully aligned saturating the magnetization
70
2. BOGOLIUBOV THEORY OF A DILUTE BOSE GAS
to its maximal value, this is not the case for the staggered sublattice
magnetization of the antiferromagnet.
We are going to use the quantities introduced in the HolsteinPrimakoff treatment of antiferromagnetic magnons to calculate the deviation ∆M from the fully polarized sublattice magnetization N S/2
of the antiferromagnet. It is given by
*
+
X
N
∆M :=
S−
Ŝr .
(2.170)
2
r∈Λ
A
Here, h·i denotes the ground state expectation value, which is defined
by the occupation numbers α̂k† α̂k and β̂k† β̂k from Eq. (2.155a) being
zero for all k from the first Brillouin zone of ΛA .
Exercise 3.3: Using Eqs. (2.152), (2.153), and the inverse transform of Eq. (2.155a) for the case when B = 0, show that
!
1X
1
p
∆M =
−1 .
(2.171)
2 k
1 − (Re γk )2
Hint: Use that hα̂k† α̂k i = 0, hβ̂k† β̂k i = 0, while all off-diagonal terms
such as hα̂k† β̂k i vanish.
Exercise 3.4: Evaluate Eq. (2.171) numerically for a simple cubic
lattice with lattice spacing a = 1 in the thermodynamic limit, i.e.,
evaluate the integral
Z
1
1
N
+
d3 k p
∆M = −
3
4
2(2π)
1 − (Re γk )2
(2.172)
N
≈ 0.0784 .
2
CHAPTER 3
Non-Linear-Sigma Models
Outline
The O(N ) non-linear-σ model (NLσM) is defined as an effective
field theory that encodes the pattern of symmetry breaking of an O(N )
classical Heisenberg model defined on a square lattice. A geometric
and group-theoretical interpretation of the O(N ) NLσM is given. The
notions of fixed points and the notions of relevant, marginal, and irrelevant perturbations are introduced. The real-valued scalar (free) field
theory in two-dimensional Euclidean space is taken as and example of
a critical field theory and the Callan-Symanzik equations obeyed by
(m + n) point correlation functions are derived for the two-dimensional
O(2) NLσM. The Callan-Symanzik equations are generalized to include
relevant coupling constants. The Callan-Symanzik equation obeyed by
the spin-spin correlator in the (d = 2)-dimensional O(N > 2) NLσM is
derived. The beta function and the wave-function renormalization are
computed up to second order in the coupling constant of the (d = 2)dimensional O(N > 2) NLσM. It is shown that the free-field fixed point
is IR unstable in that the coupling constant flows to strong coupling
upon coarse graining and that there exists a finite correlation length
that increases exponentially fast upon approaching the free-field unstable fixed point. The beta function for the O(N ) NLσM is also computed
in more than two-dimensions, in which case it exhibits a non-trivial
fixed point at a small but finite value of the coupling constant.
3.1. Introduction
The so-called non-linear-sigma-models (NLσMs) will occupy us for
this and the following chapters. NLσM were first introduced in highenergy physics in the context of chiral symmetry breaking. NLσM also
play an essential role in condensed matter physics where they appear
naturally as effective field theories describing the low-energy and longwavelength limit of numerous microscopic models. We begin by defining the O(3) NLσM. We then proceed with NLσM whose target manifolds are Riemannian manifolds, homogeneous spaces, and symmetric
spaces.
71
72
3. NON-LINEAR-SIGMA MODELS
H
Z2
R3
i
Si
e3
e2
Figure 1. Heisenberg model on a square lattice with
O(3)/O(2) symmetry in the presence of a uniform magnetic field H. A symmetry breaking magnetic field
enforces the pattern of spontaneous-symmetry breaking
into the ferromagnetic state. The symmetry group of
the Heisenberg exchange interaction is O(3). A uniform
magnetic field H breaks the O(3) symmetry down to
the subgroup O(2) of rotations about the axis pointing
along H. There is a one-to-one correspondence between
elements of the coset space O(3)/O(2) and points of the
two-sphere, i.e., the surface of the unit sphere in R3 .
3.2. Non-Linear-Sigma-Models (NLσM)
We shall set the Boltzmann constant kB = 1.38065 × 10−23 J/K to
unity, in which case β ≡ 1/(kB T ) = 1/T is the inverse temperature
throughout this chapter.
3.2.1. Definition of O(N ) NLσM. An example for an unfrustrated classical O(N ) Heisenberg magnet at the inverse temperature β
and in the presence of an external uniform magnetic field H is given
by the classical partition function


Z
X
X
ZN ,βJ,βH := D[S] exp β J
Si · Sj + β
H · S i  , (3.1a)
hiji
i
where the local degrees of freeedom
S i = S i+Leµ ∈ RN ,
S 2i = 1,
i ∈ Zd ,
H ∈ RN ,
N = 1, 2, · · · ,
(3.1b)
are defined on a lattice spanned by the basis vectors
eTµ = (0 · · · 0 a 0 · · · 0),
µ = 1, · · · , d.
(3.1c)
Accordingly, on each site i ∈ Zd of a d-dimensional hypercubic lattice
made of N = (L/a)d sites, a unit vector S i of RN interacts with its 2d
nearest-neighbors S i±eµ , µ = 1, · · · , d, [hiji ≡ hi(i + eµ )i] through the
ferromagnetic Heisenberg coupling constant (also called spin stiffness)
J > 0 as well as with an external uniform magnetic field H. Periodic
3.2. NON-LINEAR-SIGMA-MODELS (NLσM)
73
boundary conditions are imposed. The measure D[S] is the product
measure over the infinitesimal surface element of the unit sphere in
RN . Alternatively and in anticipation of taking the continuum limit,
the lattice spacing a can be made explicit through rescaling,


+∞
ad−2 P s ·s +ad P h·s
YZ
i j
i
1
i
hiji
ZN ,βJ,βH ∝ 
dN si δ
− s2i  e
, (3.2a)
2
g
i
−∞
or

ZN ,βJ,βH ∝ lim 
+∞
YZ
λ→∞

dN si  e
ad−2
P
hiji
si ·sj +ad
P
i
h·si −λ
1
−s2i
g2
2 ,
i −∞
(3.2b)
or

ZN ,βJ,βH ∝ 

i
Ph
P
Z+∞
1
d−2
d
2
dλi  a hiji si ·sj +a i h·si +iλi g2 −si
N
d si
,
e
2π
+∞
YZ
i −∞
−∞
(3.2c)
where
r
d−2
1
β
g 2 := ad−2
,
h := a−d+ 2
H = a−d βgH.
(3.2d)
βJ
J
Both the measure and the Heisenberg interaction are invariant under any global rotation Q ∈ O(N ) of the spins S i 1
e,
S = QS
∀i.
(3.4)
i
i
The uniform magnetic field H breaks this global O(N ) invariance down
to the subgroup O(N −1) of global rotations in the (N −1)-dimensional
subspace of RN orthogonal to H (see Fig. 1). A magnetic field should
here be thought of as either a formal device to break the O(N ) symmetry down to O(N −1) when it is uniform or as a source term inserted for
mathematical convenience to compute correlation functions, in which
case it can be taken to be non-uniform. In both interpretations, it must
be set to zero at the end of the day [see Eq. (3.12)].
Invariance of the partition function under the transformation
J → −J,
S i → (−)||i/a|| S i ,
where
||i/a|| ≡
d
X
µ=1
iµ ,
i≡
H → (−)||i/a|| H,
d
X
iµ e µ ,
iµ ∈ Z,
(3.5a)
(3.5b)
µ=1
1
The group of orthogonal matrices O(N ) is made of all N × N matrices Q
with real-valued matrix elements, non-vanishing determinant, and obeying
QT Q = Q QT = 1N .
(3.3)
Equation (3.3) implies that the determinant of an orthogonal matrix is ±1. The
subgroup SO(N ) ⊂ O(N ) is made of all orthogonal matrices with determinant one.
74
3. NON-LINEAR-SIGMA MODELS
defines absence of frustration. In general, a lattice is said to be geometrically frustrated when it cannot be decomposed into two interpenetrating sublattices. For frustrated lattices, ferromagnetic (J > 0)
and antiferromagnetic (J < 0) couplings are not equivalent. A nextnearest-neighbor coupling constant on a square lattice is another way
by which frustration arises.
The spin configuration with the lowest energy has all N spins parallel to the external magnetic field H, i.e., is fully polarized into the
ferromagnetic ground state. The uniform magnetization
1 X
M :=
Si
(3.6)
N i
is maximal in magnitude in the ferromagnetic ground state,
|M | ≤ |M ferro | = 1,
M ferro :=
H
.
|H|
(3.7)
The expectation value of the magnetization
1
∂H ln ZN ,βJ,βH
|H|→0 N β
lim hM iZN ,βJ,βH := lim
|H|→0
(3.8)
vanishes for any finite N . 2 In the thermodynamic limit, the expectation value of the magnetization (3.8) depends crucially on the order
in which the two limits N → ∞ and |H| → 0 are taken. On the one
hand, if the limit |H| → 0 is taken before the thermodynamic limit
N → ∞, then the expectation value of the magnetization vanishes at
any temperature. On the other hand, if the limit |H| → 0 is taken
after the thermodynamic limit N → ∞, then the expectation value
of the magnetization need not vanish anymore (the answer depends
on the dimensionality d of the lattice), since any two configurations
e } d of the spins that differ by a global or rigid rota{S i }i∈Zd and {S
i i∈Z
tion Q 6= 1N ∈ O(N ) of all spins,
e,
Si = Q S
i
∀i ∈ Zd ,
(3.9)
differ in energy by the infinitely high potential barrier
lim [N × |H · (1N − Q) M |] = ∞.
N →∞
2
(3.10)
This is so because only the magnitude of the magnetization is fixed in the
ferromagnetic ground state when the external magnetic field has been switched
off. The direction in which the magnetization points is arbitrary. Hence, the
path integral over all spin configurations can be restricted to a path integral over
all spins pointing to the northern hemisphere of the N -dimensional unit sphere in
some arbitrarily chosen spherical coordinate system provided −M is added to +M
between the brackets on the left-hand side of Eq. (3.8).
3.2. NON-LINEAR-SIGMA-MODELS (NLσM)
75
How should one decide if spontaneous symmetry breaking at zero temperature as defined by
1 = lim lim lim hM iZN ,βJ,βH ,
|H|→0 N →∞ β→∞
0 = lim lim lim hM iZN ,βJ,βH ,
N →∞ |H|→0 β→∞
(3.11)
extends to finite temperature, i.e.,
0 6=
lim lim hM iZN ,βJ,βH ,
|H|→0 N →∞
lim lim hM iZN ,βJ,βH ?
0=
N →∞ |H|→0
(3.12)
Since the thermodynamic limit must matter for spontaneous-symmetry
breaking to take place, we can limit ourselves to very long wavelengths.
Since we want to know whether or not zero-temperature spontaneoussymmetry breaking is destroyed by thermal fluctuations for arbitrarily
small temperatures, we can limit ourselves to low energies. If we are
after some sort of perturbation theory, the dimensionless bare coupling
constant
1
a−(d−2) g 2 =
(3.13)
βJ
might be a good candidate at very low temperatures and very large
spin stiffness J.
As the simplest possible effective field theory sharing the global
O(N ) symmetry of Eqs. (3.1) or (3.2) in the absence of a symmetry
breaking external magnetic field, we might try the Euclidean field theory 3


Z
Z
h
i
2
1
ZβJ,βH := D[n] δ 1 − n2 exp − 2 dd x ∂µ n − 2a−d βg 2 H · n 
2g
Rd
(3.14a)
that defines the O(N ) NLσM when H = 0. The partition function (3.14a) is proportional to the partition function


Z
Z
h
i
1
1
2
− m2 exp −
dd x ∂µ m − 2h · m 
ZβJ,βH ∝ D[m] δ
2
g
2
Rd
(3.14b)
that is obtained by rescaling the dimensionless
field n through a divip
sion by the positive square root g = + g 2 of the coupling constant g 2 .
In turn, we can exponentiate the constraint on the length of the vector
3
For any pair of directed nearest-neighbor sites hiji, we write S i · S j =
2
S i − S j + 12 S 2i + 21 S 2j in Eq. (3.1). We may then replace finite differences
by derivatives in the spirit of a naive continuum limit.
− 21
76
3. NON-LINEAR-SIGMA MODELS
field m in two different ways. First,


2 +∞
R d
2
− 12
d x (∂µ m) −2 h·m−λ 12 −m2
Y Z
g
ZβJ,βH ∝ lim 
dN m(x) e Rd
.
λ→∞
x∈Rd−∞
(3.14c)
Second,

ZβJ,βH

h
io
+∞
R d n
2
Z+∞
1
1
2
−2
h·m−iλ
−m
−
d
x
∂
m
Y Z
(
)
µ
2
2
dλ(x) 
g
∝
.
dN m(x)
e Rd
2π
d
x∈R −∞
−∞
(3.14d)
3.2.2. O(N ) NLσM as a field theory on a Riemannian manifold. To better understand the relationship between the theory (3.14)
in the continuum and the theory (3.2) on the lattice, choose a coordinate system of RN in which the symmetry breaking magnetic field h is
aligned along the direction e1 (see Fig. 1),
h · m = |h|m1 .
(3.15)
We first observe that 4


+∞
+∞
Z
Z
Z


Y
1
2
D[m] δ
− m (· · · ) =
d[m2 (x)] · · ·
d[mN (x)]
 d

g2
x∈R −∞
−∞
!"
#
N
X
(·
·
·
)
(·
·
·
)
1
−
m2 (x)
+
×Θ
g 2 j=2 j
2 m1 (x) m (x)=+σ(x) 2|m1 (x)| m (x)=−σ(x)
1
1
(3.16a)
where Θ(x) is the Heaviside step function and we have introduced
v
u
N
X
u1
t
σ(x) :=
−
m2 (x).
(3.16b)
g 2 j=2 j
Motivated by Eq. (3.16), we shall restrict all the local configurations
m(x) entering in the path integral (3.14) to configurations called spin
waves which are defined by the conditions that:
(1) Local longitudinal fluctuations σ(x) about the ferromagnetic
state
1
∀x,
eT1 = (1 0 · · · 0),
(3.17a)
m(x) = e1 ,
g
are strictly positive
0 < σ(x) ≡ m1 (x) ≤ 1/g,
4
Use δ(x2 − a2 ) = δ[(x − a)(x + a)] =
1
± 2|±a| δ[x
P
∀x.
− (±a)].
(3.17b)
3.2. NON-LINEAR-SIGMA-MODELS (NLσM)
77
(2) Local transverse fluctuations π,
πi (x) ≡ mi+1 (x),
i = 1, · · · , N − 1,
(3.17c)
about the ferromagnetic state are smaller in magnitude than
1/g,
r
1
1
0 < σ(x) = +
(3.17d)
− π 2 (x) ≤ .
2
g
g
(3) Transverse fluctuations π are smooth, i.e., the Taylor expansion
∞
X
1 l i = 1, · · · , N −1,
µ = 1, · · · , d,
πi (x+yeµ ) =
∂µ πi (x)y l ,
l!
l=0
(3.17e)
converges very rapidly when |y| is of the order of the lattice
spacing a.
Only the northern-half hemisphere of the surface of the sphere with
radius 1/g in RN is thus parametrized in the spin-wave approximation. Accessing configurations of spins in which the field m points
locally, say at x, towards the southern hemisphere, m1 (x) < 0, is
impossible within the spin-wave parametrization (3.17d). In the spinwave approximation, the second additive term on the right hand side
of Eq. (3.16a) is neglected. This approximation is good energetically
since configurations of spins in which m1 (x) is negative over some large
d
but bounded
region Ω of R is suppressed by the exponential factor of
R
order exp +|h| dd x m1 (x) in (· · · ) of Eq. (3.16a). However, this
Ω
argument fails to account for the entropy of the excursions of m1 into
the southern hemisphere, i.e., the multiplicity of spin configurations
that are suppressed by an exponentially small penalty in energy for
pointing antiparallel to h in some region of Rd . The spin-wave approximation breaks down whenever the entropy of defects by which m1 is
antiparallel to h overcomes the loss of energy incurred by this excursion
of m1 into the southern hemisphere.
In the spin-wave approximation, the Euclidean action of the NLσM
becomes
Z
h
i
2
2
1
Ssw β,h :=
dd x ∂µ σ + ∂µ π − 2|h|σ
2
Rd
"
#
r
Z
(π
∂
π
)(π
∂
π
)
1
1
2
i
µ
i
j
µ
j
=
dd x
+ ∂µ π − 2|h|
− π2
1
2
2
2
g
−
π
2
g
Rd
r
Z
1
1
d
=
d x (∂µ πi ) gij (∂µ πj ) − 2|h|
− π2 ,
2
g2
Rd
(3.18a)
78
3. NON-LINEAR-SIGMA MODELS
where the symmetric (metric) tensor
g 2 (πi πj )(x)
gij (x) :=
+ δij ,
1 − g 2 π 2 (x)
i, j = 1, · · · , N − 1,
(3.18b)
has been introduced and summation over repeated indices is understood.
The metric tensor transforms according to
(∂µ πi ) gij (∂µ πj ) = (∂µ π
ek ) e
gkl (∂µ π
el ),
(3.19a)
where
e
gkl = Rik gij Rjl
= RT ki gij Rjl ,
k, l = 1, · · · , N − 1,
(3.19b)
under the global rotation R ∈ O(N − 1) of the transverse modes π
under which
e (x).
π(x) = R π
(3.19c)
In matrix form, Eq. (3.19) reads
e
g(x) = RT g(x) R,
∀R ∈ O(N −1) ⇐⇒ g(x) = R e
g(x) RT ,
∀R ∈ O(N −1).
(3.20)
A useful invariant under global O(N − 1) rotations of the transverse
modes π is the determinant of the metric tensor (3.18b),
det[g(x)] = det R e
g(x) RT
= det (R) det [e
g(x)] det RT
= [det (R)]2 det [e
g(x)]
= det [e
g(x)] , ∀R ∈ O(N − 1).
(3.21)
Equation (3.21) also extends to the situation when the matrix R ∈
O(N − 1) is allowed to vary in space, although it should then be remembered that Eq. (3.19) does not hold anymore. This observation is
useful in that it allows to compute det[g(x)] by choosing the local rotation R(x) ∈ O(N − 1) that rotates π(x) along e2 , say, in which case
g(x) is purely diagonal with the eigenvalue 1 (N − 2)-fold degenerate
g2 π2
and the eigenvalue 1−g
2 π 2 + 1. Thus, we infer that
det[g(x)] =
1
,
1 − g 2 π 2 (x)
∀x ∈ Rd .
(3.22)
3.2. NON-LINEAR-SIGMA-MODELS (NLσM)
79
We are now ready to write in a compact manner the spin-wave
approximation


Y Z dN −1 π(x) p
−1

Zsw β,h := 
g 2 det g(x) Θ g 2 det g(x)
2
d
x∈R RN −1


Z
h
i
−1/2
1

× exp −
dd x (∂µ πi )gij (∂µ πj ) − 2|h| g 2 det g
2
Rd
(3.23a)
to the partition function of the O(N ) NLσM,


Z
N
−1
Y
−1
d
π(x) p 2

g det g(x) Θ g 2 det g(x)
Zβ,h := 
2
x∈RdRN −1


Z
h
i
1
−1/2

dd x (∂µ πi )gij (∂µ πj ) − 2|h| g 2 det g
× exp −
2
Rd


Y Z dN −1 π(x) p
−1

+
g 2 det g(x) Θ g 2 det g(x)
2
x∈RdRN −1


Z
h
i
−1/2
1
.
dd x (∂µ πi )gij (∂µ πj ) + 2|h| g 2 det g
× exp −
2
Rd
(3.23b)
We made two approximations to reach Eq. (3.23a). First, we performed the naive continuum limit (3.14) consisting in expanding the
lattice action to Gaussian order in the derivative of the spin field. Second, we ignored the field configurations of the spins that have, locally,
any antiparallel component with respect to the external magnetic field
in Eq. (3.23b). What is left out from the naive continuum limit is
the possibility that “singular” lattice configurations of the spins matter in the thermodynamic limit. 5 These “singular” configurations of
the spins on the lattice correspond in the continuum approximation
to spin fields whose orientations along the external magnetic field can
change from parallel to antiparallel as a function of space. It turns
out that singular lattice configurations of the spins are essential to the
understanding of the phase diagram of the O(2) NLσM in d = 2 as we
shall see in the chapter devoted to the Kosterlitz-Thouless transition.
The usefulness of the spin-wave approximation is that it allows for an
5
A configuration {si } of spins on the lattice is said to be singular if its naive
continuum limit counterpart m is not smooth everywhere in Rd , i.e., is singular at
isolated points. On the lattice there is no notion of smoothness.
80
3. NON-LINEAR-SIGMA MODELS
answer to the question of whether thermal fluctuations in the form of
spin waves are sufficient to destroy spontaneous-symmetry breaking at
zero temperature.
Representation (3.23b) of the O(N ) NLσM is geometric in nature.
The (N −1)–sphere is an example of a Riemannian manifold on which a
special choice of coordinate system, encoded by the metric (3.18b), has
been made. The action in representation (3.23b) is covariant under a
change of coordinate system of the (N − 1)-sphere. The determinant of
the metric in the functional measure of integration over the fields π(x)
guarantees that the functional measure is a geometrical invariant under
O(N ) induced transformations. If Q denotes an element of O(N ), one
can always define the matrix-valued function Q(x) that relates m(x)
to e1 through
p
g m(x) =: Q(x) e1 ,
g = + g 2 ≥ 0,
(3.24a)
The subgroup of O(N ) that leaves e1 invariant is called the little group
(or stabilizer) of e1 . Here, it is the subgroup O(N − 1) of O(N ). If
R(x) takes values in the little group of e1 ,
Q(x) R(x) e1 = Q(x) e1 = g m(x).
(3.24b)
Relations (3.24a) and (3.24b) exhibit the isomorphism between the
coset (homogeneous) space O(N )/O(N − 1) and the (N − 1)–sphere
SN −1 . More generally, Eq. (3.23b) can be taken as the definition of
a NLσM on a (N − 1)–dimensional Riemannian manifold with local
metric gij .6 This definition of a NLσM is more general than that of
the O(N ) NLσM (3.14) as it is not always possible to establish an
isomorphism between any given Riemannian manifold and some coset
(homogeneous) space. Appendix C is devoted to a detailed study of a
NLσM on a generic Riemannian manifold.
3.2.3. O(N ) NLσM as a field theory on a symmetric space.
Representation (3.23) puts the emphasis on the geometrical structure
behind NLσM. The initial question on spontaneous-symmetry breaking is cast in the language of group theory. Is there a representation
of the O(N ) NLσM that puts the emphasis on the underlying group
theoretical structure, i.e., renders the pattern of symmetry breaking
explicit?
On the one hand, Eq. (3.23) say that, for any given x ∈ Rd , (N − 1)
real parameters π1 (x), · · · , πN −1 (x), are needed to parametrize the
6
A Riemannian manifold is a smooth manifold on which a continuous 2covariant symmetric and non-degenerate tensor field called the metric tensor can
be defined, i.e., for any point p on the manifold there exists a symmetric and nondegenerate bilinear form gp from the tangent vector space at x to the real numbers.
3.2. NON-LINEAR-SIGMA-MODELS (NLσM)
81
(N − 1)-sphere. 7 On the other hand, the number of independent
generators of the coset space O(N )/O(N − 1) is also N − 1. 8
This agreement is not coincidental as we saw in Eqs. (3.24a) and
(3.24b). Indeed, we recall that for any point g m(x) on the unit sphere
SN −1 with the coordinates g π(x), there exists the N × N orthogonal
matrix Q(x) ∈ O(N ) such that
g π(x) ∼ g m(x) =: Q(x) e1 .
(3.29)
Evidently, the relation between the point g m(x) of the unit sphere
SN −1 and the N × N rotation matrix Q(x) is one to many, since right
multiplication of Q(x) by any element R(x) from the little group O(N −
1) that leaves the north pole e1 unchanged yields
Q(x) e1 = Q(x) R(x) e1 ,
∀R(x) ∈ O(N − 1).
(3.30)
The one-to-one relationship that we are seeking is between the unit
sphere SN −1 and the quotient space of N × N matrices
O(N )/O(N − 1) := {[Q] | Q ∈ O(N )},
(3.31a)
where the equivalence class
[Q] := {Q0 ∈ O(N ) | Q ∼ Q0 }
(3.31b)
is defined through the equivalence relation Q ∼ Q0 if and only if there
exists R ∈ O(N ) such that Q = Q0 R and R e1 = e1 . In other words,
7The
(N − 1)-sphere is the (N − 1)-dimensional surface embedded in RN and
defined by


N
−1
X
σ
2 2
2
g
σ +
πj = 1,
∀
∈ RN .
(3.25)
π
j=1
The (N − 1)-sphere is often denoted SN −1 ⊂ RN .
8 For any Q ∈ O(N ) det Q = ±1. If det Q = 1, i.e., Q ∈ SO(N ), it is always
possible to write Q = exp(A) and QT = exp(AT ) where A is also a N × N matrix
with real-valued matrix elements. Equation (3.3) in foootnote 1 implies that A and
AT obey
A + AT = 0,
(3.26)
i.e., that A is a N × N real-valued antisymmetric matrix. The number of independent real-valued matrix elements in A equals the number of entries above the
diagonal, i.e.,
1
1
N 2 − N = N (N − 1).
(3.27)
2
2
As real vector spaces, the dimensionality of O(N ) is thus 12 N (N − 1) and the
dimensionality of O(N − 1) is 21 (N − 1)(N − 2). The dimensionality of the coset
space O(N )/O(N − 1) is, by definition, the difference between the dimensionality
of O(N ) and O(N − 1),
dim O(N )/O(N −1) := dim O(N )−dim O(N −1) =
1
(N −1) [N − (N − 2)] = N −1.
2
(3.28)
82
3. NON-LINEAR-SIGMA MODELS
the coordinate g π(x) of the point g m(x) on the unit sphere is identified with the set
{Q(x)R(x)|g m(x) = Q(x) e1 and R(x) e1 = e1 }.
(3.31c)
We are now going to represent the O(N ) NLσM in terms of the
elements of O(N ). To this end, observe that any real-valued N × N
antisymmetric matrix A(x) can be written as 9
g X
α (x) Tij ,
Tij := Eij − Eji ,
(3.34)
A(x) =
2 1≤i<j≤N ji
where the N × N matrices Eij has one single non-vanishing matrix element equal to 1 for line i and column j, αji (x) are real-valued numbers,
and αji are smooth functions Rd → R. The factor of 1/2 is convention
[see Eq. (3.33) in footnote 9] and we have endowed αij (x) with the
dimensions of g −1 . We shall assume that g is infinitesimal, in which
case A is also infinitesimal. According to Eqs. (3.26) and (3.33) in footnotes 8 and 9, respectively, for any infinitesimal A = −AT we deduce
the following chain of equalities
Q(x) = eA(x) ≈ 1N + A(x) ∈ SO(N ),
(3.35a)
g X
QT ∂µ Q (x) ≈
∂µ αji (x)Tij ,
(3.35b)
2 1≤i<j≤N
T
g X
∂µ αji (x)Tij ,
QT ∂µ Q (x) = − QT ∂µ Q (x) ≈ −
2 1≤i<j≤N
(3.35c)
and
tr
9
h
QT ∂µ Q
T
i g2
(x) QT ∂µ Q (x) ≈
2
X
∂µ αji
2
(x). (3.35d)
1≤i<j≤N
The algebra
1 ≤ k < l ≤ N,
(3.32)
defines the Lie algebra of the Lie group SO(N ). Since Tij is antisymmetric with
only two non-vanishing entries, +1 for line i and column j and −1 for line j and
column i,
[Tij , Tkl ] = δik Tlj + δjl Tki + δil Tjk + δjk Til ,
tr Tij Tkl =
N
X
Tij
mn
1 ≤ i < j ≤ N,
(Tkl )nm
m,n=1
=
N
X
δim δjn − δin δjm (δkn δlm − δkm δln )
(3.33)
m,n=1
= 2 δil δjk − δik δjl ,
i, j, k, l = 1, · · · , N.
The scaling factor of 1/2 in Eq. (3.34) insures that the trace of Tij /2 with itself
gives −1/2.
3.2. NON-LINEAR-SIGMA-MODELS (NLσM)
83
The right-hand side of Eq. (3.35d) is positive and can thus be used to
construct a Boltzmann weight.
Next, we define the partition function


h
i
R
T
Z
− dd x 12 tr (QT Dµ Q) (QT Dµ Q)−H0 I1,N −1 Q
Y
g


Zg2 ,H0 := 
dQ(x) e Rd
,
x∈RdO(N )
(3.36a)
where we are making use of the covariant derivative
Dµ Q := ∂µ Q − Q Aµ ,
(3.36b)
with the gauge field defined by
Aµ (x) := Projection of QT ∂µ Q (x) onto the little group.
(3.36c)
[The little group was defined in Eq. (3.24).] The (Haar) measure of
O(N ) accounts for the fact that O(N ) is not simply connected, for it
is impossible to smoothly change the sign of the determinant of an orthogonal matrix. 10 In other words, one must sum separately over the
two connected components of O(N ), i.e., over those pure rotations with
determinant +1, and those rotations that have been composed with the
inversion of one and only one coordinate (the combined operation thus
has determinant −1). This observation is nothing but a manifestation
of the additive decomposition (3.16a). In turn, the measure on either
of the two connected components of O(N ) is the Haar measure of the
group SO(N ), an example of a simple and compact Lie group. In this
book, it will be sufficient to know that a Haar measure can be constructed for any compact Lie group and that this measure is invariant
under left or right group multiplication. The so-called covariant derivative Dµ expresses the fact that, out of the N (N − 1)/2 degrees of
freedom associated to the connected subgroup SO(N ), (N −1)(N −2)/2
of them are redundant. Indeed, the “magnetic field” H0 I1,N −1 , here
represented by the real number H0 multiplying the diagonal matrix


+1 0
0 ··· 0
 0 −1 0 · · · 0 
I1,N −1 := 
(3.37)
..
..
..
.. 
 ...
.
.
.
. 
0
0
···
0
−1
in O(N ), defines the subgroup O(N − 1) ⊂ O(N ) made of all matrices
from O(N ) that commute with I1,N −1 . This is the little group. Any two
local elements Q1 (x) ∈ SO(N ) and Q2 (x) ∈ SO(N ) from the target
manifold that differ by the right multiplication with the local matrix
R(x) from SO(N − 1),
Q1 (x) = Q2 (x) R(x),
10
(3.38)
A gentle introduction to the mathematics of Haar measures can be found in
chapter 15 of Ref. [30].
84
3. NON-LINEAR-SIGMA MODELS
are physically equivalent and should only be counted once in the path
integral. To put it differently, the covariant derivative insures that the
path integral over SO(N ) reduces to a path integral over all equivalence
classes in the coset space SO(N )/SO(N − 1). This redundancy under
local right multiplication is an example of a local gauge symmetry.
The transformation laws of the non-Abelian gauge field and covariant
derivative under the right multiplication
Q(x) → Q(x) R(x),
R(x) ∈ SO(N − 1),
(3.39)
are
Aµ → RT Aµ R + RT ∂µ R
(3.40)
Dµ Q → Dµ Q R,
(3.41)
and
respectively. Hence, when H0 = 0, the action in Eq. (3.36a) is locally
gauge invariant due to the cyclicity of the trace. Evidently, when H0 =
0, the action in Eq. (3.36a) is also invariant under the O(N ) global left
multiplication
Q(x) → L0 Q(x)
(3.42)
since
QT ∂µ Q (x) → QT ∂µ Q (x)
(3.43)
while
Dµ Q (x) → L0 Dµ Q (x).
(3.44)
When H0 6= 0, by the cyclicity of the trace, the global symmetry group
is the transformation
Q(x) → LT0 Q(x) L0
(3.45)
where L0 is any N × N matrix from the subgroup O(N − 1) of matrices
in O(N ) that commute with I1,N −1 .
The partition function (3.36) shares the same global symmetries as
the partition function (3.23b). Deriving the partition function (3.23b)
from the partition function (3.36) requires an explicit parametrization
of the N × N orthogonal matrices Q. One possible choice can be found
in chapter 6 from Ref. [31]. However, the message of section 3.2.3 is
that a classical partition function can be interpreted as a gauge theory
if a redundant description of the degrees of freedom is chosen.
We close this discussion with a brief description of some mathematical background.
An N -dimensional Riemannian manifold can be pictured as a smooth
N -dimensional surface embedded in some Euclidean (flat) space through
the imposition of a constraint (see appendix C). For example, the unit
sphere SN −1 is the set of all N -dimensional real-valued vectors with
unit length.
Riemannian manifolds are endowed with a metric, i.e., a notion of
distance (see appendix C). For the case of the unit sphere SN −1 with
3.2. NON-LINEAR-SIGMA-MODELS (NLσM)
85
the coordinates π1 , · · · , πN −1 , and the metric (3.18b), the distance
between any two points follows from minimizing the length
Z1 r
dπ dπj
L[c] := g dt gij i
(3.46)
dt dt
0
with respect to the choice made for the curve c(t) parametrized by
0 ≤ t ≤ 1 that connects the two points on the sphere. The minimal
curve is called a geodesic.
The unit sphere SN −1 has, however, more than a metric. Any rotation of the Cartesian coordinate system in the embedding Euclidean
space RN leaves the distance between any two points from the unit
sphere unchanged. As a corollary, Eq. (3.29) holds.
This property of the unit sphere SN −1 can be generalized as follows.
A Riemannian manifold M is said to be homogeneous if it can be
associated to a Lie group G in such a way that for any two point
x and y in M (i) there exists an element from g ∈ G with g x = y
(transitivity) and (ii) the distance between x and y is the same as the
distance between g x and g y (isometry).
For example, the unit sphere SN −1 is an homogeneous Riemannian
manifold with the transitive isometric group O(N ).
An homogeneous Riemannian manifold M is characterized by the
coset G/H where H is the subgroup of G that leaves an arbitrary point
x of M invariant,
x = hx
(3.47)
for any h ∈ H. An homogeneous Riemannian manifold M is said to be
symmetric if its symmetry group G (a semi-simple compact Lie group)
is also characterized by a mapping on itself that preserves the group
structure (an automorphism) and is involutive (it becomes the identity
mapping if composed with itself). All elements of the Lie algebra of G
are then either odd or even under this involution. The little group H
is then generated by all the even generators from the Lie algebra of G
under the involutive automorphism.
For example, a family of involutive automorphisms on O(N ) are
the mappings
−1
Q → Ip,q
Q Ip,q
(3.48a)
where the N × N diagonal matrices Ip,q are
Ip,q = diag(+1, · · · , +1, −1, · · · , −1),
|
{z
} |
{z
}
p-times
q -times
N = p + q.
(3.48b)
The family of subgroups of O(N ) left invariant by these automorphisms
is
{O(p) × O(N − p) | p = 1, · · · , N − 1}.
(3.49)
86
3. NON-LINEAR-SIGMA MODELS
The corresponding family of coset spaces
{Gp ≡ O(N )/O(p) × O(N − p) | p = 1, · · · , N − 1},
(3.50)
are called Grassmannian manifolds. The case p = 1 corresponds to
the choice (3.37) that we made for the symmetry-breaking term in the
partition function (3.36a). Thus, the O(N ) NLσM is the special case
when the target space is the p = 1 Grassmannian manifold G1 . A
one-to-one realization of Gp in O(N ) is given by
x → T (x) Ip,N −p T −1 (x),
T (x) ∈ O(N ),
(3.51)
since right multiplication
T (x) → T (x) R(x),
R(x) ∈ O(p) × O(N − p) ⊂ O(N )
(3.52)
leaves T (x) Ip,N −p T −1 (x) unchanged. The action
Z
i
T T
1 h
Sg2 ,H0 [Q] := dd x 2 tr QT ∂µ Q
Q ∂µ Q − H0 Ip,N −p Q (3.53)
g
Rd
where Q(x) ∈ O(N ) is parametrized according to Eq. (3.51) delivers
the Riemannian metric of Gp once the trace has been evaluated (see
Ref. [25]).
3.2.4. Other examples of NLσM.
(1) Classical ferromagnetism with the group O(3).
(2) Liquid crystals.
(3) Quantum antiferromagnets on a square lattice with the group
O(3).
(4) Spin-1/2 quantum spin chains with the group SU (2) which is
locally isomorphic to O(3).
(5) Anderson localization, polymers, and other disordered systems.
(6) Strongly correlated systems with the groups SO(5) and CPN −1 ,
the latter being locally isomorphic to SU (N )/U (N − 1).
3.3. Fixed point theories, engineering and scaling
dimensions, irrelevant, marginal, and relevant
interactions
For notational simplicity, we shall consider the case N = 1 in this
section. No fluctuations about the ferromagnetic state is allowed, irrespective of temperature, in the O(1) NLσM,


R d
Z
Z
1
h
i
+ |g|
d xh
2
1
1
2
d
−

Rd
D[ϕ]δ
−
ϕ
exp
d
x
∂
ϕ
−
2hϕ
=
e
.
µ
g2
2
Rd
(3.54)
3.3. FIXED POINT THEORIES, ENGINEERING AND SCALING DIMENSIONS, IRRELEVANT, MARGINAL, A
Instead of the O(1) NLσM, we consider the interacting field theory
for the real-valued scalar field ϕ defined by the (Euclidean) partition
function
Z
Z = D[ϕ]e−(S0 +S1 ) ,
(3.55a)
with the non-interacting action (Lagrangian)
Z
2
1
S0 = dd x L0 ,
L0 =
∂µ ϕ ,
2
(3.55b)
Rd
and the interacting action (Lagrangian)
Z
S1 = dd x L1 ,
L1 = V (ϕ).
(3.55c)
Rd
As usual units are chosen such that the fundamental constants ~ = c =
1. If so, the action
S = S0 + S1
(3.56)
must be dimensionless, sitting as it is in the argument of an exponential.
Consequently, the Lagrangian density
L = L0 + L1
(3.57)
[L] = (length)−d .
(3.58)
has dimension
By convention (stemming from high-energy physics whereby dimensions are counted in inverse powers of length, i.e., in powers of momentum) the engineering dimension of L is d. The engineering dimension
of the scalar field can be read from the kinetic energy L0 ,
[ϕ] = (length)−(d−2)/2 .
(3.59)
Thus, ϕ has engineering dimension (d − 2)/2. In particular, the engineering dimension of the scalar field is: (i) −1/2 if d = 1, (ii) 0 if d = 2,
(iii) +1/2 if d = 3, and (iv) 1 if d = 4.
When the interaction potential vanishes,
V (ϕ) = 0,
(3.60)
any rescaling of space
e ⇐⇒ a = κ ã,
x = κx
0 < κ < ∞,
(3.61)
(a is the initial microscopic length scale, say the lattice spacing, ã is the
rescaled microscopic length scale) can be compensated by the rescaling
ϕ = κ−(d−2)/2 ϕ
e
(3.62)
88
3. NON-LINEAR-SIGMA MODELS
of the scalar field ϕ so as to insure the invariance of the action S = S0
under this rescaling,
Z
Z
2
1
1 e 2 e
d
e
S= d x
∂µ ϕ = dd x
∂ ϕ
e = S.
(3.63)
2
2 µ
Rd
Rd
Equation (3.63) encodes the property of scale invariance. The action of
the free scalar field theory is scale invariant. The partition function Z ≡
Z0 of the free scalar field theory is not scale invariant since Eq. (3.62)
changes the partition function by an infinite multiplicative factor (the
factor κ−(d−2)/2 for each x). However, this infinite multiplicative factor
drops out of all correlation functions
R
D[ϕ] ϕ(x1 ) · · · ϕ(xm )ϕ(y 1 ) · · · ϕ(y n ) e−S0
hϕ(x1 ) · · · ϕ(xm )ϕ(y 1 ) · · · ϕ(y n )iZ := R
0
D[ϕ]
e−S0
∂ m+n Z0 J
1
= (−)m+n
,
Z0 ∂J(x1 ) · · · ∂J(y n ) J=0
(3.64a)
where


Z
Z0 J :=
Z
D[ϕ] exp −S0 −
dd x Jϕ .
(3.64b)
Rd
Scale invariance of the action S0 fixes the engineering dimension of
(m + n)–point correlation functions of the free scalar field,
h
i
hϕ(x1 ) · · · ϕ(xm )ϕ(y 1 ) · · · ϕ(y n )iZ = (length)−(d−2)(m+n)/2 . (3.65)
0
Equation (3.65) suggests the guess that, up to some dimensionless multiplicative prefactor,
(d−2)/2
1
hϕ(x)ϕ(y)iZ ∝
,
d = 1, 3, 4, · · · .
(3.66)
0
|x − y|2
This guess is confirmed by direct computation of the Fourier transform
of the free scalar field propagator 1/k2 in momentum space,
Z
dd k 1 ik·x
D(x) :=
e
(2π)d k2
(3.67)
(d−2)/2
Γ d−2
1
2
=
,
d = 1, 3, 4, · · · ..
4π d/2
|x|2
The case of two space-time dimension, d = 2, is very special in that
ϕ is itself scale invariant. This is reflected by the singularity of the
3.3. FIXED POINT THEORIES, ENGINEERING AND SCALING DIMENSIONS, IRRELEVANT, MARGINAL, A
gamma function at the origin. 11 We will devote a subsection to this
case below.
Predictions from scale invariance of S0 can be circumvented by other
symmetries of S0 . For example, S0 is also invariant under the discrete
symmetry
ϕ = −ϕ.
e
(3.69)
This symmetry implies that the (m + n)–point correlation function
(3.64a) vanishes whenever m + n is odd.
Scale invariance of S0 has a limited predictive power for the spatial
dependence of 2m–point correlation function, m > 1. One must rely
on a direct calculation to show that the 2m–point correlation function
reduces to a sum over m products of two-point functions. This result is
known as the Wick theorem in an operator representation of quantum
field theory. With path integral techniques this result simply follows
from application of the product rule for differentiation. 12
3.3.1. Fixed-point theories. Consider the family of partition
functions {Z}V labeled by the interaction potential V (ϕ) in Eq. (3.55).
A fixed-point theory Z ? from this family of theories has an action
S ? = S0 + S1? that is scale invariant under simultaneous rescaling of
space-time x and ϕ. We have already encountered one fixed-point theory, the free-field-fixed-point theory when the interaction potential V (ϕ)
vanishes. One could imagine that there are other potentials for which
scale invariance is realized. At a fixed point, scale invariance dictates
that (m + n)–point correlation functions [Eq. (3.64) with S0 → S0 + S1? ]
are algebraic functions in any dimensions other than d = 2. The 2point function can then be used to define the scaling dimension δϕ of
the scalar field at a fixed point,
δϕ
a2
−(d−2)
hϕ(x)ϕ(y)iZ ? ∝ a
,
d = 1, 3, 4, · · · . (3.70)
|x − y|2
11
The gamma function has the integral representation
Z∞
Γ(z) := dt tz−1 e−t ,
z ∈ C.
(3.68)
0
The gamma function is single valued and analytic over the entire complex plane,
save for the points z = 0, −1, −2, −3, · · · where it possesses single poles with
residues (−1)n /n!.
12 Correlation functions in a local field theory defined out of the local field ϕ
are obtained from the partition function Zj in the presence of a source field j that
couples linearly to the local field ϕ. For a free-field theory, the generating function
Zj is proportional to exp(+j G j/2) where G is the free-field Green function, since
the path integral is Gaussian. Repeated differentiation with respect to the source
field of the generating function yields all correlation functions once the limit j → 0
is taken. Wick theorem is just an application of the product rule to the n-th order
differentiation of exp(+j G j/2) with respect to j.
90
3. NON-LINEAR-SIGMA MODELS
The engineering dimension of the correlation function is made explicit
by the introduction of the microscopic length scale a, say the lattice
spacing. The proportionality constant is some dimensionless numerical
factor. The free-field-fixed-point theory is characterized by the fact
that engineering and scaling dimensions coincide. This need not be
true anymore at some putative interacting fixed-point theory where
V ? (ϕ) 6= 0.
The physical significance of a fixed-point theory depends on the
way any perturbation to the fixed-point theory behaves under rescaling.
Consider for example the perturbation
1
(3.71)
Vm (ϕ) :=
λ ϕ2m ,
m = 1, 2, · · · ,
0 < λm ∈ R,
2m m
to the free-field fixed point theory Z ? = Z0 . At the free-field fixed
point, we need not distinguish engineering from scaling dimensions.
The dimension of the coupling constant λm is
[λm ] = (length)−d+(d−2)m
(3.72)
since
Z
S1 =
dd x
1
λ ϕ2m
2m m
(3.73)
Rd
is dimensionless and the scaling dimension of ϕ is fixed by Eq. (3.59).
Thus, under length rescaling (3.61),
λm = κ−d+(d−2)m λf
m.
(3.74)
Choose the rescaling factor 0 < κ < 1. The rescaled coupling constant
d−(d−2)m
λf
λm ,
m = κ
0 < κ < 1,
(3.75)
• is smaller than the original one if
(d − 2)m < d,
(3.76)
(d − 2)m = d,
(3.77)
• is unchanged if
• is larger than the original one if
(d − 2)m > d.
(3.78)
Correspondingly, the interaction Vm is said to be
• UV irrelevant if
(d − 2)m < d,
(3.79)
(d − 2)m = d,
(3.80)
• marginal if
3.3. FIXED POINT THEORIES, ENGINEERING AND SCALING DIMENSIONS, IRRELEVANT, MARGINAL, A
• UV relevant if
(d − 2)m > d.
(3.81)
The terminology of irrelevance and relevance depends on the choice
between 0 < κ < 1 or 1 < κ < ∞. Irrelevant interactions when
0 < κ < 1 become relevant interactions when 1 < κ < ∞ and vice
versa. The choice 0 < κ < 1 consists in zooming into the microscopic
scale in that the new microscopic length scale ã = a/κ by which all
lengths are measured now appears larger than the original one a. The
choice 0 < κ < 1 is made if one is interested in the asymptotic behavior
of correlation functions at short distances or short wavelengths. This
is the ultraviolet (UV) limit of primary interest in high-energy physics.
The choice 1 < κ < ∞ consists in zooming away from the microscopic
scale in that the new microscopic length scale ã = a/κ by which all
lengths are measured now appears smaller than the original one a.
The choice 1 < κ < ∞ is made if one is interested in the asymptotic
behavior of correlation functions at long distances or long wavelengths.
This is the infrared (IR) limit of primary interest in condensed matter
physics. 13 The rescaled coupling constant
d−(d−2)m
λf
λm ,
m = κ
1 < κ < ∞,
(3.85)
• is larger than the original one if
(d − 2)m < d,
(3.86)
(d − 2)m = d,
(3.87)
• is unchanged if
• is smaller than the original one if
(d − 2)m > d.
(3.88)
13
Assume that the short-distance cutoff is a − da to begin with. Imagine that
one integrates over all length scales between a − da and a, say by breaking up
integrals into
Z∞
Za
Z∞
dr · · · =
dr · · · + dr · · · .
(3.82)
a−da
a
a−da
Integration over the interval [a−da, a] can sometimes be absorbed into a redefinition
of the coupling constants of the theory. If so, one is left with
Z∞
Z∞
dr · · · =
dr̃ · · · .
(3.83)
a
a−da
Here, form invariance has been restored on the right-hand side with the help of the
rescaling [compare with Eq. (3.61)]
a
r=
r̃.
(3.84)
a − da
Hence, κ = a/(a − da) is indeed larger than unity.
92
3. NON-LINEAR-SIGMA MODELS
Correspondingly, the interaction Vm is said to be
• IR relevant if
(d − 2)m < d,
(3.89)
(d − 2)m = d,
(3.90)
(d − 2)m > d.
(3.91)
• marginal if
• IR irrelevant if
Observe that a mass term is relevant in any dimensions in the IR limit
[(Eq. (3.89) with m = 1].
At a generic IR fixed-point, it is the scaling dimension δO of a
field O, not the engineering dimension − log[O] in units of length, that
decides of the relevance, marginality, or irrelevance of the “small perturbation” O, whereby it is imagined that
Z
SO = dd x λO O,
0 ≤ ad+log[O] λO 1,
(3.92)
Rd
has been added to the fixed-point action S ? . Under rescaling a = κã,
Z
e
e κd−δO λO O,
SO = dd x
(3.93)
Rd
and the perturbation O is said to be
• IR relevant if
δO < d,
(3.94)
δO = d,
(3.95)
δO > d.
(3.96)
• marginal if
• IR irrelevant if
3.3.2. Two-dimensional O(2) NLσM in the spin-wave approximation. To illustrate the peculiarities of two-dimensional spacetime, consider the O(2) NLσM in d = 2 with the partition function


Z
Z
h
i
1
2
ZβJ,βH := D[n]δ 1 − n2 exp − 2 d2 x ∂µ n − 2a−2 βg 2 H · n  .
2g
R2
(3.97)
3.3. FIXED POINT THEORIES, ENGINEERING AND SCALING DIMENSIONS, IRRELEVANT, MARGINAL, A
The surface of the unit sphere in R2 , the one-sphere, is simply the
unit circle. Planar spins of unit length can be represented as complex
numbers of unit length, i.e., phases,
n1
n=
,
n1 = cos φ = Re ϕ,
n2 = sin φ = Im ϕ,
ϕ = eiφ .
n2
(3.98)
14
Hence,
Z
ZβJ,βH ∝
D[ϕ∗ , ϕ]δ 1 − |ϕ|
2
−
e
1
2g 2
R
R2
h
∗
∗ i
2
d2 x |∂µ ϕ| −2a−2 βg 2 H1 ϕ+ϕ
+H2 ϕ−ϕ
2
2i
(3.100)
Without loss of generality, spontaneous-symmetry breaking into the
ferromagnetic ground state is enforced by taking the external magnetic
field to be H = |H|e1 and letting |H| → 0 at the end of the day.
Hence, the ferromagnetic state is
ϕ(x) = 1,
∀x ∈ R2 .
(3.101)
At non-vanishing temperature, the path-integral representation of
the partition function will be restricted to small deviations ϕ(x) about
the ferromagnetic state (3.101). To be more precise, the spin-wave
approximation by which the angular field φ(x) = arg[ϕ(x)] is rotation
free,
I
∀x ∈ R2 ,
(3.102a)
0 = dxµ µν ∂ν φ,
x
is made. Here,
H
denotes any closed line integral that encloses x and
x
12 = −21 = 1,
11 = 22 = 0,
(3.102b)
i.e., φ is smooth and single valued everywhere. Inclusion in the path integral of multi-valued configurations of φ leads to the so-called KosterlitzThouless transition. However, We will ignore this important aspect of
the problem as our goal is to illustrate how a perturbative RG procedure can be performed on the O(N ) NLσM whereas the physics of the
Kosterlitz-Thouless transition is non-perturbative (with respect to g 2 )
by nature.
14
The proportionality constant results from the change of the normalization
of the measure in the path integral,
dn1 (x) dn2 (x) →
dn1 (x) dn2 (x)
dϕ∗ (x) dϕ(x)
≡
.
π
2πi
(3.99)
In this way a Gaussian integral is normalized to the inverse of the determinant of
the kernel.
.
94
3. NON-LINEAR-SIGMA MODELS
We want to compute the (m + n)-point correlation function
(m,n)
Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y n ) :=
hϕ(x1 ) · · · ϕ(xm )ϕ∗ (y 1 ) · · · ϕ∗ (y n )isw g2 ,H=0
(3.103a)
within the spin-wave approximation of the O(2) NLσM in (d = 2)dimensional space, i.e., the angular brackets denotes averaging with
the partition function (3.100) whereby the periodic nature of argϕ is
neglected or, equivalently, the (m + n)-point correlation function
(m,n)
Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y n ) =
+iφ(x )
1 · · · e+iφ(xm ) e−iφ(y 1 ) · · · e−iφ(y n )
e
sw g 2 ,H=0
(3.103b)
within the spin-wave approximation of the O(2) NLσM in (d = 2)dimensional space, i.e., the angular brackets denotes averaging with
the partition function
i
R 2 h
2
Z
− 12
d x (∂µ φ) −2a−2 βg 2 (H1 cos φ+H2 sin φ)
2g
R2
Zsw βJ,βH ∝ D[φ]e
(3.103c)
whereby the periodic nature of φ is ignored.
Observe that the partition function (3.103c) is unchanged under
φ = φe + const
(3.104)
in the thermodynamic limit and when H vanishes. This immediately
implies that the correlation function (3.103a) vanishes unless
m = n.
With a vanishing external magnetic field, the identity
D R 2
E
R 2 R 2
1
e+i d x J(x) φ(x)
= e− 2 d x d yJ(x) G(x,y) J(y)
sw g 2 ,H=0
(3.105)
(3.106a)
holds for any source J(x). Here, G(x, y) is the Green function defined
by
1 2
(3.106b)
(−) 2 ∂µ G(x, y) = δ(x − y).
g
In other words,
Z
d2 k e+ik·(x−y)
2
G(x, y) = lim
g
M 2 →0
(2π)2 k2 + M 2
1
2
= lim
g − ln (M |x − y|) + const + O(M |x − y|) .
M 2 →0
2π
(3.106c)
Strictly speaking, the Green function is ill-defined because of the logarithmic singularities in the infrared limit M → 0 and in the ultraviolet
3.3. FIXED POINT THEORIES, ENGINEERING AND SCALING DIMENSIONS, IRRELEVANT, MARGINAL, A
limit |x − y| → 0. The ultraviolet singularity can be removed by introducing a high-energy cut-off, say the inverse lattice spacing 1/a. The
infrared cut-off M then drops out of the difference
g 2 |x − y|
ln
+ O(M |x − y|) (3.106d)
2π
a
(b
r is some unit length vector) to leading order in |x − y|/a.
To compute the correlation function (3.103a) it suffices to choose
the source
m
n
X
X
J(x) :=
δ(x − xi ) −
δ(x − y j )
(3.107)
b) = −
G(x, y) − G(x, x + a r
i=1
j=1
in Eq. (3.106a). This gives
(m,n)
Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y n ) =
e
− 21
m
P
i,j=1
(M a)
G(xi −xj )
×e
− 21
1≤k6=l≤n
Y g2
(m+n)
+ 4π
n
P
k,l=1
G(y k −y l )
+2× 12
×e
m P
n
P
i=1 l=1
G(xi −y l )
=
m n
g2 Y Y
+ 4π
g2
(M |xi − y l |)− 2π =
M |xi − xj | (M |y k − y l |)
i=1 l=1
1≤i6=j≤m
Q


g2
g2
2
M + 4π (m−n) a+ 4π (m+n) 


g2
! + 2π
!

1≤i<j≤m
Q
|y k − y l | 


m n

QQ

|xi − y l |
|xi − xj |
1≤k<l≤n
.
i=1 l=1
(3.108)
This expression remains well defined when the infrared cut-off is
removed, M → 0, as long as the short distance cut-off a is kept nonvanishing, in which case
(m,n)
Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y n ) =

0,
if m 6= n,



g2


+ 2π

1≤k<l≤m
Q
(|xi −xj |)(|yk −yl |) 
g2
+ 2π m  1≤i<j≤m

!
a
, if m = n.



m

Q

|x −y |

i,j=1
i
j
(3.109)
The case m = n = 1 gives the two-point function
g2
+ 2π
a
(1,1)
Gsw g2 ,H=0 (x, y) =
.
|x − y|
(3.110)
All correlation functions of the form (3.103a) are thus algebraic functions for any given non-vanishing value of g 2 . At zero temperature, i.e.,
when g 2 = 0, all correlation functions are constant as it should be if the
96
3. NON-LINEAR-SIGMA MODELS
ground state supports ferromagnetic long-range order (LRO). Within
the spin-wave approximation, LRO at zero temperature (g 2 = 0) is
downgraded to algebraic order or quasi-long-range order (QLRO) at
any non-vanishing temperature. Equation (3.108) defines a critical
phase of matter for any given g 2 . At criticality scale invariance manifests itself by algebraic decaying correlation functions. Here, the critical
phase of matter is called the spin-wave phase. Direct inspection of the
two-point function (3.110) allows us to infer that the scaling dimension
δϕ of the field ϕ is given by
δϕ =
g2
.
4π
(3.111)
This scaling dimension is a smooth function of g 2 and is different from
the engineering dimension of ϕ which is zero.
Correlation functions in the spin-wave phase are ambiguous in the
limit a → 0 in which the ultraviolet cut-off is removed. This ambiguity can be interpreted as follows. The accuracy of the spin-wave
approximation improves at low energies and long distances, i.e., scaling
exponents controlling the algebraic decay of correlation functions can
be thought of as being exact or, more precisely, universal in that they
do not depend on the prescription used to regularize the theory at short
distances. Short-distance regularizations in condensed matter physics
are much more than a mathematical artifact as they refer to a specific lattice or microscopic model. The mathematical ambiguity in the
choice of an ultraviolet cut-off reflects the property that lattice models
that differ on the microscopic scale might nevertheless share the same
properties at low energies and long distances. From the point of view
of physics this is a very important property called universality without
which the task of classifying and predicting phases of condensed matter
would otherwise be hopeless.
The mathematical ambiguity in the choice of an ultraviolet cut-off
can be encoded in a differential equation obeyed by correlation functions. This differential equation is called the Callan-Symanzik equation. The construction of the Callan-Symanzik equation in the spinwave phase proceeds as follows. The ambiguity in the choice of the
3.3. FIXED POINT THEORIES, ENGINEERING AND SCALING DIMENSIONS, IRRELEVANT, MARGINAL, A
ultraviolet cut-off 1/a can be quantified by introducing a renormalization point or renormalization mass µ through
g2
+ 2π
 1≤k<l≤m
Q
|xi − xj ||y k − y l | 

g2

+ 2π
m  1≤i<j≤m
a
=


m
Q


|xi − y j |
i,j=1
g2
 1≤k<l≤m
+ 2π
Q
µ|xi − xj | (µ|y k − y l |) 

g2

m  1≤i<j≤m
+ 2π
(aµ)


m
Q


µ|xi − y l |
i,j=1
(3.112)
for the (2m)-point function (3.109) and
a
|x − y|
g2
+ 2π
g2
+ 2π
= (aµ)
1
µ|x − y|
g2
+ 2π
(3.113)
for the 2-point function (3.110). Define the renormalized field
1
ϕ(R) := √ ϕ,
Z
whereby
√
g2
(3.114)
g2
Z := (aµ)+ 4π = e+ 4π ln(aµ) .
(3.115)
The original field ϕ is called the bare or unrenormalized field. The
dimensionless number Z is called the wave-function renormalization
factor. Correlation functions (3.108) and (3.110) can be expressed in
terms of the renormalized fields as
(m,m)(R)
Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y m ) :=
(R)
ϕ (x1 ) · · · ϕ(R) (xm )ϕ(R)∗ (y 1 ) · · · ϕ(R)∗ (y m ) sw g2 ,H=0 =
Q





g2
! + 2π
!

Q
µ|xi − xj |
1≤i<j≤m
µ|y k − y l | 


!

m
Q

µ|xi − y j |
1≤k<l≤m
i,j=1
(3.116)
and
(R)
ϕ (x)ϕ(R)∗ (y) sw g2 ,H=0 =
1
µ|x − y|
g2
+ 2π
,
(3.117)
98
3. NON-LINEAR-SIGMA MODELS
respectively. We have thus traded the ultraviolet cut-off 1/a for µ. The
Callan-Symanzik equation obeyed by the correlation function (3.109)
follows from the observation that Eq. (3.109) does not depend on µ,
d (m,m)
(x , . . . , xm , y 1 , . . . , y m )
G 2
dµ sw g ,H=0 1
i
d h m (m,m)(R)
=µ
Z Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y m ) by Eqs. (3.116) and (3.114)
dµ

√ 
∂
∂ ln Z  (m,m)(R)
Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y m )
= Z m µ
+ 2m µ
∂µ
∂µ
| {z }
∂
g2
(m,m)(R)
m
Eq. (3.115) = Z
µ
+ 2m
Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y m )
∂µ
4π
∂
(m,m)(R)
2
m
Gsw g2 ,H=0 (x1 , . . . , xm , y 1 , . . . , y m ).
+ 2m γ(g )
≡Z µ
∂µ
(3.118)
0 =µ
The anomalous scaling dimension
√
∂ ln Z
γ(g ) := µ
∂µ
2
g
=
4π
2
(3.119)
has been introduced. It is the difference between the scaling and the
engineering dimension of the field ϕ.
The lessons learned from the example of the (d = 2)-dimensional
O(2) NLσM are:
(1) The vanishing of the m 6= n correlator (3.103a) as the infrared
cut-off M → 0 guarantees that the O(2) symmetry is not spontaneously broken at any non-vanishing temperature. This is
an example of the Hohenberg-Mermin-Wagner-Coleman theorem that asserts that no continuous global symmetry can be
spontaneously broken in d ≤ 2.
(2) Correlator (3.103a) depends on an ultraviolet cut-off. This dependence can be quantified by the Callan-Symanzik equation
obeyed by renormalized fields.
(3) Anomalous scaling dimensions that appear in the Callan-Symanzik
equation are universal in that they are independent of the
choice of the ultraviolet cut-off.
(4) The spin-wave phase is a critical or QLRO phase in which
correlator (3.103a) is an algebraic decaying function at any
non-vanishing temperature. Anomalous scaling dimensions
are continuous functions of the temperature.
3.4. GENERAL METHOD OF RENORMALIZATION
99
3.4. General method of renormalization
In this section, we are going to set up the Callan-Symanzik equation
obeyed by correlation functions in all generality. Consider some bare
correlation function
(m,n)
GB
(z; gB , Λ)
(3.120a)
between (m + n) local fields. Here, z denotes collectively the (m + n)
space arguments of the local fields,
z = {x1 , . . . , xm , y 1 , . . . , y n } ,
(3.120b)
and gB denotes collectively all (IR) relevant coupling constants at the
free-field fixed point,
n
o
(1) (2)
gB = gB , gB , . . . .
(3.120c)
It is commonly assumed that the number of relevant coupling constants is finite but this need not be so, for example when dealing with
disordered systems. The inverse of the lattice spacing
Λ=
1
a
(3.121)
is taken as the UV cut-off. We now assume that it is possible to express
the correlator (3.120a) in terms of a wave-function renormalization factor Z, renormalized coupling constants gR , and a new renormalization
point µ according to
(m,n)
GB
(z; gB , Λ) = [Z(gR , µ/Λ)]+
m+n
2
(m,n)
× GR
(z; gR , µ).
(3.122)
Equation (3.122) is certainly not correct when a pair of spatial arguments of the correlator is within a distance of the order of the lattice
spacing a as the renormalized correlator must then also depend on Λ.
However, Eq. (3.122) becomes plausible when all spatial arguments are
separated pairwise by an amount much larger than the lattice spacing
a. In any case Eq. (3.122) is to be verified by explicit computation as
we did for the spin-wave phase of the (d = 2)-dimensional O(2) NLσM.
Assumption (3.122) implies the Callan-Symanzik equation
h
i
(m,n)
0 = µ∂µ + β(gR )∂gR + (m + n)γ(gR ) GR (z; gR , µ),
β(gR ) := µ∂µ gR
√
γ(gR ) := µ∂µ ln Z
(3.123)
at fixed gB and Λ,
at fixed gB and Λ.
Observe that we could have equally well written
(m,n)
GR
(z; gR , µ) = Z −
m+n
2
(m,n)
(gB , µ/Λ) × GB
(z; gB , Λ)
(3.124)
100
3. NON-LINEAR-SIGMA MODELS
instead of Eq. (3.122) to derive the Callan-Symanzik equation
h
i
(m,n)
e )∂ − (m + n)e
0 = Λ∂Λ + β(g
γ
(g
)
GB (z; gB , Λ),
B gB
B
e ) := Λ∂ g
β(g
B
Λ B
√
γ
e(gB ) := Λ∂Λ ln Z
(3.125)
at fixed gR and µ,
at fixed gR and µ.
e )] quantifies the rate of change of the renorThe function β(gR ) [β(g
B
malized (bare) coupling constants as the renormalization point (lattice
spacing) is varied. The flow of the coupling constants under an infinitesimal change in the renormalization point (lattice spacing) is thus
controlled by the so-called beta function. For the (d = 2)-dimensional
O(2) NLσM in the spin-wave approximation there is only one coupling
constant g 2 that does not flow, i.e., the beta function of g 2 vanishes
identically as it should be at a critical point.
3.5. Perturbative expansion of the two-point correlation
function up to one loop for the two-dimensional O(N )
NLσM
We are after the expansion of
(1,1)
Gsw g2 ,H=0 (x, y; a) := hn(x) · n(y)isw;a
= g 2 hm(x) · m(y)isw;a
r
r
1
1
2
− π 2 (x)
− π 2 (y)
+ g 2 hπ(x) · π(y)isw;a
=g
2
2
g
g
sw;a
(3.126)
up to order g 4 , where the expectation value h(· · · )isw;a is defined by
R
h(· · · )isw;a :=
R
d[π]Θ(1 − g 2 π 2 )e
d[π]Θ(1 − g 2 π 2 )e
− 21
− 21
R
R2
R
R2
d2 x
a2
ln(1−g 2 π 2 )− 12
d2 x
a2
ln(1−g 2 π 2 )− 12
R
d2 x(∂µ πi )
R2
R
R2
d2 x(∂µ πi )
g 2 πi πj
1−g 2 π 2
+δij (∂µ πj )
g2 π π
i j
1−g 2 π 2
+δij (∂µ πj )
(3.127)
(· · · )
.
3.5. PERTURBATIVE EXPANSION OF THE TWO-POINT CORRELATION FUNCTION UP TO ONE LOOP F
Observe that the Jacobian
q
Y
Y
p
det gij (x) : =
x∈R2
x∈R2
1
1 − g2π2
!
1 X
= exp −
ln 1 − g 2 π 2
2
x∈R2


Z 2
1
dx
= exp −
ln 1 − g 2 π 2 
2
a2
R2


Z
1
2 
= exp −δ(r = 0)
d2 x ln 1 − g 2 π(3.128a)
2
R2
depends explicitly on the short distance cut-off a that regularizes the
delta function in position space,
0, if r 6= 0.
δ(r) =
(3.128b)
1
, if r = 0.
a2
In the sequel, we can forget the Heaviside step function in the measure
for the spin waves as it plays no role in perturbation theory in powers
of g 2 .
To organize the perturbative expansion, note that we need to expand
• the argument of the expectation value in powers of g 2 , i.e., we
need
√
1
11 2
1−x=1− x−
x + ··· .
(3.129)
2
24
• the action in powers of g 2 , i.e., we need
1
ln(1 − x) = −x − x2 + · · · .
2
(3.130)
• the Boltzmann weight in powers of g 2 , i.e., we need
1
e−x = 1 − x + x2 + · · · .
2
(3.131)
• the inverse of the partition function in powers of g 2 , i.e., we
need
1
= 1 + x + x2 + · · · .
(3.132)
1−x
102
3. NON-LINEAR-SIGMA MODELS
Expansion in powers of g 2 of the argument in the expectation value
(3.126) gives
1
1
(1,1)
2
Gsw g2 ,H=0 (x, y; a) = 1 + g − π(x) · π(x) − π(y) · π(y) + π(x) · π(y)
2
2
sw;a
1
1
1
+ g 4 + π 2 (x)π 2 (y) − π 2 (x)π 2 (x) − π 2 (y)π 2 (y)
4
8
8
sw;a
+ O(g 6 ).
(3.133)
Before proceeding with the expansion, we introduce the notation
2
1
g πi πj
1
2 2
Lsw;a := δ(r = 0) ln 1 − g π + (∂µ πi )
+ δij (∂µ πj )
| {z } 2
2
1 − g2π2
=1/a2
≡ L0 + g 2 L1,1 + g 2 L1,2;a + O(g 4 ),
(3.134a)
where
1
L0 := (∂µ π) · (∂µ π),
2
1
L1,1 :=
π · ∂µ π π · ∂µ π ,
2
1
L1,2;a := − δ(r = 0) π 2 .
| {z } 2
(3.134b)
=1/a2
The four actions obtained from the four Lagrangians Lsw;a , L0 , L1,1 ,
and L1,2;a , are denoted Ssw;a , S0 , S1,1 , and S1,2;a , respectively. We will
also need the expansion
R
−S0 −g 2 S1
AB
2
2 R d[θ] e
g hABi := g
2S
−S
−g
1
d[θ] e 0
R
d[θ] e−S0 AB (1 − g 2 S1 + · · · )
= g2 R
d[θ] e−S0
(1 − g 2 S1 + · · · )
(3.135a)
= g 2 hABi0
+ g 4 [− hABS1 i0 + hABi0 hS1 i0 ]
+ O(g 6 ),
where S1 = S1,1 + S1,2;a and
R
d[θ] e−S0 (· · · )
h(· · · )i0 := R
.
d[θ] e−S0
(3.135b)
3.5. PERTURBATIVE EXPANSION OF THE TWO-POINT CORRELATION FUNCTION UP TO ONE LOOP F
Altogether, the final expansion for the spin-spin correlator reads
(1,1)
Gsw g2 ,H=0 (x, y; a)
1
1
= 1 + g − π(x) · π(x) − π(y) · π(y) + π(x) · π(y)
2
2
S0
1 2
1 2
1 2
4
2
2
2
+ g + π (x)π (y) − π (x)π (x) − π (y)π (y)
4
8
8
S0
1
1
+ g 4 + π 2 (x) + π 2 (y) − π(x) · π(y) S1,1 + S1,2;a
2
2
S0
1
1
S1,1 + S1,2;a S
+ g 4 − π 2 (x) − π 2 (y) + π(x) · π(y)
0
2
2
S
2
0
+ O(g 6 ).
(3.136)
This is the expansion of the two-point function in the O(N ) NLσM up
to order g 4 in the coupling constant.
If we recall that the limit g 2 → 0 corresponds to zero temperature,
we infer that the two-point function is constant to zero-th order in
g 2 . This is the signature of spontaneous-symmetry breaking through
ferromagnetic LRO.
Spin waves disturb the ferromagnetic LRO at any non-vanishing
temperature. Noting that
Z
d2 k eik·(x−y)
πi (x)πj (y) S → δij
0
(2π)2 k2 + M 2
(3.137)
1
ln (M |x − y|) + · · ·
= − δij
2π
≡ δij G(x, y) + · · · ,
i, j = 1, · · · , N − 1,
we see that the deviations from ferromagnetic LRO induced by spin
waves are logarithmically large to order g 2 ,
1
1
(1,1)
Gsw g2 ,H=0 (x, y; a) = 1 + g 2 − π(x) · π(x) − π(y) · π(y) + π(x) · π(y)
+ O(g 4 )
2
2
S0
1
1
= 1 + g 2 − (N − 1)G(x, x) − (N − 1)G(y, y) + (N − 1)G(x, y) + O(g 4 )
2
2
= 1 + g 2 (N − 1) [G(x, y) − G(0, 0)] + O(g 4 ),
(3.138)
where it is understood that the IR cut-off M drops out from
1
G(x, y) − G(0, 0) = −
[ln(M |x − y|) − ln(M a)] + · · ·
2π
(3.139)
1
|x − y|
=−
ln
+ ··· .
2π
a
Perturbation theory in powers of g 2 thus appear to be hopeless except
for the possibility that the contribution of order g 4 to the expansion be
proportional to
[G(x, y) − G(0, 0)]2 .
(3.140)
104
3. NON-LINEAR-SIGMA MODELS
Indeed, this possibility could signal that the inclusion of spin waves
renders the anomalous scaling dimensions of π non-vanishing, as was
the case for the O(2) NLσM, 15 and that an expansion in powers of g 2
could be reinterpreted in a sensible way through a RG analysis based
on a Callan-Symanzik equation.
There are several contributions to account for to order g 4 . The
second line of Eq. (3.136) gives, with the application
hABCDi0 = hABi0 hCDi0 + hACi0 hBDi0 + hADi0 hBCi0 (3.141)
of Wick’s theorem and with the help of translation invariance,
+g
4
1 2
1 2
1 2
2
2
2
+ π (x)π (y) − π (x)π (x) − π (y)π (y)
=
4
8
8
S
0
1
1
4
2
2
+ g + π (x) S π (y) S +
π (x)πj (y) S πi (x)πj (y) S
0
0
0
0
4
2 i
1
1
2 2 4
π (x)πj (x) S πi (x)πj (x) S
+ g − π (x) S π (x) S −
0
0
0
0
8
4 i
1
1
2 2 4
+ g − π (y) S π (y) S −
π (y)πj (y) S πi (y)πj (y) S =
0
0
0
0
8
4 i
1
1
4
π (0)πj (0) S πi (0)πj (0) S =
+ g + πi (x)πj (y) S πi (x)πj (y) S −
0
0
0
0
2
2 i
1
+ g 4 (N − 1) G2 (x, y) − G2 (0, 0) .
2
(3.142)
15
2
of g .
This can be seen by expanding the right-hand side of Eq. (3.110) in powers
3.5. PERTURBATIVE EXPANSION OF THE TWO-POINT CORRELATION FUNCTION UP TO ONE LOOP F
The third line of Eq. (3.136) demands the evaluation of
g4
+
4
Z
d2 r πi (x)πi (x)[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
g4
+
4
Z
d2 r πi (y)πi (y)[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
g4
−
2
Z
d2 r πi (x)πi (y)[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
(3.143)
g4
+
4
Z
1
− 2
d2 r πi (x)πi (x)πj (r)πj (r) S
0
a
R2
Z
1
g4
d2 r πi (y)πi (y)πj (r)πj (r) S
− 2
+
0
4
a
R2
Z
g4
1
−
− 2
d2 r πi (x)πi (y)πj (r)πj (r) S .
0
2
a
R2
The fourth line of Eq. (3.136) subtracts
g4
+
4
Z
d2 r hπi (x)πi (x)iS
d2 r hπi (y)πi (y)iS
d2 r hπi (x)πi (y)iS
0
[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
g4
+
4
Z
0
[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
g4
−
2
Z
0
[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
R2
g4
+
4
Z
1
− 2
d2 r hπi (x)πi (x)iS πj (r)πj (r) S
0
0
a
R2
Z
g4
1
+
− 2
d2 r hπi (y)πi (y)iS πj (r)πj (r) S
0
0
4
a
R2
Z
g4
1
−
− 2
d2 r hπi (x)πi (y)iS πj (r)πj (r) S
0
0
2
a
0
(3.144)
R2
from the third line of Eq. (3.136), i.e., it is sufficient to evaluate Eq.
(3.143) with the help of Wick’s theorem with the additional rule that
106
3. NON-LINEAR-SIGMA MODELS
no Wick contraction between the two points x and x or y and y or x
and y can occur. 16
Because of translation invariance, Eqs. (3.143) and (3.144) simplify
to
g4
+
2
Z
d2 r πi (0)πi (0)[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
g4
−
2
Z
d2 r πi (x)πi (y)[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
g4
+
2
(3.147)
Z
1
− 2
d2 r πi (0)πi (0)πj (r)πj (r) S
0
a
R2
Z
1
g4
− 2
d2 r πi (x)πi (y)πj (r)πj (r) S
−
0
2
a
R2
and
g4
+
2
Z
d2 r hπi (0)πi (0)iS
d2 r hπi (x)πi (y)iS
[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
g4
−
2
Z
0
0
R2
g4
+
2
(3.148)
Z
1
− 2
d2 r hπi (0)πi (0)iS πj (r)πj (r) S
0
0
a
R2
Z
g4
1
− 2
d2 r hπi (x)πi (y)iS πj (r)πj (r) S ,
−
0
0
2
a
R2
16 Wick’s theorem reduces a Gaussian expectation value h(· · · )i
0 of 2m variables
to the sum over all possible products of two-point functions, say for m = 3,
hABCDEF i0 = hABi0 hCDEF i0 + hACi0 hBDEF i0 + hADi0 hBCEF i0 + hAEi0 hBCDF i0
+ hAF i0 hBCDEi0
(3.145)
where
hABCDi0 = hABi0 hCDi0 + hACi0 hBDi0 + hADi0 hBCi0 .
There are thus 5 × 3 = 15 contributions when m = 3.
hABCDEF i0 of hABi0 hCDEF i0 gives 12 contributions.
(3.146)
Subtraction from
3.5. PERTURBATIVE EXPANSION OF THE TWO-POINT CORRELATION FUNCTION UP TO ONE LOOP F
respectively. It is then sufficient to evaluate
Z
h
g4
2
−
d r πi (x)πi (y)[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
2
R2
− hπi (x)πi (y)iS
0
i
[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
(3.149)
and
g4
−
2
Z
h
i
1
− 2
d2 r πi (x)πi (y)πj (r)πj (r) S − hπi (x)πi (y)iS πj (r)πj (r) S
0
0
0
a
R2
(3.150)
since one can always choose x = y. Remarkably, contribution (3.150)
is contained in contribution (3.149) but with the opposite sign and
thus cancels out of the spin-spin correlator. To see this, make use of
translation invariance and of Eqs. (3.145) and (3.146) to write the Wick
decomposition
Z
h
g4
2
d r πi (x)πi (y)[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
−
0
2
R2
i
− hπi (x)πi (y)iS [πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S =
0
0
4 Z
g
−1×2×
d2 r πi (x)πj (r) S [πk (r)∂µ πk (r)] S [∂µ πj (r)]πi (y) S
0
0
0
2
R2
Z
g4
−1×2×
d2 r πi (x)πj (r) S [πk (r)∂µ πj (r)] S [∂µ πk (r)]πi (y) S
0
0
0
2
R2
g4
−1×2×
2
Z
d2 r πi (y)πj (r) S [πk (r)∂µ πk (r)] S [∂µ πj (r)]πi (x) S
0
0
0
R2
g4
−1×2×
2
Z
d2 r πi (y)πj (r) S [πk (r)∂µ πj (r)] S [∂µ πk (r)]πi (x) S
0
0
0
R2
g4
−1×2×
2
Z
d2 r πi (x)[∂µ πj (r)] S πj (r)πk (r) S [∂µ πk (r)]πi (y) S
0
0
0
R2
g4
−1×2×
2
Z
d2 r πi (x)πj (r) S [∂µ πj (r)][∂µ πk (r)] S hπk (r)πi (y)iS .
0
0
0
R2
(3.151)
108
3. NON-LINEAR-SIGMA MODELS
Insertion of the unperturbed Green function (3.137) turns Eq. (3.151)
into
g4
−
2
Z
d2 r
h
πi (x)πi (y)[πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S
0
R2
−1×2×
−1×2×
−1×2×
−1×2×
−1×2×
−1×2×
i
− hπi (x)πi (y)iS [πj (r)∂µ πj (r)][πk (r)∂µ πk (r)] S =
0
0
4 Z
g
e)δji ∂rµ G (r, y)
d2 rδij G(x, r)δkk lim ∂reµ G (r, r
e→r
r
2
R2
4 Z
g
e)δki ∂rµ G (r, y)
d2 rδij G(x, r)δkj lim ∂reµ G (r, r
e→r
r
2
R2
Z
g4
e)δji ∂rµ G (r, x)
d2 rδij G(y, r)δkk lim ∂reµ G (r, r
e→r
r
2
R2
Z
g4
e)δki ∂rµ G (r, x)
d2 rδij G(y, r)δkj lim ∂reµ G (r, r
e→r
r
2
R2
Z
g4
2
d rδij ∂rµ G (x, r)δjk G(0, 0)δki ∂rµ G (r, y)
2
R2
4 Z
g
e)δki G(r, y).
d2 rδij G(x, r)δjk lim ∂rµ ∂reµ G (r, r
e→r
r
2
R2
(3.152)
The first four lines on the right hand side of Eq. (3.152) vanish since
Z
e) ∼
lim ∂reµ G (r, r
e→r
r
R2
= 0.
d2 q qµ
(2π)2 q 2
(3.153)
3.5. PERTURBATIVE EXPANSION OF THE TWO-POINT CORRELATION FUNCTION UP TO ONE LOOP F
The fifth line gives (with the help of −∆G(r, y) = δ(r − y) ⇔ q 2 Gq =
1)
g4
−1×2×
2
Z
2
d rδij
∂rµ G (x, r)δjk G(0, 0)δki ∂rµ G (r, y) =
R2
4
g
− 1 × 2 × (N − 1)G(0, 0)
2
Z
d2 r ∂rµ G (x, r) ∂rµ G (r, y) =
R2
4
g
− 1 × 2 × (N − 1)G(0, 0)
2
Z
h
i
d2 r G(x, r) (−)∂rµ ∂rµ G (r, y) =
R2
g4
− 1 × 2 × (N − 1)G(0, 0)
2
Z
d2 r G(x, r)δ(r − y) =
R2
4
− g (N − 1)G(0, 0)G(x, y).
(3.154)
The last line gives
g4
−1×2×
2
Z
e)δki G(r, y) =
d2 rδij G(x, r)δjk lim ∂rµ ∂reµ G (r, r
e→r
r
R2
4
Z
− g (N − 1)
e)G(r, y) =
d2 r G(x, r) lim δ(r − r
e→r
r
R2
Z
1
d2 r G(x, r)G(r, y).
+ g (N − 1) − 2
a
4
R2
(3.155)
This is nothing but the same as contribution (3.150) up to an overall
sign. As promised contribution (3.150) cancels out. Adding up all nonvanishing contributions of order g 4 to the spin-spin correlator gives
1
1
+g 4 (N − 1) [G(x, y) − G(0, 0)]2 = + g 4 (N − 1) G2 (x, y) − G2 (0, 0)
2
2
4
− g (N − 1)G(0, 0)G(x, y)
+ g 4 (N − 1)G(0, 0)G(0, 0).
(3.156)
110
3. NON-LINEAR-SIGMA MODELS
In summary, the expansion of the spin-spin correlator up to order
g is
4
(1,1)
Gsw g2 ,H=0 (x, y; a) = 1
+ g 2 (N − 1) [G(x, y) − G(0, 0)]
1
+ g 4 (N − 1) [G(x, y) − G(0, 0)]2
2
+ O(g 6 ).
(3.157)
As a check, we recognize the first two terms in the expansion in powers
of g 2 of
g2
+ 2π
a
= exp +g 2 [G(x, y) − G(0, 0)]
(3.158)
|x − y|
if we set N = 2.
The origin of the divergent logarithms occurring in the expansion in
powers of g 2 is the existence in two dimensions of very strong fluctuations. Spin waves destroy ferromagnetic LRO. In mathematical terms,
the engineering dimension of the spin degrees of freedom differs from
the scaling dimension. Correspondingly, LRO is downgraded to QLRO.
The factor N − 1 counts all the “Goldstone modes”, i.e., those
independent degrees of freedom that parametrize small fluctuations orthogonal to the ferromagnetic magnetization axis of the ferromagnetic
ground state. The lattice spacing a (1/a) plays the role of a short
distance (ultraviolet) cutoff. Perturbative expansion (3.157) suggests
that the expansion parameter is not simply g 2 but g 2 ln |x − y|/a. This
hypothesis is verified when N = 2. Correspondingly, expansion (3.157)
of the spin-spin correlator is not uniformly convergent as a function
of |x − y|/a. The most likely interpretation of this mathematical difficulty is that spin-wave fluctuations destroy the ferromagnetic longrange order (LRO) of the ground state at any finite temperature as it
does when N = 2. However, the destruction of ferromagnetic LRO by
spin waves when N > 2 is qualitatively different from the N = 2 case
as we shall argue that ferromagnetic LRO at zero temperature (g 2 = 0)
is replaced by paramagnetism at any finite temperature (g 2 > 0) using
renormalization-group (RG) methods when N > 2. More precisely,
we will derive the RG flow obeyed by g 2 and show that the coupling
constant g 2 is infrared (IR) relevant relative to the ferromagnetic fixed
point g 2 = 0 up to order g 4 when N > 2. Assuming that this relevance
holds for all g 2 , this implies that the infinite temperature fixed point
g 2 = ∞ is stable. But the attractive fixed point g 2 = ∞ is, on physical grounds, nothing but the paramagnetic phase with a correlation
length of the order of the lattice spacing. To put it in more quantitative
terms, there must exist a correlation length ξ which is a function of g 2
that diverges as g 2 → 0 and is of the order of the lattice spacing a as
3.6. CALLAN-SYMANZIK EQUATION OBEYED BY THE SPIN-SPIN CORRELATOR IN THE d = 2-DIMENSI
g 2 → ∞ such that
hn(x) · n(y)isw g2 ,H=0 = c e−
|x−y|
ξ
(3.159)
when N > 2. The proportionality constant c is a dimensionless number
that depends on g 2 among others. We shall show how one can compute
the correlation length ξ as an expansion in powers of g 2 up to order
g 4 , thereby obtaining a non-perturbative result with respect to the
expansion (3.157). The RG approach that we will follow is based on the
observation that the short-distance cutoff a in the spin-spin correlator
(3.157) can be arbitrarily chosen for very large separations |x − y|/a 1.
3.6. Callan-Symanzik equation obeyed by the spin-spin
correlator in the d = 2-dimensional O(N > 2) NLσM
We shall derive the Callan-Symanzik equation obeyed by the spinspin correlator in the (d = 2)-dimensional O(N ) NLσM. We will follow
the conventions used in high-energy physics, i.e., we will work with the
UV momentum cutoff
1
Λ≡
(3.160a)
a
and the bare spin-spin correlator
GB (x) := hn(x) · n(0)isw g2 ,H=0;Λ
= 1 − (N − 1)
gB2
1
g4
ln(Λ|x|) + (N − 1) B 2 ln2 (Λ|x|) + O(gB6 ).
2π
2
(2π)
(3.160b)
As a warm up, we multiply and divide the UV cutoff Λ by the new UV
cutoff µ so as to trade the bare spin-spin correlator GB (x) that depends
on the original UV regulator Λ for a renormalized spin-spin correlator
GR (x) that depends on the new UV regulator µ. This is done up to
order gB2 for which
GB (x) = hn(x) · n(0)isw g2 ,H=0;Λ
gB2
ln(Λ|x|) + O(gB4 )
2π gB2
Λ
gB2
= 1 − (N − 1) ln
− (N − 1) ln(µ|x|) + O(gB4 )
2π
µ
2π
2
gR
Λ
gR2
4
4
= 1 − (N − 1) ln
+ O(gB ) 1 − (N − 1) ln(µ|x|) + O(gB )
2π
µ
2π
2
≡ Z(gR ) GR (x),
(3.161a)
= 1 − (N − 1)
112
3. NON-LINEAR-SIGMA MODELS
where
gR2 := gB2 + O(gB4 ),
gR2
Λ
:= 1 − (N − 1) ln
+ O(gB4 ),
2π
µ
2
g
GR (x) := 1 − (N − 1) R ln(µ|x|) + O(gB4 ).
2π
Z(gR2 )
(3.161b)
Observe that
• The wave-function renormalization is given by
Z(gR2 ) = lim GB (x)
(3.162)
|x|→1/µ
up to order gR2 . Equivalently,
lim GR (x) = 1
(3.163)
|x|→1/µ
up to order gR2 .
• The renormalized coupling constant is given by
∂GR (x)
g2
= −(N − 1) R
|x|→1/µ ∂ ln |x|
2π
lim
(3.164)
up to order gR2 .
We shall first extend the expansion of the renormalized coupling
constant gR2 , the wave-function renormalization Z(gR2 ), and the renormalized spin-spin correlator GR (x) up to order gR4 . We shall then
compute the Callan-Symanzik equation obeyed by the spin-spin correlator. However, we need to define the renormalized coupling constant
gR2 and the wave-function renormalization Z(gR2 ) non-perturbatively to
begin with.
3.6.1. Non-perturbative definitions of the renormalized coupling constant and the wave-function renormalization. The wavefunction renormalization Z(gB2 ) is defined, to all orders in gB2 , by demanding that
Z(gB2 ) := GB (x) when |x| = µ1 .
(3.165)
This definition is equivalent to demanding that
GR (x) = 1 when |x| =
1
µ
(3.166)
since
1
G (x).
(3.167)
Z B
Observe that this definition is consistent with Eq. (3.161). For given
µ, the condition
|x| = 1/µ
(3.168)
GR (x) =
3.6. CALLAN-SYMANZIK EQUATION OBEYED BY THE SPIN-SPIN CORRELATOR IN THE d = 2-DIMENSI
defines the renormalization point. The definition of the renormalized
coupling gR2 is also motivated by Eq. (3.161) as it is given by the condition
∂GR (x)
g2
lim
= −(N − 1) R ,
(3.169)
|x|→1/µ ∂ ln |x|
2π
which must now hold non-perturbatively in powers of gB2 .
3.6.2. Expansion of the renormalized coupling constant,
the wave-function renormalization, and the renormalized spinspin correlator up to order gB4 . Inputs are
gB2
1
g4
ln(Λ|x|) + (N − 1) B 2 ln2 (Λ|x|) + O(gB6 ),
2π
2
(2π)
(3.170a)
∂GR 2π
gR2 (gB2 ) := −
N − 1 ∂ ln |x| |x|=1/µ
2π ∂ (Z −1 GB ) =−
N − 1 ∂ ln |x| |x|=1/µ
2π
1
gB4
Λ
gB2
6
=−
+ (N − 1)
ln
+ O(gB ) ,
−(N − 1)
N − 1 Z(gB2 )
2π
(2π)2
µ
(3.170b)
GB (x) = 1 − (N − 1)
Z(gB2 ) := GB (x)||x|=1/µ
g2
= 1 − (N − 1) B ln
2π
Λ
gB4
1
Λ
2
ln
+ (N − 1)
+ O(gB6 ),
2
µ
2
(2π)
µ
(3.170c)
and
gB2
1
=
1+(N
−1)
ln
Z(gB2 )
2π
Λ
Λ
gB4
2
ln
+(N −1)(N −3/2)
+O(gB6 ),
µ
(2π)2
µ
(3.170d)
from which follows that
2
4
gB
gB
Λ
−(N
−
1)
+
(N
−
1)
ln
+ O(gB6 )
2π
(2π)2
µ
2π
2
2
gR (gB ) = −
2
gB
N −1
1 − (N − 1) 2π ln Λµ + O(gB4 )
g4
+gB2 − 2πB ln Λµ + O(gB6 )
=
g2
1 − (N − 1) 2πB ln Λµ + O(gB4 )
gB4
Λ
2
= gB + (N − 2) ln
+ O(gB6 ),
2π
µ
(3.171)
114
3. NON-LINEAR-SIGMA MODELS
on the one hand, and
GR (x) := Z −1 (gB2 ) GB (x)
=1
gB2
gB2
Λ
− (N − 1) ln(Λ|x|) + (N − 1) ln
2π
2π
µ
1
gB4
gB4
Λ
2
2
+ (N − 1)
ln (Λ|x|) + (N − 1)(N − 3/2)
ln
2
2
2
(2π)
(2π)
µ
4
g
Λ
+ O(gB6 ),
− (N − 1)2 B 2 ln(Λ|x|) ln
(2π)
µ
(3.172)
on the other hand.
3.6.3. Expansion of the bare coupling constant, the wavefunction renormalization, and the renormalized spin-spin correlator up to order gR4 . Inverting
gB4
Λ
2
2
2
gR (gB ) = gB + (N − 2) ln
+ O(gB6 ),
(3.173)
2π
µ
gives
gB2 (gR2 )
=
gR2
g4
− (N − 2) R ln
2π
Λ
+ O(gR6 ).
µ
(3.174)
Insertion of Eq. (3.174) into the right-hand sides of Eqs. (3.170a),
(3.170c), and (3.170d) gives
GB (x) = 1
1
g4
Λ
2
6
gR
− (N − 2) R ln
+ O(gR
) ln(Λ|x|)
2π
2π
µ
4
1
g
6
+ (N − 1) R 2 ln2 (Λ|x|) + O(gR
)
2
(2π)
=1
− (N − 1)
2
gR
ln(Λ|x|)
2π
4
4
1
gR
gR
Λ
2
6
+ (N − 1)
ln (Λ|x|) + (N − 1)(N − 2)
ln(Λ|x|) ln
+ O(gR
)
2
(2π)2
(2π)2
µ
=1
− (N − 1)
2
gR
ln(Λ|x|)
2π
h
i g4
4
1
gR
Λ
2
2
6
R
+ (N − 1)
ln (Λ|x|) + (N − 1) − (N − 1)
ln(Λ|x|) ln
+ O(gR
),
2
2
2
(2π)
(2π)
µ
− (N − 1)
(3.175a)
3.6. CALLAN-SYMANZIK EQUATION OBEYED BY THE SPIN-SPIN CORRELATOR IN THE d = 2-DIMENSI
2
Z(gR
) =1
4
1
gR
Λ
Λ
2
6
− (N − 1)
gR − (N − 2)
ln
+ O(gR ) ln
2π
2π
µ
µ
4
Λ
1
gR
6
ln2
+ (N − 1)
+ O(gR
)
2
(2π)2
µ
g2
Λ
Λ
g4
6
= 1 − (N − 1) R ln
+ (N − 1)(N − 3/2) R 2 ln2
+ O(gR
),
2π
µ
(2π)
µ
(3.175b)
and
1
1
2
4
2) =
gR
gR
2 Λ
Λ
Z(gR
6
1 − (N − 1) 2π ln µ + (N − 1)(N − 3/2) (2π)
2 ln
µ + O(gR )
2
4
gR
Λ
gR
Λ
2
2
6
ln
= 1 + (N − 1)
ln
+ (N − 1) − (N − 1)(N − 3/2)
+ O(gR
)
2π
µ
(2π)2
µ
g2
Λ
g4
Λ
6
= 1 + (N − 1) R ln
+ (N − 1) (N − 1 − N + 3/2) R 2 ln2
+ O(gR
)
2π
µ
(2π)
µ
g2
Λ
g4
1
Λ
6
= 1 + (N − 1) R ln
+ (N − 1) R 2 ln2
+ O(gR
),
2π
µ
2
(2π)
µ
(3.175c)
respectively. Multiplication of Eq. (3.175a) by Eq. (3.175c) gives the
desired expansion of the renormalized spin-spin correlator
1
g4
gR2
ln(µ|x|) + (N − 1) R 2 ln2 (µ|x|) + O(gR6 ),
2π
2
(2π)
(3.176)
since the cross term of order gR4 cancels the term underlined in Eq.
(3.175a). The fact that GR is obtained from GB with the substitution
GR (x) = 1 − (N − 1)
gB2 ←→ gR2 ,
Λ ←→ µ
(3.177)
is an artifact of the expansion to order gR4 coupled with the choice of
the renormalization point made in section 3.6.1.
3.6.4. Callan-Symanzik equation obeyed by the spin-spin
correlator. The Callan-Symanzik equation obeyed by the renormalized spin-spin correlator is
∂
∂
2
2
0= µ
+ β(gR ) 2 + 2γ(gR ) GR (x),
∂µ
∂gR
(3.178)
1 = lim GR (x).
|x|→1/µ
We try the Ansatz
β(gR2 ) :=
b2 gR4 + O(gR6 ),
γ(gR2 ) := a1 gR2 + a2 gR4 + O(gR6 ).
(3.179)
116
3. NON-LINEAR-SIGMA MODELS
With the help of
∂
g2
g4
GR (x) = −(N − 1) R + (N − 1) R 2 ln(µ|x|) + O(gR6 ),
∂µ
2π
(2π)
∂
4
6
2
b2 gR + O(gR )
β(gR ) 2 GR (x) =
∂gR
1
gR2
2
6
× −(N − 1) ln(µ|x|) + (N − 1)
ln (µ|x|) + O(gR )
2π
(2π)2
gR4
= −b2 (N − 1) ln(µ|x|) + O(gR6 ),
2π
2
2
4
6
2γ(gR )GR (x) = 2 a1 gR + a2 gR + O(gR )
gR2
1
gR4
2
6
× 1 − (N − 1) ln(µ|x|) + (N − 1)
ln (µ|x|) + O(gR )
2π
2
(2π)2
gR4
4
2
= 2a1 gR + 2a2 gR − 2a1 (N − 1) ln(µ|x|) + O(gR6 ),
2π
(3.180)
µ
one needs to solve the equations
0 = −(N − 1)
gR2
+ 2a1 gR2 ,
2π
0 = 2a2 gR4 ,
0 = (N − 1)
gR4
gR4
gR4
ln(µ|x|)
−
b
(N
−
1)
ln(µ|x|)
−
2a
(N
−
1)
ln(µ|x|),
2
1
(2π)2
2π
2π
(3.181)
i.e.,
N −1
,
2π
2a2 = 0,
(3.182)
N −2
b2 = −
.
2π
We conclude that the Callan-Symanzik equation obeyed by the renormalized spin-spin correlator is given by
∂
∂
2
2
0= µ
+ β(gR ) 2 + 2γ(gR ) GR (x),
(3.183a)
∂µ
∂gR
2a1 =
∂gR2
g4
= −(N − 2) R + O(gR6 ),
∂µ
2π
√
∂ ln Z
N − 1 gR2
2
γ(gR ) ≡ µ
=+
+ O(gR6 ),
∂µ
2 2π
β(gR2 ) ≡ µ
(3.183b)
(3.183c)
3.6. CALLAN-SYMANZIK EQUATION OBEYED BY THE SPIN-SPIN CORRELATOR IN THE d = 2-DIMENSI
with the non-perturbative condition
lim GR (x) = 1.
|x|→1/µ
(3.183d)
3.6.5. Physical interpretation of the Callan-Symanzik equation. The Callan-Symanzik equation (3.183) is a set of three firstorder differential equations obeyed by the spin-spin correlator, the coupling constant, and the wave-function renormalization in the (d = 2)dimensional O(N ) NLσM. As such it has a unique solution if and only if
the value of the spin-spin correlator is specified at “one point” (µx, gR2 ),
the so-called renormalization point. The renormalization point that we
chose is
2π
∂GR
µ|x| = 1,
gR2 = −
lim
(3.184)
N − 1 |x|→1/µ ∂ ln |x|
(by translation invariance, the spin-spin correlator is a function of |x|
only) at which we took the spin-spin correlator to be unity. The numerical values taken by the expansion coefficients of the beta function
β(gR2 ) and the anomalous scaling dimension γ(gR2 ) depend on the choice
of the renormalization point [the point (µx, gR2 ) at which the renormalized spin-spin correlator is unity, say]. The signs of β(gR2 ) and γ(gR2 ) in
the vicinity of the free-field fixed point gR2 = 0 are independent of the
renormalization point.
The Callan-Symanzik equation (3.183) can be solved by the method
of characteristics by which Eq. (3.183) is recast into
d
2
e µ(t)x ,
+ 2γ gR (t) G
0=
R
dt
dµ(t)
(3.185a)
µ(t) :=
,
dt
dg 2 (t)
β gR2 (t) := R ,
dt
with some initial data at “time” t0 , say,
eR µ(t0 )x ≡ G
eB (Λx). (3.185b)
µ(t0 ) ≡ Λ,
gR2 (t0 ) ≡ gB2 ,
G
The curve parametrized by t and defined by the set of points µ(t)|x|, gR2 (t) ∈
R2 is called a characteristic of the Callan-Symanzik equation (3.185).
In this incarnation, the Callan-Symanzik equation encodes the notion
of scaling in that the spatial argument of the spin-spin correlator only
depends on the dimensionless ratio of length scales µ(t)x,
e µ(t)x := G (x),
e (Λx) := G (x),
G
G
(3.186)
R
R
B
B
as we have verified explicitly up to second order in perturbation theory
with Eqs. (3.176) and (3.160b), respectively. The coupling constant
gR2 (t) is reinterpreted as a “running” coupling constant, i.e., as a scale
dependent coupling constant.
118
3. NON-LINEAR-SIGMA MODELS
In the representation (3.185), the Callan-Symanzik equation can be
integrated to, say,


Zt
0
2 0 
e µ(t)x = G
e (Λx) × exp 
G
−2 dt γ gR (t )  ,
R
B
t0
µ(t) = Λ et−t0 ,
(3.187a)
2
ZgR
t − t0 =
dg 2
.
β(g 2 )
2
gB
By choosing the initial time t0 so that Λ|x| is at the renormalization
point
(3.188)
Λ|x| = 1,
gB2 ,
Eq. (3.187) becomes


Zt
0
2 0 
eR e+(t−t0 ) = exp 
G
−2 dt γ gR (t )  ,
t0
2
ZgR
t − t0 =
(3.189a)
2
dg
.
β(g 2 )
2
gB
By choosing the final time t so that µ(t)|x| is at the renormalization
point
µ(t)|x| = 1,
gR2 ,
(3.190)
Eq. (3.187) becomes


Zt
0
2 0 
e e−(t−t0 ) = exp 
G
+2
dt
γ
g
(t
)
,

B
R
t0
2
ZgR
t − t0 =
(3.191a)
2
dg
.
β(g 2 )
2
gB
Qualitative RG characteristics for the spin-spin correlators are displayed in Fig. 2, whereby it is assumed that β(g 2 ) < 0 for all g 2 > 0.
Figure 2 gives a pictorial answer to the question of what range of
e µ(t0 )x from its initial reference value
g 2 is needed to integrate G
R
e
1 = GR µ(t0 )|x| = 1 (an open circle in Fig. 2 along the horizontal line
e µ(t)x at some given
µ|x| = 1 at which g 2 ≡ gB2 ) to its final value G
R
gR2 (t) [a closed circle in Fig. 2 along the vertical line g 2 = gR2 (t)]. We
can distinguish two families of characteristics.
3.6. CALLAN-SYMANZIK EQUATION OBEYED BY THE SPIN-SPIN CORRELATOR IN THE d = 2-DIMENSI
µ|x|
µ|x| = 1
2
gR
g2
Figure 2. RG characteristics for the Callan-Symanzik
equation in an asymptotically free theory at short distances. Initial data are depicted as open circles along
the constant line µ|x| = 1 in the µ|x|-g 2 plane. Final
data are depicted as filled circles along the characteristics emanating from the initial data.
There are those characteristics that intercept the fixed vertical line
g = gR2 (t) at a value of µ(t)|x| < 1. The range of g 2 interpolating
between the initial gB2 at t0 (open circle) and the final gR2 (t) is then the
finite segment [0, gR2 (t)] as µ(t)|x| → 0.
There are those characteristics that intercept the fixed vertical line
g 2 = gR2 (t)) at a value of µ(t)|x| > 1. The range of g 2 interpolating
between the initial gB2 at t0 (open circle) and final gR2 (t) is then the
semi-infinite segment [gR2 (t), ∞[ as µ(t)|x| → ∞.
e µ(t)x with µ(t)|x| 1 given G
e µ(t )x
If we seek values of G
0
R
R
perturbation theory is thus condemned to failure. On the other hand,
e µ(t)x
perturbation theory can be accurate if we seek values of G
R
e
given GR µ(t0 )x with µ(t)|x| 1.
As scale invariance implies that we can equally well regard variations of µx as being variations of x at fixed µ or conversely, we infer that
the accuracy of perturbation theory improves as the Callan-Symanzik
equation is integrated to probe the spin-spin correlation function at
arbitrary small |x|, a property called asymptotic freedom at short distances. Conversely, the accuracy of perturbation theory diminishes
(breaks down) as the Callan-Symanzik equation is integrated to probe
the spin-spin correlation function at arbitrary large |x|.
2
3.6.6. Physical interpretation of the beta function. This
property of the Callan-Symanzik equation follows from the fact that
the renormalized coupling constant flows away from the free-field fixed
point g 2 = 0 in the IR limit. Indeed, the beta function β(gR2 ) encodes
the rate of change of the coupling constant of the (d = 2)-dimensional
O(N ) NLσM as the separation |x| in the spin-spin correlator is effectively reduced since an increasing µ implies a decreasing |x| at the
120
3. NON-LINEAR-SIGMA MODELS
renormalization point. As β(gR2 ) is negative with increasing µ for
N > 2, the renormalized coupling constant gR2 effectively decreases
at shorter distances. At shorter distances, the NLσM resembles more
and more the free-field fixed point gR2 = 0. Conversely, the renormalized coupling constant gR2 effectively increases at longer distances.
Within the RG terminology, the coupling constant of the (d = 2)dimensional O(N > 2) NLσM is UV irrelevant, or, equivalently, IR
relevant at the free-field fixed point. The free-field fixed point gR2 = 0
is UV stable, or, equivalently, IR unstable when N > 2. Our perturbative RG analysis can thus only be trusted in the close vicinity of
the UV limit limt→∞ µ(t) = ∞. As perturbation theory breaks down
in the IR limit limt→∞ µ(t) = 0 one must rely on alternative methods
[Bethe Ansatz, numerical simulations on the underlying lattice model,
high temperature (g 2 1) expansions] to probe the physics of the
(d = 2)-dimensional O(N ) NLσM at long distances.
As the RG trajectories flow out of the regime of applicability of
perturbation theory in g 2 , we cannot infer from our calculation the
behavior of the spin-spin correlator for very large separations. The
most economical hypothesis is to imagine that the flow is to an IR
stable fixed point describing a paramagnetic phase as g 2 → ∞. In the
paramagnetic phase, the exponential decay
|x|
(3.192)
exp −
ξ
with large separation |x| of the spin-spin correlator allows the identification of the length scale ξ, the so-called paramagnetic correlation
length, which is of the order of the lattice spacing. Although our RG
analysis cannot alone establish the existence of the IR stable paramagnetic phase and of the concomitant finite correlation length of order
of the lattice spacing, it can predict the small g 2 dependence of a finite
(though large) correlation length in the close vicinity of the ferromagnetic IR unstable fixed point.
By dimensional analysis, the rescaling
a
da
a → ≡ a + da,
− O (da/a)2 = ln(1/b),
(3.193)
b
a
implies that the correlation length ξ(ga2 ) calculated with the lattice
2
spacing a is related to the correlation length ξ(ga/b
) calculated with
the lattice spacing a/b by
1 2
ξ(ga2 ) = ξ(ga/b
).
(3.194)
b
To proceed, integrate (note the sign difference relative to a variation
with respect to the momentum cutoff µ)
a
∂g 2
g4
= +(N − 2)
+ O(g 6 )
∂a
2π
(3.195)
3.6. CALLAN-SYMANZIK EQUATION OBEYED BY THE SPIN-SPIN CORRELATOR IN THE d = 2-DIMENSI
to find
2
Zga/b
0=
dg 2 N − 2
−
g4
2π
Za/b
da
a
a
ga2
=
1
1
− 2
2
ga
ga/b
!
!
=
1
1
− 2
2
ga
ga/b
−
N −2
[ln (a/b) − ln a]
2π
−
N −2
ln(1/b).
2π
(3.196)
2
Assume now that a/b is chosen so that ξ(ga/b
) in Eq. (3.194) is of order
of the lattice spacing so that, when combined with Eq. (3.196), it is
found that
1 2
ξ(ga2 ) = ξ(ga/b
)
b
≈ exp ln(1/b) × a
(3.197)
"
!#!
2
ga
2π 1
1+O
× a.
≈ exp +
2
2
N − 2 ga
ga/b
The very rapid divergence of the correlation length as g 2 → 0 corresponds to a weak singularity of the free energy (the logarithm of the
partition function). We have uncovered a second important property of
the (d = 2)-dimensional O(N > 2) NLσM aside from UV asymptotic
freedom, namely that of dimensional transmutation, whereby a macroscopic length scale in the form of a correlation length is generated out
of a field theory depending on one dimensionless coupling constant and
one microscopic UV cutoff.
Before closing this section observe that the prefactor to the exponential dependence on g 2 in the correlation length (3.197) can also be
g 2 dependent. To see this it suffices to include the first non-vanishing
contribution of higher order than g 4 to the beta function in the (d = 2)dimensional O(N > 2) NLσM, say the term β3 g 6 , in the expansion
∞
X
N −2
,
2π
n=0
(3.198)
where β3 is yet to be calculated. Assuming that β3 is non-vanishing,
one finds
!(2π)2 β3 /(N −2)2
"
!#!
2
2
2π
g
1
g
a
a
ξ(ga2 ) ≈
exp +
1+O
× a.
2
2
ga/b
N − 2 ga2
ga/b
(3.199)
2
β(g ) =
βn g 2n ,
β0 = 0,
β1 = 0,
β2 = −
122
3. NON-LINEAR-SIGMA MODELS
3.7. Beta function in the d > 2-dimensional O(N > 2) NLσM
The derivation of the Callan-Symanzik equation obeyed by the spinspin correlator in section 3.6 was done in the spirit of RG approach used
in high-energy physics in the 50’s and early 70’s. High-energy physics
in the 50’s and in the 70’s relied heavily on quantum field theory to
describe the electromagnetic, weak, and strong interactions. Locality,
causality, and relativistic invariance were elevated to the status of fundamental principles of nature. The mathematical starting point was an
action for local fields describing elementary (i.e., point-like) relativistic
particles interacting through gauge fields.
The price to be paid in this approach is the occurrence of divergences caused by the point-like nature of the quantum fields, i.e., the
absence of a high-energy (UV) cutoff. The severity of the UV divergences plaguing quantum field theories is measured by the notion of
whether or not a theory can be renormalized. The idea behind the
program of renormalization of quantum field theories is to demand
that the scattering cross sections be finite so as to allow a comparison
with measured cross sections in colliders.
This selection criterion for quantum field theories describing the
fundamental interactions of nature led in the 50’s and 70’s to the realization that all the divergences associated to the point-like nature
of renormalizable local quantum field theories can be consistently absorbed into a redefinition of a finite number of bare coupling constants
in the Lagrangian, while leaving all measurable cross sections finite.
Absorbing all UV divergences of a renormalizable local quantum field
theory into a redefinition of the coupling constants means that the coupling constants depend on the UV cutoff whereas physical quantities
are cutoff independent. Fundamental (physical) objects are, typically,
gauge-invariant correlation functions made up of the local fields entering the theory. The independence on the UV cutoff of physical
correlation functions implies that they obey a Callan-Symanzik equation through the implicit dependence of the coupling constants on the
UV cutoff. From this point of view, the Lagrangian, action, and partition function are not considered to be as fundamental as correlation
functions that can be measured in a collider.
In statistical physics the partition function (intensive free energy)
plays a much more fundamental role than in quantum field theory. It
can be considered as a fundamental physical quantity as it is well defined in the thermodynamic limit due to the presence of UV cutoff
such as the lattice spacing. Statistical models that correspond to unrenormalizable field theories if the UV cutoff were to be removed are
not a priori ruled out. Correspondingly, it is desirable to compute the
partition function (intensive free energy) and to decide on a case by
case basis if and how some correlation functions become independent
3.7. BETA FUNCTION IN THE d > 2-DIMENSIONAL O(N > 2) NLσM
123
of the UV cutoff as the thermodynamic limit is taken. With the advent
of powerful computers it is possible to compute the partition function
(intensive free energy) for very large system sizes. It is thus not surprising that RG approaches were developed by the condensed matter
and statistical physics communities to evaluate directly the partition
function (intensive free energy). One popular method is to integrate
high-energy degrees of freedom through a momentum-shell integration.
The momentum-shell integration can be easily implemented on the
O(N > 2) NLσM if one is only after the beta function up to order
g 4 .[32] This is the method that we will use to derive the IR RG equation obeyed by g 2 in the O(N > 2) NLσM in dimensions larger than
2. Another method consists in performing the RG analysis in position
space, as will be illustrated in the chapter on the Kosterlitz-Thouless
transition.
The RG analysis of the O(N > 2) NLσM in dimensions d larger
than 2 can be performed on the partition function
2
Z
1
dd x(∂µ n)
− 2ad−2
g2
2
d
R
d[n] δ n − 1 e
Z
d[m] δ m2 −
Z :=
∝
R
1
e
ad−2 g 2
− 12
R
(3.200)
2
dd x(∂µ m)
.
Rd
Observe that the partition function depends explicitly on the lattice
spacing a that plays the role of the UV cutoff as we have chosen to
keep g 2 dimensionless when d 6= 2. Choose the parametrization
s
1 − ad−2 g 2 π 2
cos θ,
ad−2 g 2
s
1 − ad−2 g 2 π 2
sin θ,
ad−2 g 2
m1 :=
m2 :=
(3.201)
m3 := π1 ,
..
.
mN := πN −2 ,
motivated as we are by the O(2) NLσM, under which the Lagrangian
L=
2
1
∂µ m
2
1
=
2
∂µ m1
2
+ ∂µ m2
2
+
N
X
j=3
!
∂µ mj
2
(3.202)
124
3. NON-LINEAR-SIGMA MODELS
becomes
2
2
2
1 1 − ad−2 g 2 π 2
ad−2 g 2
L=
∂µ θ +
π · ∂µ π + ∂µ π
−ln |J (π)|.
2
ad−2 g 2
1 − ad−2 g 2 π 2
(3.203)
Here, J (π) is the Jacobian of the transformation
1
θ
2
m whereby m = d−2 2 −→
.
(3.204)
π
a g
As J (π) does not depend on π, it will be dropped.
make the replacement
L −→ L0 + L1 + L2 ,
2
1 1
L0 = d−2 2 ∂µ θ ,
2a g
2 1
2
1
L 1 = ∂µ π − π 2 ∂µ θ ,
2
2
∞
2 X
n
d−2 2
L 2 = a g π · ∂µ π
ad−2 g 2 π 2 .
17
Thus, we can
(3.206a)
n=0
Finally, we can neglect L2 if we are only after RG equations up to order
g 4 , in which case we need to perform an RG analysis of the partition
function
R
Z
− dd x(L0 +L1 +O(g 2 ))
.
(3.206b)
Zsw := d[θ, π] e Rd
Observe that the field θ is much more rigid or stiff than the fields π
in the limit of very low temperatures g 2 1. In other words, θ varies
appreciably on much longer length scales than π does.
We would like to integrate over the fast modes in the partition
function. To this end, we choose the asymmetric Fourier convention
Z
Z
dd k +ik·x
f (x) :=
e
f (k),
f (k) := dd x e−ik·x f (x), (3.207)
(2π)d
Rd
17
Y
x∈Rd
Rd
The Jacobian J (π) can be read from
Z
π
N −2
d
π=
Z
Y
(2π)
x∈Rd
RN −2
=
+∞
Z
Y
d
+∞
r
Z
r
1
2
−π
π
dr
δ r−
|2r|
ad−2 g 2
0
RN −2
Z2π
dr r
x∈Rd 0
=
N −2
Z
dθ
0
dN −2 πδ r2 −
1
ad−2 g 2
−∞
−∞
RN −2
+∞
+∞
+∞
Z
Z
Y Z
dm1
dm2 · · ·
dmN δ m2 −
x∈Rd
+ π2
1
ad−2 g 2
.
−∞
(3.205)
3.7. BETA FUNCTION IN THE d > 2-DIMENSIONAL O(N > 2) NLσM
125
for some complex-valued function f . Without an UV cutoff in momentum space, the momentum-space representation of
Z
S0 + S1 ≡ dd x (L0 + L1 )
(3.208a)
Rd
is then
Z
S0 +S1 =
dd k 1
(2π)d 2
1
k θ(+k)θ(−k) + k π(+k) · π(−k) − f (+k)g(−k) ,
2
2
ad−2 g
2
Rd
(3.208b)
where
dd q
π(k + q) · π(−q)
(2π)d
Z
f (+k) :=
(3.208c)
Rd
and
Z
g(+k) :=
dd q
[−i(k + q)] · [−i(−q)] θ(k + q)θ(−q)
(2π)d
(3.208d)
Rd
are the Fourier transforms of
f (x) ≡ π 2 (x)
(3.208e)
g(x) ≡ (∂µ θ)2 (x),
(3.208f)
and
respectively.
When g 1, the same variation of θ and π takes place on vastly
different characteristic length scales. In momentum space this means
that θ is much more strongly peaked about k = 0 than π is. This fact
suggests the introduction of a thin momentum shell
bΛ < |k| < Λ,
b = 1 − ,
a positive infinitesimal number,
(3.209)
below the UV momentum cutoff Λ and to perform the approximation
by which k2 θ(+k)θ(−k) is negligible relative to k2 π(+k)π(−k) and
g(−k) ≈ (2π)d g(0) δ(k) in the momentum shell (3.209), i.e.,
Z
dd k 1
bd−2
S0 + S1 ≈
k2 θ(+k)θ(−k)
(2π)d 2 (ba)d−2 g 2
|k|<bΛ
!
+ k2 π(+k) · π(−k) − fbΛ (+k)gbΛ (−k)
Z
+
dd k 1
(2π)d 2
2
k π(+k) · π(−k) −π(+k) · π(−k)gbΛ (q = 0) .
bΛ<|k|<Λ
(3.210)
126
3. NON-LINEAR-SIGMA MODELS
The Fourier transforms fbΛ (+k) and gbΛ (−k) are defined as in Eqs.
(3.208c) and (3.208d) except for the sharp UV momentum cutoff bΛ,
i.e., it is understood that the replacements
θ(k) −→ θ(k)Θ(bΛ − |k|)
(3.211)
and
π(k) −→ π(k)Θ(bΛ − |k|)
have been made [Θ(x) is the Heaviside step function].
Integration over the partial measure
Y
dπ(+k) =
bΛ<|k|<Λ
kY
1 >0
kY
1 >0
dπ(−k)dπ(+k) =
bΛ<|k|<Λ
(3.212)
dπ ∗ (+k)dπ(+k)
bΛ<|k|<Λ
(3.213)
of the partition function with action (3.210) is Gaussian and given by
Z
0
0
Zsw ∝
d[θ, π]e−S0 −S1 −δS ,
|k|<bΛ
S00
Z
=
dd k 1 1
1
k2 θ(+k)θ(−k),
d
d−2
(2π) 2 b
(a/b)d−2 g 2
|k|<bΛ
S10
Z
=
dd k 1 2
k π(+k) · π(−k) − fbΛ (+k)gbΛ (−k) ,
d
(2π) 2
(3.214)
|k|<bΛ
dd k N − 2
ln
(2π)d 2
Z
δS = −
bΛ<|k|<Λ
1
2
k − gbΛ (q = 0)
.
With the estimate
Z
δS = −
"
dd k N − 2
1
ln 2 + ln
(2π)d 2
k
1
1−
bΛ<|k|<Λ
Z
=+
dd k
(N − 2) ln |k| −
(2π)d
bΛ<|k|<Λ
Z
=+
bΛ<|k|<Λ
Z
bΛ<|k|<Λ
!#
gbΛ (q=0)
k2
n
∞
dd k N − 2 X 1 gbΛ (q = 0)
(2π)d 2 n=1 n
k2
∞
d−2n
d k
Ω(d) N − 2 X 1
− (bΛ)d−2n
n Λ
[g
(q
=
0)]
,
(N
−
2)
ln
|k|
−
bΛ
(2π)d
(2π)d 2 n=1 n
d − 2n
d
(3.215)
where it is understood that Ω(d) is the area of the d-dimensional unit
sphere and that integration over the momentum shell gives a logarithm
and not a power law when d = 2n, the original action with the sharp
UV cutoff Λ is modified in three ways:
• The new sharp UV cutoff is bΛ, i.e.,
Λ −→ bΛ = eln b Λ = 1 + ln b + O(ln2 b) Λ.
(3.216)
3.7. BETA FUNCTION IN THE d > 2-DIMENSIONAL O(N > 2) NLσM
(a) Case d
127
2=0
g2
(b) Case d
2=✏
g2
g?2
Figure 3. Qualitative IR flow of the coupling constant
g 2 in the O(N > 2) NLσM as a function of dimensionality d − 2 ≥ 0. The filled circle depicts the ferromagnetic
fixed point at zero temperature, i.e., at g 2 = 0. The
empty circle depicts the infinite temperature paramagnetic fixed point. The star depicts a finite temperature
g?2 := N2π
critical point below which the system devel−2
ops ferromagnetic LRO and above which the system is
paramagnetic.
• The action has changed by an additive constant
Z
+
dd k
(N − 2) ln |k|.
(2π)d
(3.217)
bΛ<|k|<Λ
• To leading order in the UV momentum cutoff Λ = 1/a the
coupling constant ad−2 g 2 has changed by
Ω(d) N − 2
d−2
−→
−
1
−
b
ad−2 g 2
ad−2 g 2 (2π)d d − 2
1
1
d−2
1
.
a
(3.218)
The beta function for the coupling constant g 2 is obtained from
Ω(d) N − 2
d−2
:=
−
1
−
b
(a0 )d−2 (g 2 )0
ad−2 g 2 (2π)d d − 2
1
1
d−2
1
a
(3.219a)
whereby
a0 ≡ a + da + O[(da)2 ]
a
:=
b
= 1 − ln b + O(ln2 b) a
(3.219b)
and
g2
0
≡ g 2 + d(g 2 ) + O
n
2 o
d(g 2 )
.
(3.219c)
128
3. NON-LINEAR-SIGMA MODELS
On the one hand, we have the expansion
1
1
=
2
(a0 )d−2 (g 2 )0 ad−2 1 − (d − 2) ln b + O(ln b) g 2 + d(g 2 ) + O [d(g 2 )]2
1
=
2
2
2 )]2
+
O(ln
b)
+
O
[d(g
ad−2 g 2 1 − (d − 2) ln b + dg
2
g
n
o
1
dg 2
2
2 2
= d−2 2 1 + (d − 2) ln b − 2 + O(ln b) + O d(g )
a g
g
n
o
da dg 2
1
2
2 2
= d−2 2 1 − (d − 2) − 2 + O (da/a) + O d(g )
a g
a
g
(3.220)
of the left-hand side of Eq. (3.219a). On the other hand, we have the
expansion
1
Ω(d) N − 2 2 1
2
=
g (d − 2) ln b + O(ln b)
1+
(a0 )d−2 (g 2 )0 ad−2 g 2
(2π)d d − 2
1
Ω(d) N − 2 2
da
2
= d−2 2 1 −
g (d − 2) + O (da/a)
a g
(2π)d d − 2
a
(3.221)
of the right-hand side of Eq. (3.219a). At last we arrive at the beta
function
Ω(d)
dg 2
= +(d − 2)g 2 −
(N − 2)g 4 + O(g 6 ).
(3.222)
−a
da
(2π)d
The UV beta function for the O(N > 2) NLσM in
d = 2 + ,
reduces to
−a
as
a positive infinitesimal number,
(3.223)
dg 2
N −2 4
= + g 2 −
g + O(g 4 , g 6 )
da
2π
(3.224)
Ω(2 + ) = 2π + O(),
(2π)2+ = (2π)2 + O().
(3.225)
The UV beta function
β(g 2 ) := + g 2 −
vanishes when
N −2 4
g
2π
(3.226)
2π
.
(3.227)
N −2
For 0 < g 2 < g?2 , the UV beta function is positive, i.e., g 2 is UV
relevant (IR irrelevant). For g 2 = g?2 , the theory is critical as the beta
function vanishes. For g 2 > g?2 , the UV beta function is negative, i.e.,
g 2 is UV irrelevant (IR relevant). The critical point g 2 = g?2 is an IR
unstable fixed point. To the right of this fixed point the system flows
in the IR limit to the infinite temperature paramagnet fixed point. To
g 2 = g?2 :=
3.7. BETA FUNCTION IN THE d > 2-DIMENSIONAL O(N > 2) NLσM
129
the left of this fixed point the system flows in the IR limit to the zero
temperature ferromagnetic fixed point. Within the spin-wave approximation, the ferromagnetic LRO at zero temperature thus extends to a
finite critical temperature which is proportional to = d − 2. Figure 3
depicts the qualitative behavior of the beta function in the O(N > 2)
NLσM as a function of dimensionality.
A final comment is of order. We analytically continued dimensionality d = 1, 2, · · · of space to real values d = 2 + . If so, one might
also wonder if it makes sense to analytically continue N = 2, 3, · · · to
real values below 2 and, in particular, to the limit N = 0. This limit
is interesting as it changes the sign of the beta function in d = 2. It
turns out that some problems in statistical physics such as polymers or
the problem of Anderson localization demand analytical continuations
of the type N → 0. We would like to give a geometric interpretation
to this change in sign.
Let
Z d
d
N
d x 1 XX
S[φ] :=
g (φ) ∂µ φa ∂µ φb
(3.228a)
ad−2 2 a,b=1 µ=1 ab
denote the action of a NLσM on the Riemannian manifold M. It is
shown in appendix C that it is the Ricci curvature tensor of the target
space that controls the RG flow of the beta function up to first order
in the loop expansion,
∂
1
gab = gab −
R ,
infinitesimal.
(3.228b)
∂a
2π ab
Imagine that it is permissible to analytically continue the coordinates
on the target space M from real values to imaginary values according
to
φa = iφ? a ,
a = 1, · · · , N,
(3.229a)
a
so as to obtain a NLσM on the Riemannian manifold M? defined by
the metric tensor 18
g?ab (φ? ) := −gab (φ? )
(3.229b)
and the action
?
?
S [φ ] :=
Z
N
d
dd x 1 X X ? ?
g (φ )∂µ φ? a ∂µ φ? b .
ad−2 2 a,b=1 µ=1 ab
(3.229c)
Then, there follows the one-loop RG flow
a
18
∂ ?
1 ?
gab = g?ab −
R ,
∂a
2π ab
infinitesimal.
(3.229d)
Observe that the step (3.229b) is not equivalent to the transformation
law g?ab (φ? ) := −gab (iφ? ) under the reparametrization φa = iφ? a . The metric
g?ab (φ? ) := −gab (iφ? ) with the action (3.229c) deliver the RG equations (3.228b).
130
3. NON-LINEAR-SIGMA MODELS
We ignore the important question of the convergence of the path integral upon this analytic continuation.
We need to answer two questions. When does the one-loop RG
flow (3.228b) reduce to a one-loop RG flow of the form (3.224)? What
lessons do we learn from the analytical continuation (3.229) in this
case?
For a generic N -dimensional Riemannian manifold, the one-loop
RG flow (3.228b) involves at most N +N (N −1)/2 independent running
couplings, for the metric tensor is a symmetric matrix. Symmetry
properties of a N -dimensional Riemannian manifold can reduce the
number of independent running couplings. For symmetric spaces, this
reduction in the number of independent running couplings is the most
dramatic. For any compact symmetric space M, it is possible to rewrite
the UV one-loop RG flow (3.228b) as
∂ 2
c
g = g2 − v g4,
infinitesimal.
(3.230)
∂a
4π
Here, the positive number cv is the quadratic Casimir invariant of the
global symmetry group associated to the symmetric space M. Upon
the analytic continuation (3.229), the UV one-loop RG flow (3.230)
turns into [see Eq. (C.64)]
−a
∂ 2
c
g = g2 + v g4,
infinitesimal.
(3.231)
∂a
4π
The analytic continuation (3.229) has induced a change of the sign by
which the contribution arising from the Ricci curvature tensor enters
in the one-loop RG flow (3.231). This sign change is interpreted as
the fact that the symmetric space M? defined by the analytic continuation (3.229) on the symmetric space M is non-compact. For example, the analytic continuation (3.229) on the O(N ) NLσM delivers the
O(1, N − 1) NLσM. Hereto, we would like to interpret the change of
sign of the term proportional to g 4 in the beta function of the O(N )
NLσM when N → 0 as the fact that this limit “defines” a non-compact
target manifold.
Symmetric spaces have been classified by Cartan into families of
triplets. [33] Two of the three symmetric spaces making up a triplet
have sectional curvatures of opposite sign. The third member of the
triplet has vanishing sectional curvature. The O(2) NLσM is an example of a target Riemannian manifold with vanishing sectional curvature.
In random matrix theory, the statistical properties of diverse quantities are controlled by statistical ensemble of matrices closely related
to symmetric spaces. Statistical correlations of random energy eigenvalues follow from choosing symmetric spaces with vanishing sectional
curvature. Statistical correlations of the random eigenvalues of unitary
matrices follow from choosing symmetric spaces with positive sectional
curvature. Statistical correlations of the eigenvalues of pseudo-unitary
−a
3.8. PROBLEMS
131
matrices (the so-called Lyapunov exponents) follow from choosing symmetric spaces with negative sectional curvature. In turn, the global
symmetry group characterizing the symmetric space is dictated by the
intrinsic symmetries respected by the ensemble of statistical matrices. These intrinsic symmetries are the presence or absence of timereversal symmetry, of spin-rotation symmetry, of particle-hole symmetry in both its unitary and antiunitary incarnations as defined by the
application on spinors (fermions). More generally, the NLσM’s that
describe the physics of Anderson localization when the dimensionality
of base space is d = 0, 1, 2, · · · have (supersymmetric) target spaces
with both a compact and non-compact component.
As will be illustrated in the chapter on the Kosterlitz-Thouless
phase transition, the global structure of the target manifold plays no
role in the perturbative analysis (spin-wave approximation) that we
have performed so far. For example, the O(2) NLσM has the circle as
a target manifold. Locally the real line and the circle cannot be distinguished. The spin-wave approximation on the (d = 2)-dimensional
O(2) NLσM neglects the fact that the circle is a compact manifold, i.e.,
replaces the circle by the line as a target manifold. Any perturbative
treatment in powers of the coupling g 2 on a symmetric space, even if
it leads to non-perturbative results such as the essential singularity in
the dependence of the correlation length on g 2 , is bound to ignore the
compact nature of the symmetric space and to fail if this property is
essential. A classification by mathematicians of the global structures
of Riemannian manifolds has been undertaken and applied to physics.
When the Riemannian manifold has a non-trivial global structure (nontrivial topology) such as is the case for a circle in opposition to the real
line, it is possible, on a case by case basis, to supplement the action of
the NLσM by a new term or by new degrees of freedom that account
for the superseding global structure. The inclusion of these “global”
degrees of freedom leads to a finite temperature transition from the
spin-wave phase to a paramagnetic phase called the Kosterlitz-Thouless
phase transition for the case of the (d = 2)-dimensional O(2) NLσM.
For Heisenberg quantum spin chains with nearest-neighbor antiferromagnetic exchange interactions, the addition of a topological term (i.e.,
a contribution to the action that is necessarily quantized) to the O(3)
NLσM induces critical behavior when the spin degrees of freedom carry
half-integer representations of SU (2).
3.8. Problems
3.8.1. The Mermin-Wagner theorem for quantum spin Hamiltonians.
Introduction. We are going to prove the Mermin-Wagner theorem
as it is stated in the original paper by Mermin and Wagner. [34] It says
132
3. NON-LINEAR-SIGMA MODELS
that the quantum Heisenberg Hamiltonian
X
X
Ĥ := −
Jr−r0 Ŝ r · Ŝ r0 − h
Ŝrz e−iK·r
r,r 0 ∈Λ
(3.232)
r∈Λ
has no long-range ferromagnetic or antiferromagnetic order in d = 1
and d = 2 dimensions of position space at non-vanishing temperature
T > 0, if the field h is taken to zero and the interaction Jr is shortranged, that is, at long distances it decays faster than |r|−d−2 . Here,
the sum over r runs over the sites of a lattice Λ with periodic boundary conditions in place and the operators Ŝ r = (Ŝrx , Ŝry , Ŝrz )T describe a
quantum spin-S, thus obeying the SU (2) algebra (2.134) at every lattice site r ∈ Λ. Translation invariance of the Hamiltonian is manifest
in the fact that Jr−r0 depends only on the difference of lattice sites.
We shall further assume
Jr = J−r ,
J0 = 0.
(3.233)
To probe the tendency toward ferromagnetic order, we choose K =
0 to describe a homogeneous field. To probe the tendency toward
antiferromagnetic order, we assume that Λ can be bipartitioned into
two sublattices and we choose K such that e−iK·r = −1 if r connects
sites on different sublattices, while e−iK·r = +1 if r connects sites on
the same sublattice. The amplitude h is thus that of a source field that
selects a colinear order parameter characterized by the wave vector K.
Magnetic colinear long-range order is signaled by a non-vanishing
value of the order parameter
1 X D z −iK·r E
sz (β, h) :=
Ŝr e
,
(3.234a)
N r∈Λ
where N is the number of lattice sites and the expectation value of any
operator  is given by
Tr e−β Ĥ Â
,
hAi :=
(3.234b)
Tr e−β Ĥ
with β := 1/(kB T ) and Tr the trace over the Hilbert space. Our task is
thus to compute the function sz (β, ·) and consider it in the limit h → 0.
If sz (β, ·) remains non-vanishing in this limit, spontaneous colinear
long-range order takes place. Otherwise we can rule out spontaneous
symmetry breaking. It turns out that sz (β, h) cannot be computed
exactly. Instead, we will be able to find an upper bound to sz (β, h)
and show that this upper bound vanishes in the limit h → 0 at any
non-vanishing temperature.
Observe that the Hamiltonian has SU (2) spin-rotation invariance
in the limit h → 0. Even though we will not explicitly make use of
this symmetry in our calculation, its presence is of crucial importance
for the result to hold. The use of the terminology Mermin-Wagner
3.8. PROBLEMS
133
theorem often refers to the following generalization of the statement
above. There exists no spontaneous breaking of a continuous symmetry
group in d = 1 and d = 2 dimensions of space at at non-vanishing
temperature T , if the Hamiltonian has only short-ranged interactions.
In contrast, discrete symmetries can very well be broken spontaneously
in d = 2 dimensions for a non-vanishing temperature T by short-range
interactions. A prominent example is the Ising model.
Proof of Bogoliubov’s inequality. Before we turn to the proof of
the Mermin-Wagner theorem itself, we want to establish an inequality
due to Bogoliubov on which the proof relies. (This inequality will
allow us to establish an upper bound on the order parameter sz .) The
(Bogoliubov) inequality states that
h
i 2
Tr Â, Â e
Tr [D̂, Ĉ], Ĉ e
≥ 2 Tr Ĉ, Â eD̂ ,
(3.235)
where Â, Ĉ, and D̂ are bounded linear operators on a Hilbert space H,
D̂ = D̂† is Hermitean, the brackets [·, ·] and {·, ·} denote the commutator and the anticommutator, respectively, and the trace Tr is taken
over H.
To prove the inequality (3.235), we use a basis of H in which D̂
is diagonal. We denote the orthonormal eigenvectors of D̂ with |ii,
i = 1, 2, · · · , and the corresponding eigenvalues with di , i = 1, 2, · · · .
The same basis also diagonalizes the operator eD̂ and its eigenvalues
are given by wi = edi , i = 1, 2, · · · . In summary, we have
n
†
o
D̂
h
†
i
D̂
eD̂ |ii = wi |ii,
D̂ |ii = di |ii,
i = 1, 2, · · · .
(3.236)
We can now define an inner product (·, ·) between any two bounded
linear operators  and B̂ on H as
X D E D E wi − wj
j † i i B̂ j
(Â, B̂) :=
,
di − dj
i,j
(3.237)
di 6=dj
where it is understood that the sum runs over all pairs i, j, except for
those which have degenerate eigenvalues di = dj .
Exercise 1.1: Show that the definition Eq. (3.237) has the properties of an inner product, that is
(Â, B̂) = (B̂, Â)∗
conjugate symmetry,
(Â, β1 B̂1 + β2 B̂2 ) = β1 (Â, B̂1 ) + β2 (Â, B̂2 )
(Â, Â) ≥ 0
positive definiteness,
for any pair β1 and β2 of complex numbers.
(3.238a)
linearity,
(3.238b)
(3.238c)
134
3. NON-LINEAR-SIGMA MODELS
With Eq. (3.237) defining an inner product, we conclude that the
Schwarz inequality
2
(3.239)
(Â, Â) (B̂, B̂) ≥ (Â, B̂)
holds.
Exercise 1.2: Show that the inequality
n
o 1
(Â, Â) ≤ Tr
Â, † eD̂
2
(3.240)
holds. Hint: Start from the definition (3.237) of the inner product
(Â, Â) and show that
wi − wj
wi + wj
≤
,
di − dj
2
di 6= dj .
(3.241)
Exercise 1.3: Show that the two equalities
(Â, [Ĉ † , D̂]) = Tr [† , Ĉ † ] eD̂
(3.242a)
([Ĉ † , D̂], [Ĉ † , D̂]) = Tr [[D̂, Ĉ], Ĉ † ] eD̂
(3.242b)
hold. Using them, as well as the inequality (3.240), and the Schwarz
inequality (3.239), prove the Bogoliubov inequality (3.235).
Application of Bogoliubov’s inequality to the quantum Heisenberg
Hamiltonian. Define the operators
Ŝr− := Ŝrx − iŜry ,
Ŝr+ := Ŝrx + iŜry ,
r ∈ Λ,
(3.243)
and the Fourier transform fk of any operator or function fr that is
defined on the lattice
X
fk :=
e−ik·r fr
(3.244)
r∈Λ
such that
fr =
1 X +ik·r
e
fk ,
N k∈BZ
(3.245)
where k takes values in the first Brillouin zone (BZ).
Exercise 2.1: Rewrite the Hamiltonian (3.232) in terms of the
Fourier transformed operators Ŝk+ , Ŝk− , Ŝkz and the function Jk .
Exercise 2.2: Familiarize yourself with the algebra obeyed by the
operators Ŝk+ , Ŝk− , and Ŝkz by computing the commutators
[Ŝk+ , Ŝk−0 ],
[Ŝk+ , Ŝkz 0 ],
[Ŝk− , Ŝkz 0 ].
(3.246)
3.8. PROBLEMS
135
Then, use this algebra and the momentum-space representation of
Hamiltonian (3.232) that was obtained in exercise 2.1 to verify that
D
E
+
−
gk := [[Ŝk , Ĥ], Ŝ−k ]
E
D + −
1 X
− +
z z
=
Jq − Jq+k Ŝq Ŝ−q + Ŝq Ŝ−q + 4 Ŝq Ŝ−q + 2 h N sz .
N q
(3.247)
Exercise 2.3: Use the Bogoliubov inequality (3.235) with the following choice for the operators
−
 = Ŝ−k−K
,
Ĉ = Ŝk+ ,
D̂ = −β Ĥ,
(3.248)
to show that
oE 4N 2 |sz |2
1 Dn −
+
Ŝ−k−K , Ŝk+K
.
(3.249)
≥
2
β gk
We can already anticipate that the inequality (3.249) might allow
us to establish an upper bound on the order parameter sz . The idea is
to sum both sides over k ∈ BZ and use the identity
X X X
X
0
e−ik·r eik·r Ŝ r · Ŝ r0
Ŝ k · Ŝ −k =
k∈BZ
k∈BZ r∈Λ r 0 ∈Λ
X
=N
(3.250)
Ŝ r · Ŝ r
r
= N 2 S (S + 1),
to establish an upper bound for the left-hand side of inequality (3.249)
oE 4N 2
X 1 Dn
X 1
−
+
Ŝ−k−K
|sz |2
.
N 2 S (S + 1) ≥
, Ŝk+K
≥
2
β
g
k
k∈BZ
k∈BZ
(3.251)
P
To evaluate the right-hand side, that is, k∈BZ gk−1 , is essentially intractable. Rather, we will establish an appropriate lower bound to this
quantity by finding an upper bound to gk .
Exercise 2.4: Show that
X
gk ≤ 4N S (S + 1) k2
r 2 |Jr | + 2N |hsz |
(3.252)
r∈Λ
and use this result to rewrite the inequality (3.251) as
|sz |2 ≤
where
Θ :=
S (S + 1) β
,
2Θ
1 X
1
,
2
N k∈BZ 2 J k + |h sz |
(3.253a)
(3.253b)
and
J := S (S + 1)
X
r∈Λ
r 2 |Jr |.
(3.253c)
136
3. NON-LINEAR-SIGMA MODELS
In the final step, we are going to the thermodynamic limit, in which
we can replace the summation
Z d
1 X
d k
f (k),
(3.254)
fk →
N k∈BZ
Ω
where Ω is the volume of the BZ. As we are only after an upper bound
for Eq. (3.253), and the integrand of Θ is positive definite, we can
restrict the integration to a ball of radius k0 > 0 that entirely fits in
the BZ. Physically, this is a valid approximation, as the tendency to
long-range order is determined by the contributions at small momenta
only.
Exercise 2.5: Show that within this approximation, one obtains
the following leading expansions for small fields h

2/3
1/3

β |h| × const., d = 1,

|sz | <
(3.255)

β 1/2

√

× const.,
d = 2.
| ln |h||
Equation (3.255) shows that the order parameter sz vanishes in the
limit h → 0 in one and two dimensions. Note that we have implicitly
used the fact that only short-range interactions are permitted, by assuming that the constant J remains finite when the thermodynamic
limit is taken. This is indeed the case if Jr decays faster than |r|−d−2
as the distance |r| tends to infinity.
3.8.2. Quantum spin coherent states and the O(3) QNLσM.
Introduction. We are after a path-integral representation of a quantum spin Hamiltonian in terms of the coherent-state representation
of the irreducible representations of the group SU (2). The lack of a
version of Wick theorem for quantum spin degrees of freedom complicates enormously perturbation theory. One way out is to represent
the spin algebra in terms of fermions or bosons. However, the price
paid is the enlargement of the Hilbert space, i.e., the introduction of
gauge degrees of freedom. Another way out is to work with a basis of
the Hilbert space that mimics the classical limit of the quantum spin
system as closely as possible. Our first goal is to show how the latter approach can be achieved. Our second goal is to derive the O(3)
Quantum Non-Linear-σ-Model (QNLσM) representation of a quantum
antiferromagnet whose classical ground state supports colinear antiferromagnetic long-range order.
The O(3) QNLσM is a long-wavelength and low-energy effective
field theory that is believed to capture qualitatively the properties of
a quantum antiferromagnet at very low temperatures, provided the
quantum ground state supports colinear antiferromagnetic correlations
on the scale of few lattice spacings.
3.8. PROBLEMS
137
Although quantum antiferromagnets have a long and illustrious history dating back to the Bethe solution to the quantum spin-1/2 Heisenberg chain in the early days of quantum mechanics, it is only through
the work of Haldane in the early 80’s that the connection between the
QNLσM and quantum spin Hamiltonians on bipartite lattice was established in Ref. [35]. The insights brought by this connection were
revolutionary.
It had been believed for one generation, based on the Bethe Ansatz
solution to the spin-1/2 antiferromagnetic Heisenberg chain and numerical simulations thereof, that all quantum spin S = 1/2, 1, 3/2, · · ·
antiferromagnetic Heisenberg chains were characterized by quasi-longrange order in their ground states and that the excitation spectrum
above these ground states were gapless.
Haldane deduced from his mapping of the quantum spin-S antiferromagnetic Heisenberg chain to the O(3) QNLσM that the case of
integer spin chains differs qualitatively from the case of half-odd-integer
spin chains. To the contrary of the half-odd-integer case, the integer
case was conjectured by Haldane to display a ground state without
quasi-long-range order for the spin degrees of freedom and supporting
a gap to all spin excitations.
The prediction of Haldane was initially controversial as it relied on a
mapping to the O(3) QNLσM that is approximate with an error of order
1/S. As we shall see, this is a semi-classical approximation, one reason
for which it is surprising that this approximation captures a quantum
manifestation as dramatic as the distinction between integer and halfodd-integer spins. Exactly soluble models, numerical simulations, and
the discovery of quasi-one dimensional quantum antiferromagnets in
“real life” have vindicated Haldane since then.
We shall consider the quantum lattice model
ĤS,H [Ŝ] := −
X
1 X
H i · Ŝ i .
Jij Ŝ i · Ŝ j −
2 i,j∈Λ
i∈Λ
(3.256a)
Here, the sites i and j belong to a lattice Λ. There is a classical local
magnetic field H i that couples to the local spin operator Ŝ i through
the Zeeman term. The three components Ŝia with a = 1, 2, 3 ≡ x, y, z
of the local spin operator Ŝ i satisfy the commutation relations
h
i
Ŝia , Ŝjb = iδij abc Ŝjc ,
a, b, c = 1, 2, 3,
i, j ∈ Λ.
(3.256b)
We fix the irreducible representation of this algebra defined by the
Casimir operator taking the value
2
Ŝ i = S(S + 1),
i ∈ Λ.
(3.256c)
138
3. NON-LINEAR-SIGMA MODELS
Here, S is either a positive half odd integer or a positive integer. The
Heisenberg exchange couplings obey
Jij = Jji
(3.256d)
for any pair of sites i, j ∈ Λ. The Heisenberg exchange interaction
Jij Ŝ i · Ŝ j is the simplest interaction between two quantum spins that
is invariant under a global SU (2) rotation of all the quantum spins. The
Zeeman term breaks the local SU (2) symmetry in spin space down to
the subgroup U (1) of local rotations around the direction in spin space
corresponding to H i .
We shall limit ourselves to the case when the lattice Λ is assumed to
be bipartite and made of N 1 sites. More precisely, the lattice will
be taken to be a macroscopically large subset of the hypercubic lattice
Zd with lattice spacing a. We shall also assume that the couplings Jij
are only non-vanishing if i belongs to one sublattice, while j belongs
to the other sublattice, in which case they are taken negative Jij < 0.
The latter condition (Jij ≤ 0) defines a quantum spin-S Heisenberg
antiferromagnet, while the former condition insures the absence of geometric frustration. The interaction |Jij | Ŝ i · Ŝ j favors the singlet state
for the two-site problem. If the degrees of freedom Ŝ i and Ŝ j were not
operator-valued vectors but classical vectors in R3 of a fixed magnitude, the classical interaction |Jij | Ŝ i · Ŝ j would favor an antiparallel
alignment of these classical vectors. If so, the classical configuration
that minimizes the classical energy when H i = 0 for all i ∈ Λ of the
classical counterpart to Eq. (3.256a) has all spins pointing along one
direction on one sublattice and all spins pointing in the opposite direction on the other sublattice for any d ≥ 1. This is the so-called
Néel colinear antiferromagnetic state. The fundamental question to be
addressed at the quantum level when H i = 0 for all i ∈ Λ is what is
the fate of the classical long-range order in the quantum ground state
as a result of quantum fluctuations.
The two-site problem. Exercise 1.1:
(a) Compute exactly the partition function
Zβ,S,H := tr e−β ĤS,H [Ŝ]
(3.257)
with ĤS,H [Ŝ] given by Eq. (3.256a), when the lattice is made
of two sites and Jij = J.
(b) Comment on the difference when J > 0 and J < 0.
Semi-classical limit. Exercise 2.1:
(a) Perform the rescaling
Ŝ i =: S ŝi
(3.258)
3.8. PROBLEMS
139
and deduce the algebra obeyed by the operators ŝi . From the
algebra obeyed by the ŝi justify why the limit S → ∞ can be
interpreted as the semi-classical limit.
(b) We now consider the classical counterpart to the Hamiltonian (3.256a) obtained by replacing the operator-valued Ŝ i by
classical unit vectors N i from R3 .
– Assume that H i = 0 for all i ∈ Λ, that the lattice is the
square lattice, and that Jij is non-vanishing and negative
on nearest-neighboring sites only. Construct the classical
manifold of configurations that minimizes the classical energy (3.256a).
– Assume that H i = 0 for all i ∈ Λ, that the lattice is the
triangular lattice, and that Jij is non-vanishing and negative on nearest-neighboring sites only. Construct the classical manifold of configurations that minimizes the classical energy (3.256a).
Single quantum spin coherent states. In this warm-up we begin with
a single quantum spin Ŝ and will therefore drop the site index. We
shall denote the quantum Hamiltonian of the single quantum spin Ŝ by
Ĥ[Ŝ]. Here, if the operator-valued argument Ŝ is replaced by a classical
vector in R3 and if we then drop the hat over Ĥ, then H should be
thought of as some smooth scalar-valued function. The Hilbert space
is spanned by the (2S + 1) orthonormal states of the quantum spin-S
irreducible representation of the group SU (2)
|S, mi ,
m = −S, −S + 1, · · · , S − 1, S,
(3.259a)
where S takes integer or half-odd-integer values, and
Ŝ z |S, mi = m |S, mi ,
2
Ŝ |S, mi = S(S + 1) |S, mi .
(3.259b)
The (2S + 1) states in Eq. (3.259a) can be constructed with the help
of the ladder operators
Ŝ + := Ŝ x + iŜ y ,
Ŝ − = Ŝ x − iŜ y .
(3.259c)
We call the state |S, Si the highest weight state. Observe that
Ŝ + |S, Si = 0,
(3.259d)
so that we can interpret the highest weight state as the counterpart to
the vacuum state for the boson annihilation operators in Eq. (2.10b).
Successive action of Ŝ − on the highest weight state |S, Si yields all
the states in Eq. (3.259a), very much in the same way as application of all powers of the boson creation operators on the bosonic vacuum generates a basis of the bosonic Fock space (2.10a). Notice that
Ŝ − |S, −Si = 0 implies that we could equally have chosen |S, −Si = 0
140
3. NON-LINEAR-SIGMA MODELS
(a)
N0
N0
(b)
N2
N1
N2
N1
Figure 4. A spherical triangle with vertices N 0 , N 1 ,
and N 2 . The definition of the area of this spherical triangle is ambiguous. It can be interpreted either as the
inner area (a) or as the outer area (b).
as the highest weight state. In this basis, the resolution of the identity
is the representation
X
1=
|S, mi hS, m|
(3.259e)
m=−S,−S+1,··· ,S−1,S
of the unit (2S + 1) × (2S + 1) matrix 1.
Exercise 3.1: Using the commutation relations Eq. (3.256b), compute the commutator of Ŝ + with Ŝ − .
For the derivation of the path integral, we seek a set of states for
which the matrix elements of Ŝ are “as classical as may be”. For this
purpose, the states in Eq. (3.259a) are not convenient. Instead, we
shall use the so-called spin coherent states. These are an infinite and
over-complete set of states |N i, labeled by the points N on the surface
of the unit sphere in R3 ,
N T := (sin θ cos φ, sin θ sin φ, cos θ),
(3.260a)
that obey the following properties,
iS Φ(N 0 ,N 1 ,N 2 )
hN 1 | N 2 i = e
1 + N1 · N2
2
hN | Ŝ |N i = SN ,
Z
2S + 1
1=
dµ(N ) |N i hN | .
4π
S
, (3.260b)
(3.260c)
(3.260d)
Equation (3.260b) implies that the states {|N i} are not orthogonal for
any finite S. The phase of the overlap between states |N 1 i and |N 2 i
has a geometrical origin as Φ(N 0 , N 1 , N 2 ) is the oriented area of the
spherical triangle with vertices N 1 , N 2 , and some arbitrarily chosen
reference unit vector N 0 (see Fig. 4).
Exercise 3.2:
(a) Explain why the ambiguity in defining Φ(N 0 , N 1 , N 2 ) does
not matter in Eq. (3.260b).
(b) What is the value of the overlap (3.260b) in the limit S → ∞?
3.8. PROBLEMS
141
Equation (3.260c) defines “as classical as may be”. Equation (3.260d)
is the resolution of unity, where it is understood that the integral is over
the unit sphere,
Z
Z
dµ(N ) ≡ d3 N δ(N 2 − 1).
(3.261)
R3
Equations (3.260b), (3.260c), and (3.260d) are the only ingredients
that we will need to derive the path integral representation of a quantum spin-S Hamiltonian. We shall now construct explicitly coherent
states satisfying Eqs. (3.260).
Given the unit vector (3.260a), we define the state |N i by
|N i := eζ Ŝ
+ −ζ ∗
Ŝ −
|S, Si,
(3.262a)
where
θ
ζ := − e−iφ .
(3.262b)
2
In this representation the highest weight state |S, Si corresponds to the
north pole of the unit sphere N T = (0, 0, 1) at which θ = 0. We now
specialize to the case of a spin-1/2 for which we shall prove explicitly
Eqs. (3.260). We thus choose the representation, in units for which
~ = 1,
1
Ŝ = σ
(3.263)
2
of the spin operators in terms of the 2 × 2 Pauli matrices.
Exercise 3.3:
(a) Show that for S = 1/2 the spin coherent states Eq. (3.262)
can be written as
θ
θ
|N i = cos |1/2, 1/2i + eiφ sin |1/2, −1/2i ,
(3.264)
2
2
by making use of the identity (σ0 is the 2 × 2 unit matrix)
θ
θ
− in · σ sin
(3.265)
2
2
where n ∈ R3 is a unit vector.
(b) With the help of Eq. (3.264), show that the spin-1/2 coherent
states satisfy the completeness relation
X
σ0 =
|1/2, mi h1/2, m|
i
e− 2 θ n·σ = σ0 cos
m=− 12 , 21
Z
1
=
dµ(N ) |N i hN |
2π
Z
2S + 1
=
dµ(N ) |N i hN |
.
4π
S=1/2
(3.266)
142
3. NON-LINEAR-SIGMA MODELS
(c) With the help of Eq. (3.264) and of Eq. (3.263), show that
1
hN | Ŝ |N i = S N = N .
(3.267)
2
(d) With the help of Eq. (3.264), show that
1/2
1 + N1 · N2
(i/2) Φ(N 0 ,N 1 ,N 2 )
hN 1 | N 2 i = e
.
(3.268)
2
Here, the three orthonormal vectors N 2 , N 1 , and N 0 define
a Cartesian basis of internal spin-1/2 space R3 with the equatorial plane of the two-sphere S 2 depicted in Fig. 4 spanned
by N 2 and N 1 .
Coherent-state path integral for a single quantum spin. Having defined the spin-coherent states for a single quantum spin, we are now
ready to derive the coherent state path integral for the partition function
Z
2S + 1
−β Ĥ[Ŝ]
Zβ := Tr e
=
dµ(N ) hN | e−β Ĥ[Ŝ] |N i ,
(3.269)
4π
where β is the inverse temperature in units for which the Boltzmann
constant is unity and Ĥ is a linear function of the spin operator Ŝ.
Although we are restricting ourselves to a single quantum spin, the
generalization to many quantum spins is straightforward. As usual, we
break the above exponential into a product of exponentials of infinitesimal time evolution operators
Zβ = lim
M →∞
M
Y
e−∆τi Ĥ[Ŝ] ,
∆τi := β/M,
(3.270)
i=1
and insert the resolution of the identity Eq. (3.260d) between each
exponential.
Exercise 4.1:
(a) With the help of Eq. (3.260c), evaluate the time evolution
during the infinitesimally small “time” ∆τ ,
D
E
N (τ ) e−∆τ Ĥ[Ŝ] N (τ − ∆τ ) ,
(3.271)
where one neglects terms of order (∆τ )2 and higher.
(b) Insert this result into Eq. (3.270) and show that the functional
integral for the partition function (3.269) is given by
Rβ
Z
−SB − dτ H[S N (τ )]
0
Zβ = DN (τ ) e
,
(3.272a)
where
Zβ
SB =
0
dN
dτ N (τ ) (τ ) ,
dτ
(3.272b)
3.8. PROBLEMS
143
and periodic boundary conditions
|N (0)i = |N (β)i
(3.272c)
are used. The real-valued H(S N ) is obtained by replacing every occurrence of Ŝ in the quantum Hamiltonian Ĥ[Ŝ] by SN
and removing the hat above the functional for the quantum
Hamiltonian.
(c) Show that the first term in the argument of the exponential in
Eq. (3.272a), i.e., SB given by Eq. (3.272b), leads to a phase
factor by verifying that SB is pure imaginary.
The term SB is called the Berry phase [36]. It represents the overlap
of the coherent states at infinitesimally separated imaginary times. In
differential geometry, it is interpreted as a “gauge connection”. In
the physics literature, it is interpreted as a gauge field of geometrical
origin, with the geometry being that of the set made of an overcomplete
basis of the Hilbert space that varies adiabatically as a function of a
continuous parameter, here imaginary time. In Eq. (3.272b) the Berry
phase is given in terms of an integral along the closed curve N (τ ) on
the unit sphere. In order to bring the Berry phase into a geometrically
more transparent form, we take advantage of the properties of the spin
coherent states and transform the line integral into a surface integral .
Thereto, we make use of the identity 19
Z1
d Ô
dÔ Ô u
e = du eÔ (1−u)
e ,
(3.273)
dτ
dτ
0
x
d x
which is the generalization of dτ
e = dx
e to an operator Ô that
dτ
does not commute with its derivative dÔ/dτ .
Exercise 4.2:
(a) With the help of Eqs. (3.260b), (3.260c), (3.262b), and (3.273),
show that
Zβ
Z1
h
i
SB = −S dτ
du ζ ∂τ N + (τ, u) − ζ ∗ ∂τ N − (τ, u) , (3.274a)
0
0
where
N T (τ, u) = sin (u θ(τ )) cos φ(τ ), sin (u θ(τ )) sin φ(τ ), cos (u θ(τ )) ,
(3.274b)
and
N + := N x + iN y ,
N − := N x − iN y .
(3.274c)
(b) Show that
N ∂u N − −N − ∂u N z = −2ζ,
z
19
N z ∂u N + −N + ∂u N z = −2ζ ∗ . (3.275)
Alternatively, one could use Stoke theorem (see [36]).
144
3. NON-LINEAR-SIGMA MODELS
N (⌧ +
N (⌧, 1)
⌧, 1)
Figure 5. N (τ, u) parametrizes the area on the unit
sphere bounded by N (τ, 1).
(c) Conclude that the Berry phase can be written as
Zβ
SB = iS
Z1
du N ·
dτ
0
∂N ∂N
∧
∂u
∂τ
.
(3.276)
0
From Eq. (3.274b) we infer that N (τ, u) moves with u along the
great circle between the north pole and the physical value N (τ, 1) (see
Fig. 5). Hence the integral in Eq. (3.276) is simply the oriented area on
the unit sphere bounded by N (τ ). The value of this area depends on
the fact that N (τ, 0) corresponds to the north pole. This was a gauge
choice. By making a different choice of phase for the coherent states,
the point N (τ, 0) can be chosen anywhere on the sphere. However,
eSB in the coherent state path integral (3.272) is independent on the
location of N (τ, 0) up to a factor ei4π S . Since 2S is an integer, this
factor leaves the Boltzmann weight entering the path integral (3.272a)
unchanged.
Quantum antiferromagnets. We are going to generalize the pathintegral representation for a single-spin Hamiltonian to derive the O(3)
Quantum Non-Linear Sigma Model (QNLσM) for a quantum antiferromagnet.
We assume the nearest-neighbor Heisenberg exchange couplings
Jij =


−J < 0, if i and j are nearest-neighbor sites of Λ,

0,
(3.277)
otherwise.
We start from the representation of the partition function
ZAF := Tr e−β ĤS,H [Ŝ]
(3.278)
3.8. PROBLEMS
145
for the Hamiltonian (3.256a) as the path integral over SU (2)-coherent
states
"
#
Z
Y
ZAF = D[N ]
δ N 2i − 1 e−SB −SAF ,
(3.279a)
i∈Λ
Zβ
SB := iS
dτ
0
Zβ
SAF :=
0
Z1
du
0
X
i∈Λ
Ni ·
∂N i ∂N i
∧
∂u
∂τ

dτ S 2 J
,
(3.279b)

X
hiji
Ni · Nj − S
X
H i · N i  (, 3.279c)
i∈Λ
which is a straightforward generalization of Eq. (3.272) and Eq. (3.276)
to many spins. The local constraint N 2i = 1 for all i ∈ Λ has been
enforced by a local delta function in order to trade the local integrals
over the unit sphere S 2 for local integrals over R3 . The first sum in
(3.279c) is over all directed nearest-neighbor pairs hiji on the bipartite
lattice Λ. Periodic boundary conditions are imposed across the lattice.
We now assume that the lattice Λ is bipartite as was the case in
Eq. (2.150), from which we borrow the convention for the notation
of the two sublattices. It is known that, at zero temperature and in
the classical limit S → ∞, the ground state of a nearest-neighbor
Heisenberg antiferromagnet on a bipartite lattice has spins oriented in
opposite directions on the two sublattices of the bipartite lattice. This
classical ground state is called the Néel ordered state. We assume that
the partition function (3.279) is close to an antiferromagnetic fixed
point. At this fixed point, the ground state breaks the translation
symmetry of the lattice Λ down to the translation symmetry of any
one of its sublattices. We aim at deriving an effective action for the
Euclidean action S ≡ SB + SAF , which is valid in the long-wavelength
and low-energy limit.
From now on, we choose the bipartite lattice Λ to be hypercubic
and spanned by the orthonormal (Cartesian) unit vectors e1 , · · · , ed .
The volume of Λ is Ld = N ad where L is the length of an edge of
the hypercube measured in units of the lattice spacing a and N is the
number of sites in Λ. We are going to work with a unit cell of the
hypercubic lattice Λ with two nonequivalent sites per unit cell.
Exercise 5.1:
(a) Choose a site r i on sublattice ΛA . The site r i has 2×d nearestneighbors r i ± a eµ with µ = 1, 2, 3 sitting on sublattice ΛB .
The number 2×d is the coordination number of the hypercubic
lattice. How many next-nearest-neighbor sites has r i in d = 1,
d = 2, and d = 3 and what are their Cartesian coordinates
relative to r i ?
(b) What type of lattice is ΛA in d = 1, d = 2, and d = 3?
146
3. NON-LINEAR-SIGMA MODELS
Figure 6. Two-site unit cell centered about a site of
sublattice ΛA for d = 2 and d = 3.
(c) Define a unit cell to be hypercubic with the edge-length 2a and
volume (2a)d . Place the corners of this unit cell on sublattice
ΛB , say. This unit cell fills Rd by translations built from linear
superpositions with integer-valued coefficients of the Cartesian
unit vectors 2a eµ with µ = 1, · · · , d. How many sites from
sublattice ΛB sit in the interior and on the vertices, edges,
and faces of this unit cell in d = 1, d = 2, and d = 3? How
many sites from sublattice ΛA sit in the interior and on the
vertices, edges, and faces of this unit cell in d = 1, d = 2,
and d = 3? Deduce from these numbers that the unit cell
contains two, four, and eight nonequivalent sites in d = 1,
d = 2, and d = 3, respectively. The same conclusion follows
from (2a)d /2d = ad .
(d) Define the following unit cells in d = 1, 2, 3, · · · . Draw lines
connecting r i ∈ ΛA to all its next-nearest-neighbor sites in
Λ. For any of these connecting lines, draw the hypersurface
of dimension d − 1 normal to this connecting line in such a
way that this hypersurface intersects the line at its mid-point.
The resulting volume bounded by these (d − 1)-dimensional
hypersurface is the unit cell. It is an example of a geometrical
object called a polytope. How many vertices, edges, and faces
characterizes this example of a polytope in d = 1, d = 2, and
d = 3? Where are the nearest-neighbor sites to r i ∈ ΛA about
which this polytope is centered in d = 1, d = 2, and d = 3?
Show that this unit cell contains two nonequivalent sites for
any d = 1, 2, 3, · · · . Show that the volume of this unit cell is
2 × ad for any d = 1, 2, 3, · · · . Filling Rd with this unit cell
requires both translations and rotations. This will be the unit
cell with two nonequivalent sites, shown in Fig. 6 for d = 2 and
d = 3, that we are going to use to construct the continuum
limit.
3.8. PROBLEMS
147
For any site r i ∈ Rd from sublattice ΛA , we make the Ansatz
a
r i ∼ x ∈ Rd ,
N ri ∼ +n(x) + L(x).
(3.280a)
S
For the d sites r j ≡ r i + a eµ from sublattice ΛB that are directed
nearest-neighbors of site r i , we make the Ansatz
a
N ri +a eµ ∼ −n(x + a eµ ) + L(x + a eµ ),
µ = 1, · · · , d, (3.280b)
S
while we make the Ansatz
a
N ri −a eµ ∼ −n(x − a eµ ) + L(x − a eµ ),
µ = 1, · · · , d, (3.280c)
S
for the remaining d sites r i − a eµ from sublattice ΛB that are not
directed nearest-neighbors to site r i . Similarly, we make the Ansatz,
H ri ∼ +hs (x) + hu (x),
H ri ±a eµ ∼ −hs (x ± a eµ ) + hu (x ± a eµ ),
(3.280d)
with µ = 1, · · · , d for the magnetic field. Observe that the two vertices
r i + a eµ and r i − a eµ are a distance 2 × a along the Cartesian axis eµ
apart. The smooth vector fields n, L, hs , and hu , that assign to the
continuous position x ∈ Rd the vectors n(x), L(x), hs (x), and hu (x)
from the target space R3 , are assumed to vary slowly on the scale of
the lattice spacing a. We impose the constraints
|n(x)|2 = 1,
n(x) · L (x) = 0,
∀x ∈ Rd ,
(3.280e)
Constraints (3.280e) imply that [n(x) ± a L(x)/S]2 = 1 hold up to
first order in (a/S) |L(xi )|, an approximation that becomes exact in
the classical limit S → ∞.
Exercise 5.2:
(a) Show that
d
XX
2
HS [S N ] = + S J
N ri · N ri +a eµ + N ri −a eµ
i∈ΛA µ=1
−S
X
(3.281)
H ri · N ri + H ri +a e1 · N ri +a e1 .
i∈ΛA
(b) Show that the continuum limit of Eq. (3.281) with the Ansatz (3.280)
is (the summation convention over the repeated index µ is assumed)
Ld
HS [S N ] ∼ − S 2 J d d
a
Z d h
i
2
d x
2
2
2 2
S
J
a
∂
n
+
2d
J
a
L
(x)
+
µ
2ad
Z d
d x
−2×
(S hs · n + a hu · L) (x),
2ad
to leading order in a gradient expansion.
(3.282)
148
3. NON-LINEAR-SIGMA MODELS
The constant term in Eq. (3.282) is the exchange energy S 2 J per
directed nearest-neighbor bond times the number of directed nearestneighbor bonds d × Ld /ad on a lattice Λ made of Ld /ad sites. It only
leads to a change of the normalization of the partition function (3.279a).
Therefore, it can be dropped.
To complete the expression for the coherent-state path integral of
the quantum antiferromagnet in the continuum limit, we need to express the Berry phase SB in terms of the staggered and uniform fields.
3.8. PROBLEMS
149
Exercise 5.3:
(a) The Ansatz (3.280) applied to Eq. (3.279b) yields the expansion
SB ∼ SB0 + SB00 + SB000 + SB0000 ,
(3.283)
0
00
where SB is of first order in S, SB is of zeroth order in S, SB000
is of order S −1 , and SB0000 is of order S −2 . Show that SB00 is
" Z d Zβ
Z1
d x
∂n ∂L
00
∧
dτ
du n ·
SB ∼ + i 2 × a
2ad
∂u ∂τ
0
0
(3.284)
#
∂L ∂n
∂n ∂n
+n·
∧
+L·
∧
.
∂u ∂τ
∂u ∂τ
(b) Using the fact that the vector ∂n
∧ ∂n
∈ R3 is directed along
∂u
∂τ
n and with the help of Eq. (3.280e), show that Eq. (3.284) can
be expressed as the difference of two total derivatives,
( Z d Zβ
Z1
d
x
∂
∂n
00
SB ∼ + i 2 × a
dτ
du
∧L
n·
2ad
∂τ
∂u
0
0
(3.285)
)
∂
∂n
−
n·
∧L
.
∂u
∂τ
(c) Using the periodicity of the fields n and L in τ and the fact
that L(τ, 0) = 0 for all τ , show that Eq. (3.285) simplifies to
Z d Zβ
d x
∂n
00
(3.286)
SB ∼ − i 2 × a
dτ L · n ∧
.
2ad
∂τ
0
SB000
SB0000
We shall ignore
and
since they are subleading in the expansion
in powers of 1/S.
After combining Eqs. (3.282), (3.286), and (3.279), we obtain the
path integral for the partition function of the quantum spin-S antiferromagnet
Z
0
0
ZAF ∼ D[n, L] δ n2 − 1 δ (n · L) e−SB −SAF ,
0
SAF
1
=
2
Zβ
Z
dτ
h
i
2
dd x S 2 J a2−d ∂µ n + 2 d J a2−d L2
0
Zβ
−
Z
dτ
d
d x Sa
−d
1−d
n · hs + i a
∂n
L· n∧
− i hu .
∂τ
0
(3.287)
150
3. NON-LINEAR-SIGMA MODELS
Exercise 5.4:
(a) Compute the integration over L by completing the square and
show that ZAF simplifies to
Z
0
00
ZAF ∼ D[n] δ n2 − 1 e−SB −SAF ,
00
SAF
1
∼
2
Zβ
2
a−d
2
2
2−d
∂µ n +
(∂ n − i hu ∧ n)
d x S Ja
2dJ τ
Z
d
dτ
0
Zβ
−
Z
dτ
dd x S a−d n · hs .
0
(3.288)
(b) The Euclidean action SB0 in Eq. (3.288) arises from (i) inserting
the Ansatz (3.280) into the Berry phase (3.279b), (ii) selecting
the term of order S, (iii) and performing a gradient expansion.
Carrying this program is subtle, because step (ii) contains a
term that oscillates in sign between the two sublattices. Before
evaluating SB0 in one-dimensional position space, show that
step (ii) gives
SB0
Z
∼ + iS
dd x
2ad
Zβ
Z1
du n(x, τ, u) ·
dτ
0
"
0
− n(x + a e1 , τ, u) ·
∂n ∂n
∧
∂u ∂τ
∂n ∂n
∧
(x, τ, u)
∂u ∂τ
#
(x + a e1 , τ, u) .
(3.289)
Evaluation of SB0 in one dimension. Exercise 6.1:
(a) Take advantage of the periodic boundary conditions and assume an even number of sites in one dimension to show that
the expansion of SB0 to zeroth-order in (2a) L/S is approximately given by
SB0
S
∼ −i
2
Zβ
Z
dτ n ·
dx
∂n ∂n
∧
∂x ∂τ
.
(3.290)
0
Hint: Draw the two areas on the unit sphere of Fig. 5, one
associated to the Berry phase arising from n(x, τ, u) and one
associated to the Berry phase arising from n(x + a e1 , τ, u),
assuming that their boundaries n(x, τ, 1) and n(x+a e1 , τ, 1),
respectively, are infinitesimally far apart.
3.8. PROBLEMS
151
(b) Show that the contribution of the term SB0 to the partition
function ZAF is given by
e+i2π S Q ,
(3.291)
where
1
Q :=
4π
Zβ
Z
dτ n ·
dx
∂n ∂n
∧
∂x ∂τ
,
(3.292)
0
and discuss the cases S integer and S half-odd-integer. Convince yourself that Q is an integer. The integral Q is an example of a topological invariant from algebraic topology.
The fact that the long-wavelength and low-energy properties of the
one-dimensional quantum spin chain depend, through the Berry phase,
in a dramatic fashion on S being an integer or a half-odd integer was
conjectured by Haldane. [35] The path integral (3.288) can be interpreted as an integration over all possible spin fluctuations. Those fluctuations that do not depend on τ are the classical fluctuations. Those
fluctuations that depend on τ are the quantum fluctuations. For the
00
is such that
case of S integer, the effective action Seff = SB0 + SAF
−Seff
e
is always positive as a result of the Berry phase being inoperative. Hence, all the quantum fluctuations contribute with the same
sign. However, for the case of S a half-odd integer, the Berry phase
is operative and quantum fluctuations are suppressed owing to the destructive interference caused by the alternating sign of (3.291). For S
integer, there is a finite gap in the spin excitation spectrum (Haldane
gap). For S half-odd integer, the ground state is quasi-long-range ordered (the best the system can do short of long-range order in view of
the Mermin-Wagner theorem) and excitations are gapless.
Conjecture for d = 2-dimensional quantum antiferromagnets. Exercise 7.1: Consider two decoupled spin-1/2 antiferromagnetic chains,
each of which is described by the QNLσM in one-dimensional position
space with the topological term (3.290). Assume that the two chains are
weakly coupled by an antiferromagnetic Heisenberg exchange coupling
along the “rungs” of a “ladder” in such a way that the long-wavelength
and low-energy effective action for the ladder is, to zeroth order in the
rung coupling, two copies of the QNLσM in one-dimensional position
space and at zero temperature. Argue how should one choose the relative sign of the topological terms along each chain for an infinitesimal
antiferromagnetic Heisenberg exchange coupling along the rungs. Decide from this thought experiment whether n weakly coupled spin-1/2
antiferromagnetic chains have or do not have a topological term in their
long-wavelength and low-energy effective action.
152
3. NON-LINEAR-SIGMA MODELS
Exercise 7.2: Assume that the QNLσM that captures the physics
of n weakly coupled spin-1/2 antiferromagnetic ladders at zero temperature can be brought to the form of the classical two-dimensional O(3)
NLσM studied in chapter 3 with the effective spin 2n × S = n. What is
the dependence on n of the correlation length derived in chapter 3? If
the limit n → ∞ was taken to define a two-dimensional spin-1/2 antiferromagnetic Heisenberg model on the square lattice with anisotropic
exchange couplings, would the ground state be separated from the excitations by a gap or would it be gapless? Is your conjectured excitation
spectrum consistent with taking the two-dimensional limit by using
(2n + 1) weakly coupled spin-1/2 antiferromagnetic chains instead of n
ladders?
3.8.3. Classical O(N > 2) NLσM: One-loop RG using the
Berezinskii-Blank parametrization of spin waves.
Introduction. Our goal is to perform a one-loop RG analysis on the
QNLσM defined by Eq. (3.288) in the absence of the topological term
SB0 . The RG technique that we are going to use relies on a parametrization of spin waves introduced by Berezinskii and Blank in Ref. [37].
As a warm up, we are going to perform the one-loop RG analysis
of the classical limit of the O(N ) QNLσM using the Berezinskii-Blank
parametrization of spin waves as was done by Polyakov in Ref. [32].
Definitions. The classical O(N ) NLσM is defined by the partition
function
Z
R d
D[n] δ(n2 − 1) e− d r L ,
(3.293a)
Z :=
RN
L :=
2
1
∂µ n .
2g
(3.293b)
Summation over repeated Greek indices (µ = 1, · · · , d) is assumed
throughout. The coupling constant g has dimension
[g] = lengthd−2 .
(3.293c)
Observe that g is dimensionless if and only if d = 2. The measure
of the NLσM is defined to be D[n] δ(n2 − 1) where n : Rd → RN ,
r → n(r), is a real-valued dimensionless vector field. What makes
the NLσM non-trivial is the constraint on n that is implemented by
δ(n2 − 1).
In order to exploit the techniques of renormalization group (RG),
e.g., for the computation of the beta function, we shall use Berezinskii
and Blank’s parametrization of the vector field n(r) (see Refs. [37] and
[32]),
q
N
−1
X
n(r) := 1 − φ2 (r) n0 (r) +
φa (r) ea (r),
(3.294)
a=1
3.8. PROBLEMS
153
where n0 (r), e1 (r), · · · , eN −1 (r) is an orthonormal basis of RN for
any given r ∈ Rd . It is assumed that n0 (r) deviates only slightly
from a given fixed coordinate axis, e0 say, for all r ∈ Rd . That is,
we want to describe the effect of spin fluctuations at finite g in an
antiferromagnetically ordered system when g = 0. Therefore, we can
regard n0 (r) as a slowly varying vector, with Fourier wave vectors in
e say. The “fast” degrees of freedom are contained
the range |p| < Λ,
in φ(r) = (φ1 (r), · · · , φN −1 (r)), which have wave vectors in the range
e < |p| < Λ. Note that there is an arbitrariness in choosing the
Λ
vectors e1 (r), · · · , eN −1 (r). At each point r ∈ Rd the orthonormal
basis e1 (r), · · · , eN −1 (r) is only defined up to a O(N − 1) rotation
[O(N − 1) gauge symmetry]. One way to eliminate this arbitrariness
is by choosing the “Coulomb gauge”
a = 1, . . . , N − 1,
(3.295)
∂µ ea (r) · ∂µ n0 (r) = 0,
which is a first-order differential equation in the ea ’s.
Exercise 1.1:
(a) What is problematic, if the unit length vector n0 only has
e We
non-zero wave vectors in the restricted range |p| < Λ?
will ignore this issue
in the sequel.
0
(b) Show that ∂µ n (r) is orthogonal to n0 (r) for all r ∈ Rd .
Conclude that there exist N −1 expansion coefficients A01µ (r), A02µ (r), · · · , A0(N −1)µ (r),
such that
∂µ n
0
(r) =
N
−1
X
A0bµ (r) eb (r).
(3.296)
b=1
(c) Show that ∂µ ea (r) is orthogonal to ea (r) for a = 1, · · · , N −
1 and all r ∈ Rd . Infer that
X
∂µ ea (r) =
Aabµ (r) eb (r) − A0aµ (r) n0 (r),
a = 1, · · · , N − 1,
b6=a
(3.297)
with the expansion coefficients Aa1µ (r), · · · , Aa(N −1)µ (r), where
Aaaµ (r) = 0. In particular, verify that A0aµ (r) in Eq. (3.297)
is indeed the same field as the expansion coefficient A0aµ (r) in
Eq. (3.296).
(d) Show that Aabµ is antisymmetric, i.e.,
Aabµ = −Abaµ ,
a, b = 1, · · · , N − 1.
(3.298)
Verify that
A0aµ = ea · ∂µ n0 ,
Aabµ = eb · ∂µ ea .
(3.299)
(e) With the help of Eqs. (3.296) and (3.297), show that substituting the Berezinskii-Blank parametrization (3.294) into the
154
3. NON-LINEAR-SIGMA MODELS
Lagrangian (3.293b) of the NLσM gives
i2 i2 h
1 h
2 1/2 0
2 1/2
b
0
,
Aaµ + ∂µ φa + φb Aaµ
− φa Aaµ + 1 − φ
L =
∂µ 1 − φ
2g
(3.300)
where repeated indices are to be summed over. In order to
compute the beta function in “a quick and dirty way” we shall
only retain fast fluctuations up to second order, that is we
need to isolate the term that is quadratic in the φa ’s.
(f) Isolate in Eq. (3.300) the term that is quadratic in the φa ’s.
Show that this yields 20
2
1h
L=
∂µ φa + Abaµ φb + A0aµ A0bµ φa φb − φ2 δab + A0aµ A0aµ
2g
i
0
b
+ 2Aaµ ∂µ φa + φb Aaµ .
(3.301)
Define the Lagrangian densities
L0 := Lslow
+ Lfast
+ Lint
0
0
0 ,
1 0 0
A A ,
Lslow
:=
0
2g aµ aµ
2
(3.302)
1
Lfast
:=
∂µ φa ,
0
2g
1 0 0
Aaµ Abµ φa φb − φ2 δab .
Lint
0 :=
2g
(g) With the help of Eq. (3.299), show that the last term in
Eq. (3.301) vanishes if we work in the Coulomb gauge given
by Eq. (3.295).
(h) Show that the gauge choice Eq. (3.295) is consistent with orthonormality of the ea ’s.
We shall assume that of all the terms generated by a momentum
e < |p| < Λ the single most relevant
shell integration of φ in the shell Λ
one can be absorbed by renormalization of the coupling constant g, i.e.,
the renormalized Lagrangian takes the form
2
1
Le =
∂ µ n0 + · · ·
2e
g
2
1
=
A0aµ + · · · .
(3.303)
2e
g
Needed is the change of g induced by the momentum shell integration
as was done by Polyakov in 1975 in his pioneering work. [32] In doing
so we shall ignore the renormalization associated to the gauge fields as
the gauge invariance of the theory dictates that these effects are less
20
Terms of order φa ∂µ φb φc can be dropped.
3.8. PROBLEMS
155
important than the renormalization of g. We shall assume that the
renormalization of g comes solely from the second term in Eq. (3.301).
Exercise 1.2:
(a) A kinetic energy term for the non-Abelian gauge fields Aabµ ,
a, b = 1, · · · , N − 1, will be included in the · · · of Eq. (3.303)
1
F F .
(3.304)
ẽ µν µν
On the basis of symmetry alone, write down the explicit form
of Fµν . What is the naive scaling dimension of the effective
“charge” ẽ?
(b) Substitute in the partition function (3.293) the original Lagrangian by the one defined in Eq. (3.302). Expand the exponential of Lint
0 and integrate out the “fast” fields. If so show
that
Z
RΛ
e d
1
0
0
Z≈
D[n0 ] e− 2g d p Aaµ (+p)·Aaµ (−p)
n20 =1

e
Λ
Z
e
f (Λ, Λ)


dd k A0aµ (+k) A0bµ (−k) ,
× 1 − ab
2g

(3.305a)
where, for d = 2,
N −2
Λ
(3.305b)
g ln δab .
e
2π
Λ
Conclude that, in d = 2, the integration over the fast modes
with Fourier components restricted to the momentum shell
e < |p| < Λ in Eq. (3.305a) yields the renormalization of the
Λ
coupling constant
e =−
fab (Λ, Λ)
1
1 (2 − N )
Λ
≈ +
log .
e
ge g
2π
Λ
(3.306)
(c) Generalize Eqs. (3.305b) and (3.306) to the case of d > 2.
Hint: Account for the fact that g is dimensionful when d > 2.
(d) Show for d = 2 that after repeating the momentum shell integration a sufficient number of times the coupling constant for
momentum cut-off q is given by
g
ge(q) =
,
(3.307)
1 − g (N − 2) (1/2π) ln(Λ/q)
and compute the beta function
β(e
g) =
d ge(q)
.
d ln(Λ/q)
How does this result compare with Eq. (3.183b)?
(3.308)
156
3. NON-LINEAR-SIGMA MODELS
(e) Using the parametrization (3.294) and for d = 2, show that
the correlator
D(r; Λ) = hn(0) · n(r)i ,
r ≡ |r|,
(3.309)
at the renormalization point Λ, i.e., defined by imposing the
upper momentum cutoff Λ, and the correlator D at the renore are approximately related by
malization point Λ
N −1
Λ
e
D(r, Λ) ≈
1−
g(Λ) ln
D(r, Λ)
e
2π
Λ
e
≡ Z D(r, Λ).
(3.310)
The function Z is called the wave-function renormalization.
Compute the anomalous dimension γ(e
g)
1
dZ
.
e
2 d ln(Λ/Λ)
γ(e
g ) :=
(3.311)
How does this result compare with Eq. (3.183c)?
3.8.4. O(N > 2) QNLσM: Large-N expansion.
Introduction. It is time to study the O(N ) QNLσM. We do this in
the limit N → ∞ to begin with. [38] The one-loop RG analysis is done
in section 3.8.5.
Definitions. The O(N ) QNLσM is defined by the partition function
Z
− 2 1c g
2
D[n] δ(n − 1) e
ZΩ,g [J ] :=
Rβ
dτ
0
RL
dd r [(∂τ n)2 +c2 (∇n)2 −2 c2 g n·J ]
0
.
RN
(3.312)
Use the notation x0 ≡ c τ and x ≡ r,
Z
2
− 21g
D[n] δ(n − 1) e
ZΩ,g [J ] =
cβ
R
0
dx0
RL
0
dd x
h
2
(∂0 n)
+(∂n)2 −2 g n·J
i
.
RN
(3.313)
We set c = 1 from now on, 21 and use the short-hand notation
i
Rh
Z
2
∂0 n) +(∂n)2 −2 g n·J
− 21g
(
Ω
.
(3.314)
ZΩ,g [J ] ≡
D[n] δ(n2 − 1) e
RN
The base space that defines the domain of definition of the field n is the
volume Ω of (d + 1)-dimensional space-time. The field n takes values
21
The coordinate x0 has the same units as x, namely those of the lattice
spacing a. The (d + 1)-dimensional volume Ω has units of ad+1 . The coupling g has
units of ad−1 . The field J has units of 1/Ω.
3.8. PROBLEMS
157
in RN subject to the constraint that it has unit length, i.e., the allowed
contributions to the path integral belong to the unit sphere
S N −1 := n ∈ RN n2 = 1
(3.315)
= O(N )/O(N − 1),
that defines the target space of the QNLσM.
Periodic boundary conditions are imposed on the field n at the
boundary of Ω. The field J is a source field. We seek to define the
limit N → ∞ properly and to evaluate the partition function of the
O(N ) QNLσM as well as the relevant correlation functions in this limit.
The scaling limit N → ∞. We insert the representation
R
Z
− λ (n2 −1)
2
D[λ] e Ω
.
(3.316)
δ(n − 1) =
i×R
into Eq. (3.314). The field λ has the units of 1/Ω. We integrate over
the field n,
i
Rh
Z
2
1
− 2g
(∂µ n) +2gλ(n2 −1)−2gJ·n
ZΩ,g [J ] :=
D[n, λ] e Ω
i×RN +1
R
Z
D[λ] e
=
λ
1
− 2g
Z
D[n] e
Ω
[n·(−∂µ2 +2gλ)n−2gJ·n]
(3.317)
.
R
Ω
RN
i×R
To reach the second line, we have assumed that the path integrals over
n and λ can be interchanged.
Exercise 1.1:
(a) Show that integration over n gives
"
Z
N
D[λ] e
ZΩ,g [J ] =
R
Ω
1
N
λ+ 2gN
R
J·
Ω
1
2 +2 g λ J
−∂µ
− 21
Tr ln
2 +2 gλ
−∂µ
g
#
. (3.318)
i×R
(b) Find the transformation law relating λ, J , and g to λ0 , J 0 , and
g 0 and in terms of which
Z
(λ0 ,J 0 )+ N
Tr ln g
−F
2
ZΩ,g [J ] =
D[λ0 ] e Ω,g0
,
(3.319a)
i×R
where
"Z
0
1
λ +
2
Z
1
FΩ,g0 [λ , J ] := − N
J ·
J0
2
−∂µ + 2 g 0 λ0
Ω
Ω
#
1
− Tr ln −∂µ2 + 2 g 0 λ0 .
2
0
0
0
(c) What are the units of λ0 , J 0 , and g 0 ?
(3.319b)
158
3. NON-LINEAR-SIGMA MODELS
The partition function of the O(N ) QNLσM in the large-N limit,
defined by the limit N → ∞ with g 0 held fixed, can now be evaluated
by the method of steepest descent.
Saddle-point equations. We try the Ansatz
0 < λ0 (x) =
m2
,
2 g0
J 0 (x) = H,
x ∈ Rd+1 .
(3.320)
We then define the free energy per unit volume |Ω| and per channel
(free energy divided by N )
Z
m2
H2
1
1
2
2
2
FΩ,g0 (m , H) := − 0 +
Tr ln −∂µ + m −
2g
2 |Ω|
2 |Ω|
0 + m2
Ω
2
2
=−
1
m
Tr ln −∂µ2 + m
+
0
2g
2 |Ω|
2
−
H
2 m2
(3.321)
or, equivalently, its Legendre transform with respect to the space-time
constant staggered magnetic field H
∂FΩ,g0
2
2
VΩ,g0 (m , M ) := FΩ,g0 (m , M ) − H ·
(m2 , M
(3.322a)
),
∂H
∂FΩ,g0
2
(m2 , H).
(3.322b)
M (m , H) := −
∂H
Exercise 2.1:
(a) Show that (0 < m2 )
VΩ,g0 (m2 , M ) = −
1 2 2
m2
1
2
2
+
Tr
ln
−∂
+
m
+ m M . (3.323)
µ
2 g 0 2 |Ω|
2
The saddle-point equations are then
∂VΩ,g0
0
0 = 2g
(m2 , M ),
0 < m2 ,
∂m2
∂VΩ,g0
(m2 , M ),
0 < m2 ,
0=
∂M
(3.324a)
(3.324b)
i.e.,
1
1 =
Tr
|Ω|
0 = m2 M ,
g0
−∂µ2 + m2
0 < m2 .
+ g0 M 2,
0 < m2 , (3.325a)
(3.325b)
(b) What does a non-vanishing solution for m2 imply for the uniform and static staggered magnetization M ?
(c) What does a vanishing solution for m2 imply for the uniform
and static staggered magnetization M ?
3.8. PROBLEMS
159
We are going to treat the limit β → ∞ of zero temperature first and
then that of finite temperature, while the infinite volume limit L → ∞
is here always understood.
Saddle-point equation with m2 > 0 at vanishing temperature. Exercise 3.1:
√
(a) Assume that m := + m2 > 0, β = ∞, and that H = 0.
Show that the saddle-point equations reduce then to
Z
1=
dd+1 k
g0
,
(2π)d+1 k 2 + m2
√
m := + m2 > 0.
(3.326)
Rd+1
Define the (d + 1)-dimensional integral
0
Z
I(d, g , m) :=
g0
dd+1 k
,
(2π)d+1 k 2 + m2
√
m :=
m2 > 0.
(3.327)
Rd+1
(b) Under what conditions on d = 0, 1, 2, · · · , and m ≥ 0 is this
integral well defined?
(c) Give a prescription to tame the UV divergences that preserves
the formal O(d + 1) symmetry of the integrand. Hint: Use a
momentum cutoff Λ = π/a where a is some underlying lattice
spacing.
(d) Give a prescription to tame the UV divergences that break
the formal O(d + 1) symmetry of the integrand down to the
subgroup O(d). Hint: use a momentum cutoff Λ = π/a where
a is some underlying lattice spacing.
(e) In what way do the cases of d = 0 and d = 1 differ from the
cases of d ≥ 2?
(f) What dimension d is plagued with both IR and UV divergences? This dimension is called the lower critical dimension.
Solutions with m2 > 0 and isotropic UV cutoff for frequencies and
wave vectors. To preserve the O(d + 1) invariance of the integrand in
the saddle-point equation, define the UV regulated (d + 1)-dimensional
160
3. NON-LINEAR-SIGMA MODELS
integral
Z
0
I(d, g , m, π/a) :=
dd+1 k
g0
(2π)d+1 k 2 + m2
|k|<π/a
Zπ/a
= C(d) dkk d
g0
k 2 + m2
0
Zπ/a
m d+1
g0
= C(d) dkk d
m
k 2 + m2
0
k =: my
= C(d) m
d−1
g
0
π/(a
Z m)
dy
yd
,
y2 + 1
m >(3.328)
0.
0
The d-dependent
numerical
constant C(d) is the area of the unit sphere
d+1 2
x = 1 in (d + 1)-dimensions divided by (2π)d+1 .
x∈R
Exercise 4.1: Verify that
1
1
2π (d+1)/2
=
,
×
d+1
d
(d+1)/2
(2π)
Γ (d + 1)/2
2 π
Γ (d + 1)/2
Z∞
Γ(z) := dt e−t tz−1 ,
Re z > 0,
C(d) :=
0
Γ(1/2) =
√
π,
(3.329a)
(3.329b)
Γ(n) = (n − 1)!, n = 0, 1, 2, · · · , Γ(x + 1) = x Γ(x), x > 0.
(3.329c)
Verify that
d=0:
2π (d+1)/2
= 2,
Γ (d + 1)/2
C(0) =
1
,
π
d=1:
2π (d+1)/2
= 2π,
Γ (d + 1)/2
C(1) =
1
, (3.330b)
2π
d=2:
2π (d+1)/2
= 4π,
Γ (d + 1)/2
C(2) =
1
, (3.330c)
2π 2
d=3:
2π (d+1)/2
= 2π 2 ,
Γ (d + 1)/2
C(3) =
1
, (3.330d)
8π 2
d=4:
2π (d+1)/2
8π 2
=
,
3
Γ (d + 1)/2
C(4) =
1
.(3.330e)
12π 3
(3.330a)
Exercise 4.2:
(a) Is I(d, g 0 , m, π/a) an increasing or decreasing function of m
when holding all other variables fixed?
3.8. PROBLEMS
161
Verify the recursion relation
Z
yd
y d−1
dy 2
=
−
y +1
d−1
Z
dy
y d−2
,
y2 + 1
d = 2, 3, 4, · · · .
(3.331)
Verify that recursion relation (3.331) with the seeds
Z
1
= arctan y,
+1
Z
1
y
= ln(1 + y 2 ),
dy 2
y +1
2
d=0:
dy
d=1:
y2
(3.332a)
(3.332b)
yields
y2
= y − arctan y,
y2 + 1
Z
y3
y2 1
dy 2
=
− ln(1 + y 2 ),
y +1
2
2
Z
3
4
y
y
=
− y + arctan y.
dy 2
y +1
3
Z
d=2:
d=3:
d=4:
dy
(3.332c)
(3.332d)
(3.332e)
Verify that, for any m > 0, this gives
0
d=0:
I(d, g , m, π/a) =
d=1:
I(d, g 0 , m, π/a) =
d=2:
I(d, g 0 , m, π/a) =
d=3:
I(d, g 0 , m, π/a) =
d=4:
I(d, g 0 , m, π/a) =
g0
πm
g0
4π
mg 0
2π 2
m2 g 0
16π 2
m3 g 0
36π 3
π arctan
,
(3.333a)
am
π 2
ln 1 +
,
(3.333b)
am
h π π i
− arctan
,
(3.333c)
a
m
a
m
π 2 π 2
− ln 1 +
, (3.333d)
am
am
π π π 3
−3
+ 3 arctan
.
am
am
am
(3.333e)
(b) What are the singularities, if any, of
• Eq. (3.333a) in the IR limit m ↓ 0 and UV limit a ↓ 0?
• Eq. (3.333b) in the IR limit m ↓ 0 and UV limit a ↓ 0?
• Eq. (3.333c) in the IR limit m ↓ 0 and UV limit a ↓ 0?
• Eq. (3.333d) in the IR limit m ↓ 0 and UV limit a ↓ 0?
• Eq. (3.333e) in the IR limit m ↓ 0 and UV limit a ↓ 0?
(c) For what dimensions can the saddle-point equation 1 = I(d, g 0 , m, π/a)
admit a solution at m = 0 for some critical value of g 0 ?
162
3. NON-LINEAR-SIGMA MODELS
(d) Show that, for sufficiently small values of m, the saddle-point
equation 1 = I(d, g 0 , m, π/a) has the approximate solutions
d=0:
d=1:
m≈
g0
,
2
m ≈ e−2π/g
(3.334a)
π 0
a
,
(3.334b)
when d = 0 and d = 1, respectively.
(e) Show that, in the IR limit m ↓ 0, the saddle-point equation
0
where
1 = I(d, g 0 , m = 0, π/a) implies that g 0 = gcr
a
0
= 2π 2
d=2:
gcr
,
(3.335a)
π
a 2
0
= 16π 2
,
(3.335b)
d=3:
gcr
π
3
0
3 a
d=4:
gcr = 36π
,
(3.335c)
π
when d = 2, 3, 4, respectively.
(f) Explain why real-valued roots m > 0 to the saddle-point equa0
tion 1 = I(d, g 0 , m > 0, π/a) are only possible for g 0 > gcr
when
d = 2, 3, 4.
To solve for these roots we express Eq. (3.333c), Eq. (3.333d),
and Eq. (3.333e) in terms of Eq. (3.335a), Eq. (3.335b), and
Eq. (3.335c), respectively,
a m
π i
g0 h
d=2:
I(d, g 0 , m, π/a) = 0 1 −
arctan
,
(3.336a)
gcr
π
am
π 2 a m 2 g0
0
d=3:
I(d, g , m, π/a) = 0 1 −
ln 1 +
, (3.336b)
gcr
π
am
a m 3
π a m 2
g0
0
+3
arctan
.
d=4:
I(d, g , m, π/a) = 0 1 − 3
gcr
π
π
am
(3.336c)
(g) Show that solutions to the saddle-point equations
a m
π i
g0 h
arctan
,
(3.337a)
d=2:
1= 0 1−
gcr
π
am
a m 2 π 2 g0
d=3:
1= 0 1−
ln 1 +
,
(3.337b)
gcr
π
am
a m 2 h
a m
π i
g0
d=4:
1= 0 1−3
1−
arctan
,
gcr
π
π
am
(3.337c)
in the limit
π
m
a
(3.338)
3.8. PROBLEMS
163
are
d=2:
d=3:
d=4:
0
2 g 0 − gcr
π
m≈
,
0
π
g
a
0
1/2 0
g − gcr
π
m≈
,
0
g
a
1/2 0
0
π
1 g − gcr
,
m≈
0
3
g
a
(3.339a)
(3.339b)
(3.339c)
provided
0
g 0 > gcr
.
(3.340)
The lessons that we learn are:
(i) The nature of the root m of the saddle-point equation is
very different depending on whether the UV limit a ↓ 0 is
finite (d = 0) or diverges in a power law fashion (d ≥ 2)
with the case d = 1 being marginal.
(ii) There is a qualitative difference in the algebraic depen0
)/g 0 when d = 2 compared to when
dence of m on (g 0 − gcr
d ≥ 4 with d = 3 being the marginal case. For d = 2 the
critical exponent ν defined by
0
ν
0
g − gcr
m∼
(3.341)
0
gcr
is ν = 1 while it is always ν = 1/2 for d ≥ 4.
(h) Why is ∼ used in Eq. (3.341) instead of ≈ as in Eqs. (3.339)?
(i) Give one explanation for the following observation: “The case
d = 3 is marginal in that regard as the algebraic law with
ν = 1/2 holds up to logarithmic corrections.”
This observation has a simple explanation that follows from
isolating the UV divergent contribution from the finite one in
the saddle-point equation (3.326) in combination with dimensional analysis. To this end
Z
dd+1 k
1
0
1 = g
d+1
2
(2π)
k + m2
Z
Z
dd+1 k 1
dd+1 k
1
1
0
0
= g
−g
−
(2π)d+1 k 2
(2π)d+1 k 2 k 2 + m2
Z
g0
dd+1 k
m2
0
≡ 0 −g
(3.342)
gcr
(2π)d+1 k 2 (k 2 + m2 )
implies the relation (remember that g 0 has the dimension ad−1 )
0
1/(d−1)
0
g0
g − gcr
0 d−1
1 = 0 − (UV finite constant) × g m
⇔m∼
0
gcr
gcr
(3.343)
164
3. NON-LINEAR-SIGMA MODELS
for 1 < d < 3. When the integral on the right-hand side of
Eq. (3.342) is no longer UV finite, i.e., when d ≥ 3, application
of the recursion relation (3.331) gives the mean field exponent
ν = 1/2.
(j) The naive continuum limit is obtained as a ↓ 0. Deduce from
Eqs. (3.339) how the continuum limit should really be understood when d = 2, 3, 4. Hint: Ask yourself how the limit a ↓ 0
can be taken so that Eqs. (3.339) make sense.
Solutions with m2 > 0 and with isotropic UV cutoff for the wave
vectors only. We now give up the O(d + 1) invariance of the integrand.
We define the UV regulated (d + 1)-dimensional integral
Z
Z
d$
g0
dd k
0
I∞ (d, g , m, π/a) :=
,
(3.344)
2π
(2π)d $2 + k2 + m2
|k|<π/a
R
for d = 1, 2, · · · and m > 0.
Exercise 5.1:
(a) Why did we forbid the case d = 0?
(b) Introduce the notation
p
ω(k) := k2 + m2
(3.345)
and show that
g0
I∞ (d, g , m, π/a) =
2
0
dd k 1
(2π)d ω(k)
Z
(3.346a)
|k|<π/a
d−1 g
= C(d − 1)m
0
π/(a
Z m)
1
dy y d−1 p
2
0
with d = 1, 2, · · · , and m > 0.
Assume the recursion relation
Z
Z
d−1
y
1 d−2 p 2
d−2
y d−3
dy p
=
y
y + 1−
dy p
,
d−1
d−1
y2 + 1
y2 + 1
y2
(3.346b)
+1
d = 3, 4, · · · ,
(3.347)
with the seeds
Z
d=1:
1
dy p
y2 + 1
y
Z
d=2:
dy p
y2 + 1
p
= ln y + y 2 + 1 , (3.348a)
=
p
y 2 + 1.
(c) Show that, under the assumption that m is small,
π 0
d=1:
m≈
e−2π/g ,
a
as in Eq. (3.334b).
(3.348b)
(3.349a)
3.8. PROBLEMS
165
(d) Show that the roots
d=2:
0
gcr
=
a
,
(3.349b)
π
a 2
0
d=3:
gcr
= 8π 2
,
(3.349c)
π
a 3
0
,
(3.349d)
d=4:
gcr
= 48π 2
π
of the saddle-point equation 1 = I∞ (d, g 0 , m, π/a) in the IR
limit m ↓ 0 are smaller than their counterparts Eq. (3.335a),
Eq. (3.335b), and Eq. (3.335c) by the geometrical ratio
C(d)
,
d = 2, 3, 4,
(3.350)
C(d − 1)/2
respectively.
(e) Show that the roots of these saddle-point equations are, in the
limit
π
m,
(3.351)
a
given by the elementary functions (the case d = 3 is again
special in view of the logarithmic correction)
0
0
g − gcr
π
,
(3.352a)
d=2:
m≈
g0
a
0
1/2 0
g − gcr
π
d=3:
m≈
,
(3.352b)
g0
a
0
1/2 0
1 g − gcr
π
d=4:
m≈
,
(3.352c)
2
g0
a
provided
0
g 0 > gcr
.
(3.353)
(f) Equations (3.352) should be compared to Eqs. (3.339). In
what ways do they differ and agree?
Saddle-point equation with m2 > 0 at non-vanishing temperature.
The case of non-vanishing temperature differs from the vanishing one
through the nature of the trace in the saddle-point equation
1
g0
1 =
Tr
|Ω|
−∂µ2 + m2
Z
n∈Z
1 X
dd k
g0
=
β
(2π)d $n2 + k2 + m2
$ =2πn/β
|k|<π/a
n
Eq. (3.328)
=
1
β
4π
n∈Z
X
$n =2πn/β
p
I d − 1, g 0 , $n2 + m2 , π/a ,
m
(3.354)
> 0.
166
3. NON-LINEAR-SIGMA MODELS
Exercise 6.1:
(a) Write
Z
1
g0
dd k
1=
β
(2π)d k2 + m2
|k|<π/a
1
+
β
n∈Z\{0}
X
Z
$n =2πn/β |k|<π/a
dd k
g0
,
(2π)d $n2 + k2 + m2
(3.355)
m > 0.
With the help of Eq. (3.328),
1=
1
I(d − 1, g 0 , m, π/a)
β
Z
n∈Z\{0}
X
1
dd k
g0
+
,
β
(2π)d $n2 + k2 + m2
$ =2πn/β
n
(3.356)
m > 0.
|k|<π/a
Depending on d, when is the right-hand side of the saddlepoint equation IR or UV singular and determine the singularities (pole-like or branch cut)?
(b) For what d in the large N -limit can there be long-range order
at any finite temperature?
(c) Show that the summation over the Matsubara frequencies {$n =
2πn/β, n ∈ Z} in the saddle-point equation (3.354) can be
performed exactly to yield the saddle-point equation
Z
g0
dd k 1
1 =
coth
β
ω(k)/2
2
(2π)d ω(k)
|k|<π/a
≡ I(d, g 0 , m, π/a, β),
m > 0.
(3.357)
(d) In what two limits can the integrand of I(d, g 0 , m, π/a, β) reduce to the integrand of β1 I(d−1, g 0 , m, π/a) on the right-hand
side of Eq. (3.356)?
(e) In what limit does one recover Eq. (3.346a) and its subsequent
analysis.
(f) Explain how the existence of a root at m = 0 to the saddlepoint equation (3.357) implies the existence of a positive-valued
function βcr (g 0 ) that is a monotonously increasing function of
g0.
Exercise 6.2: We now consider the cases d = 1 and d = 2 of the
saddle-point equation (3.357).
(a) Explain why the integral over wave vectors on the right-hand
side of the saddle-point equation (3.357) can be dominated by
the IR limit k → 0 when d = 2 but not in d = 1.
3.8. PROBLEMS
167
(b) Assume that for d = 2,
lim lim β ω(k) 1
d=2:
(3.358)
β→∞ |k|→0
is self-consistent. Show under assumption (3.358) that the
estimate
d=2:
1=
1
I(d − 1, g 0 , m, π/a) + · · · ,
β
(3.359)
to the saddle-point equation (3.357) implies
m(g 0 , β) ≈
d=2:
π a
0
e−2πβ/g .
(3.360)
(c) Check the selfconsistency of assumption (3.358) implied by
Eq. (3.360).
Exercise 6.3: The analysis when d = 2 can be refined as the
saddle-point equation (3.354) or, equivalently, (3.357) can be solved in
closed form if it is modified so as to remain finite in the UV limit a ↓ 0.
To this end, we define the Pauli-Villars regularization (m > 0)
d=2:
h
i
1 = lim I(d, g 0 , m, π/a, β) − I(d, g 0 , M, π/a, β) , (3.361)
a↓0
to the saddle-point equation (3.354). The only condition made on the
choice of the momentum scale M is that it is much larger than the
temperature,
1 βM.
(3.362)
(a) Show that the integral I(d, g 0 , m, π/a, β), Eq. (3.357) , when
d = 2 can be evaluated in closed form
0
0
I(d, g , m, π/a, β) = I∞ (d, g , m, π/a) + g
0
Z
dd k 1
1
(2π)d ω(k) eβ ω(k) − 1
|k|<π/a
r
a m 2
g0
1+
first order pole in a with residue g 0 /4 as a ↓ 0
4a
π
g0
−
ln 2 sinh(βm/2)
remains finite as a ↓ 0
2πβ
√
g0
−β (π/a)2 +m2
+
ln 1 − e
,
vanishes as a ↓ 0 (3.363)
2πβ
=
168
3. NON-LINEAR-SIGMA MODELS
and that the Pauli-Villars regularization of the saddle-point
equation reduces to
2πβ
0
g
sinh(βM/2)
2
−
1=
ln
⇐⇒ m =
arcsinh e g0 sinh (βM/2)
2πβ
sinh(βm/2)
β
−2πβ
=
2
e
arcsinh
β
−2πβ
−2πβ
1
− g10
g0
cr
2
1
M
− 4π
g0
e
2
arcsinh
β
2
(b) When d = 2, show that the “critical coupling”
2πβ
0
gcr
:= ln 2 sinh (βM/2)
βM 1
ln 2 sinh(βM/2)
2
2
e
≡
arcsinh
β
1
1
− 2πβ
g0
≈
.
(3.364)
(3.365a)
only becomes truly independent of β when
βM 1
(3.365b)
in which case it is given by
4π
0
−1 −βM
gcr =
+O β e
.
(3.365c)
M
Equation (3.365c) should be compared with the one derived
in Eq. (3.349b).
(c) When d = 2, show that the asymptotic dependence of m on β
0
, 22
depends on whether g 0 is equal, larger or smaller than gcr
√

2 ln(2−1 + 2−2 +1)
0

, if either β ↓ 0 or g 0 = gcr
,

β






−2πβ g10 − g10 0
1
0
cr ,
m(g , β) = β e
(3.367)
if β → ∞ and g 0 < gcr
,







0
4π 10 − 10 ,
if β → ∞ and g 0 > gcr
.
g
g
cr
For any d = 1, 2, · · · , the Laurent expansion of the coth in the
integrand on the right-hand side of the saddle-point equation (3.357)
is of course valid for all wave vectors in the high-temperature limit
22
We here make use of
p
arcsinh x = ln x + 1 + x2 ,
x 1,
1
arcsinh x = x −
x3 ± · · · ,
2×3
x ≥ 0,
arcsinh x = ln x,
(3.366a)
(3.366b)
0 < x 1.
(3.366c)
3.8. PROBLEMS
169
β ↓ 0 and can be used to estimate the correlation length 1/m in the
high-temperature limit.
Staggered spin susceptibility when m2 > 0. Exercise 7.1:
Explain why a finite root m to the saddle-point equation can be
thought of as the inverse correlation length at the AF wave vector.
Antiferromagnetic transition (Néel) temperature when d = 3, 4. We
have shown that the saddle-point equation (3.357) can be written as
1 = I(d, g 0 , m, π/a, β)
0
= I∞ (d, g , m, π/a) + g
0
Z
|k|<π/a
dd k 1
1
(. 3.368)
d
β
ω(k)
(2π) ω(k) e
−1
170
3. NON-LINEAR-SIGMA MODELS
Exercise 8.1:
(a) What are the possible singularities as a function of d of the
contribution
Z
dd k 1
1
0
g
(3.369)
d
β
ω(k)
(2π) ω(k) e
−1
|k|<π/a
to the saddle-point equation that encodes the temperature dependence?
(b) Define the AF temperature that signals long-range order in the
large N -limit of the O(N ) NLσM by writing down the proper
limit of the saddle-point equation (3.368).
(c) Assuming that
π
m
(3.370)
a
show that
1≈
g0
+ const × β 1−d g 0 ,
0
gcr
d = 3, 4, · · · .
(3.371)
(d) Show here that the combination β 1−d g 0 follows from dimensional analysis alone under the condition that the integral
Z
dd k 1
1
(3.372)
(2π)d ω(k) eβ ω(k) − 1
|k|<π/a
is finite, i.e., d = 3, 4, · · · .
(e) Use Eq. (3.371) to solve for the onset temperature for AF
long-range order.
(f) Can there be AF long-range order when
0
g 0 > gcr
?
(3.373)
0
g 0 < gcr
.
(3.374)
Elaborate for the case
−1
increase or decrease as
(g) Does the AF (Néel) temperature βAF
0
0
g is decreased away from gcr ?
3.8.5. O(N > 2) QNLσM: One-loop RG using the BerezinskiiBlank parametrization of spin waves.
Introduction. We are ready to perform a one-loop RG analysis of
the O(N ) quantum non-linear σ model. We shall reproduce the results
obtained in Ref. [39].
3.8. PROBLEMS
171
Definitions. The O(N ) QNLσM is defined by the partition function
Z
(1)
(2)
D[n] δ(n2 − 1) e−(S +S )
(3.375a)
Z[h] :=
RN
where
S
(1)
Z
(1)
L
:=
Zβ
≡
ZL
dτ
0
dd r
c
2 ad−1
2
(3.375b)
Zh h · n .
(3.375c)
g
∂µ n
a
and
S
(2)
Z
:=
L
(2)
Zβ
≡−
ZL
dτ
0
dd r
c
ad+1
a
Here, the lattice spacing a plays the role of the microscopic ultraviolet
(UV) cutoff, i.e., Λ ∼ 1/a that of an upper cutoff on momenta. The
linear size L is the largest length scale of the problem. The derivative ∂µ = (∂cτ , ∇) depends on the spin wave velocity c in the plane,
c = a g J/~, where J is a characteristic energy scale such as a Heisenberg exchange coupling. The dimensionless coupling constant g depends on the microscopic details of the intraplane interactions. The
dimensionless background field h, where h = |h|, is the external source
for a static staggered magnetic field conjugate to the planar antiferromagnetic order parameter of the underlying lattice model. It breaks
the O(N ) symmetry of Lagrangian (3.375b) down to O(N − 1) and as
such acts as an infrared (IR) regulator. The dimensionless coupling Zh
is the field renormalization constant associated to n . The use of the
continuum limit is justified if we are after the physics on length scales
much longer than a.
We assume that the O(N ) QNLσM (3.375) with h = 0 is renormalizable in that all the effects induced by the integration over the fast
modes with momenta belonging to the infinitesimal momentum shell
e ≤ |k| ≤ Λ
Λ
(3.376)
can be absorbed in a redefinition of the two dimensionless coupling
constants
g
and
t ≡ 1/(J β)
(3.377)
to leading order in g/c. We use the Berezinskii Blank parametrization
(3.294).
Exercise 1.1:
(a) Explain why g/t can be interpreted as the dimensionless slab
thickness in the imaginary-time direction.
(b) With the same level of rigor as before show that the renormale induced by averaging over
ized values ge and e
t at the scale Λ
172
3. NON-LINEAR-SIGMA MODELS
the fast modes φ with momenta in the infinitesimal momentum shell (3.376) are given by
−1
ge
Λ
g
=
,
(3.378)
e
e
t
t
Λ
d−1 a b
1
1 Λ
2 ab
,
(3.379)
=
1+ φ φ −φ δ
e
ge
g Λ
or, equivalently, by
d−1 a b
1
1 Λ
2 ab
=
,
1+ φ φ −φ δ
e
ge
g Λ
d−2 a b
1
1 Λ
2 ab
,
=
1+ φ φ −φ δ
e
e
t Λ
t
(3.380a)
(3.380b)
Here, when d = 2,
Λ g
g
coth .
(3.381)
e 4π
2t
Λ
These RG equations were first derived by Chakravarty, Halperin,
and Nelson in Ref. [39].
(c) Explain the sentence: Equation (3.381) is the fluctuation
bubble accounting for the quantum fluctuations induced by the
fast modes with momenta in an infinitesimal momentum shell.
In what ways does it differ from its counterpart in Eq. (3.305b).
(d) With the same level of rigor as before, show that the wavefunction renormalization is
Zeh
1 2
= 1−
φ + ··· .
(3.382)
Zh
2
hφa φb i = δ ab ln
(e) Use the notation
Λ
(3.383)
e
Λ
and turn Eqs. (3.380) into two coupled differential equations
for the rate of change of g and t with `.
(f) Consider the limits g/2 t → 0 and g/2 t → ∞ of the coupled
equations
dt
dg
= β1 (g, t),
= β2 (g, t),
(3.384)
d`
d`
that you have derived. One of these limits is the low-temperature
limit, the other the high-temperature limit. To what RG
equations that you have already encountered reduce the flows
(3.384) in these two limits?
(g) The phase diagram that follows from the RG flows (3.384) is
shown in Fig. 7
(i) What is the meaning of the black circles?
` := ln
3.8. PROBLEMS
g
d=3
g
173
d=2
gc
gc
tc
t
t
Figure 7. Phase diagram of quantum antiferromagnets
in three- and two-dimensional position space.
(ii) Why is gc represented smaller for d = 2 than for d = 3?
(iii) What is the meaning of the shaded region when d = 3?
(iv) What is the meaning of the complementary region to the
shaded one when d = 3?
(v) What is the meaning of the line that joins gc to tc when
d = 3?
(vi) Why are there arrows on the RG trajectories and how can
they be deduced from the RG equations?
CHAPTER 4
Kosterlitz-Thouless transition
Outline
The classical two-dimensional XY (2d–XY ) model is defined. The
quasi-long-range ordered phase in the continuum spin-wave approximation derived in section 3.3.2 is shown to be unstable to the deconfining transition of topological defects, i.e., vortices, that drives the
model into a high-temperature paramagnetic phase. [40, 41, 42] This
transition, which we shall call the Kosterlitz-Thouless (KT) transition
although the terminology Berezinskii-Kosterlitz-Thouless (BKT) transition is also used in the literature, is studied within a perturbative
renormalization-group analysis. To set up the renormalization-group
analysis, the vortex sector of the 2d–XY is first shown to be equivalent
to a 2d–Coulomb gas. [40] In turn the 2d–Coulomb gas is shown to be
equivalent to the 2d–Sine-Gordon model. The renormalization-group
analysis is made within the 2d–Sine-Gordon model.
4.1. Introduction
The classical two-dimensional XY (2d–XY ) model is extremely important, both conceptually and for its applicability to materials. Conceptually, it is perhaps the simplest example in which a phase transition
of a topological nature takes place. From a practical point of view, the
classical 2d−XY model is relevant to two-dimensional superconductivity, two-dimensional superfluidity, two-dimensional arrays of Josephson
junctions, the melting of crystalline thin films, roughening transition of
crystalline surfaces, one-dimensional metals, and quantum magnetism
in one-space dimension. This list is not exhaustive.
4.2. Classical two-dimensional XY model
Consider a square lattice with the lattice spacing a embedded in
two-dimensional Euclidean space. Assign to each site i ∈ Z2 an angle
φi ∈ S 1 where S 1 is the one-sphere, i.e., the circle in the complex plane
(see Fig. 1). The partition function of the classical XY model is defined
by
Z
ZXY := D[φ] exp (−SXY ) ,
(4.1a)
175
176
4. KOSTERLITZ-THOULESS TRANSITION
e2
Z
2
R
2
i
Si =
✓
cos
sin
i
i
◆
e1
Figure 1. Heisenberg model on a square lattice with
the lattice spacing a and with the O(2) ∼
= U (1) internal
symmetry.
with the action
SXY := K
X
1 − cos φi − φj ,
(4.1b)
hiji
and the dimensionless coupling constant K
K := βJ,
β = 1/(kB T ).
(4.1c)
We shall set the Boltzmann constant kB = 1.38065 × 10−23 J/K to
unity, in which case β is the inverse temperature. The dimensionless
coupling constant K is the reduced ferromagnetic exchange coupling
between nearest-neighbor planar spins
cos φj
cos φi
and
,
(4.2)
sin φj
sin φi
i.e., for any directed pair of nearest-neighbor sites hiji one assigns the
link energy
cos φj
J 1 − cos φi sin φi
= J 1 − cos φi − φj , (4.3)
sin φj
and the Boltzmann weight
exp − βJ 1 − cos φi − φj
,
(4.4)
respectively. The reduced ferromagnetic exchange coupling K can be
varied by changing the ferromagnetic spin stiffness J > 0 or by changing the inverse temperature β. Here, we will use the inverse temperature to vary K. The classical 2d–XY model (4.1) thus describes a
classical isotropic ferromagnet for planar spins located on a square lattice with the lattice spacing a.
The link energy (4.3) is minimized by choosing φi = φj (see Fig. 2).
Moreover, minimization of the sum over the four link energies, whereby
the four links define an elementary square unit cell (plaquette) of the
lattice, is also achieved by choosing all angles at the four corners of the
plaquette equal. The system is thus not frustrated and the configuration of angles with the lowest energy has all angles equal. This is the
4.2. CLASSICAL TWO-DIMENSIONAL XY MODEL
l
k
i
j
177
Figure 2. An elementary plaquette of the square lattice. The sublattice structure is made explicit by drawing
lattice sites with white and black circles. The four angles
φi , φj , φk , φl are all equal in the ferromagnetic state.
ferromagnetic state
φi = φferro ,
∀i ∈ Z2 .
(4.5)
Since φferro can be any real number between 0 and 2π, the ground state
of the XY model spontaneously breaks the global invariance of the XY
model under the continuous U (1) transformation
φi = φei + α,
∀i ∈ Z2 ,
0 ≤ α < 2π.
(4.6)
Thermal fluctuations disturb the long-range order of the ground
state (4.5). In fact, it is reasonable to anticipate that the long-range ferromagnetic order disappears in the limit of infinite temperature β → 0,
i.e., K → 0, as all configurations of angles {φi }i∈Z2 are equally weighted
in the partition function in this limit. Since there are “many more”
configurations of angles that deviate strongly from the ferromagnetic
state than those that do not, one is tempted to identify the limit K → 0
with a paramagnetic phase. High-temperature expansions [expansions
of the Boltzmann weight (4.4) in powers of K] are in fact consistent with
the existence of a paramagnetic phase for a finite temperature range in
the vicinity of β = 0. Hence, there must be some type of phase transition at some critical value Kc of the dimensionless coupling K that
separates a low-temperature phase related to the ferromagnetic ground
state from the paramagnetic high-temperature phase. The questions
to be answered are:
(1) Is Kc < ∞?
(2) What is the nature of the low-temperature phase, how does it
connect to the ferromagnetic long-range order at zero temperature?
(3) What is the nature of the phase transition between the lowand high-temperature phases?
A first attempt to provide an answer to these questions is to study
the stability of the ferromagnetic long-range order at zero temperature in the presence of spin-wave fluctuations. In the continuum spinwave approximation, the partition function ZXY , which is defined on
178
4. KOSTERLITZ-THOULESS TRANSITION
the lattice, is replaced by the partition function for the 2d O(2) nonlinear-sigma-model (NLσM), which is defined in the continuum by the
partition function
Z
ZXY → Zsw ,
Zsw := D[φ] e−Ssw ,
(4.7a)
with the action
Z
Ssw :=
d2 x Lsw ,
(4.7b)
and the Lagrangian density
2
K
∂µ φ .
(4.7c)
2
We have performed the continuum-spin-wave approximation that is
nothing but the naive continuum limit
X
SXY = K
1 − cos φi − φj
Lsw :=
hiji
#
" 2
4
2
φ
−
φ
φ
−
φ
a
1
i
j
i
j
−
+ ···
= K
a2
2
a
4!
a
hiji
( " 2
4 #)
X
φ
−
φ
φ
−
φ
1
i
j
i
j
= K
a2
+ O a2
2
a
a
hiji
Z
h
2
4 i
1
2
→ K dx
∂ φ + O ∂µ φ
.
(4.8)
2 µ
X
The second line is called a gradient expansion, i.e., it is an expansion in
inverse powers of the lattice spacing a. The lattice-spin-wave approximation truncates the gradient expansion to the Gaussian order as is
done on the third line. Before performing the gradient expansion, the
integration measure
Y
D[φ] :=
dφi ,
0 ≤ φi < 2π,
(4.9)
i∈Z2
encodes the compact nature of the angles φi with i ∈ Z2 , i.e., the fact
that the link interaction energy is, through cos(φi − φj ), a periodic
function of φi (φj ) with periodicity 2π. Any truncation of the gradient
expansion destroys this periodicity. To make sense of an approximation by which the gradient expansion is truncated to finite order, the
integration measure must be modified accordingly,
Y
D[φ] →
dφi ,
φi ∈ R.
(4.10)
i∈Z2
Often, the spin-wave approximation is understood to be the continuum
limit on the last line of Eq. (4.8). The gradient expansion assumes
smoothness of the spin configuration {φi }i∈Z2 in that φi − φj is small
4.2. CLASSICAL TWO-DIMENSIONAL XY MODEL
179
on the scale of the lattice spacing a, i.e., one can replace (φi − φj )/a
by the function ∂µ φ. The corresponding integration measure
Y
D[φ] :=
dφ(x),
φ(x) ∈ R,
(4.11)
x∈R2
is then restricted to smooth, i.e., differentiable, single-valued functions
φ : R2 → R
(4.12)
that vanish at infinity. The main assumption made in the spin-wave
approximation, be it before or after the continuum limit, is, as it turns
out, not so much the replacement of a non-linear theory by a Gaussian
theory than the neglect of the compactness of φi ∈ S 1 in the 2d–XY
model.
We have already studied the 2d O(2) NLσM in section 3.3.2. There,
we saw that the (2n)-point function
1
 Q
 2πK
|xi − xj ||y i − y j |

2n  1≤i<j≤n

he+iφ(x1 )+···+iφ(xn )−iφ(y1 )−···−iφ(yn ) iZsw = a 4πK 
n
n


Q Q
|xi − y j |
i=1 j=1
(4.13)
is algebraic at any finite temperature. Within the spin-wave approximation, ferromagnetic long-range order is downgraded to quasi-long-range
order (algebraic order) at any finite temperature, i.e., all spin-spin correlation functions decay algebraically fast for large separations. The
spin-wave approximation thus captures an instability of the ferromagnetic long-range order at vanishing temperature. This instability is an
example of the Mermin-Wagner theorem. However, the spin-wave approximation is deficient in that it fails to predict a phase other than
one with quasi-long-range order. 1
The failure at high temperatures of the spin-wave approximation is
rooted in that it only allows for small and smooth deviations (gradient
expansion) about the ferromagnetic ordered state. In particular, it is
ruled out that the field φ be singular at some isolated point in the sense
that it is everywhere single valued,
Z
dφ = 0
(4.14)
γx
1
Within the spin-wave approximation, we are free to absorb K into a redefinition of the non-compact lattice degree of freedom φi on the lattice (or φ(x) in the
continuum). This rescaling shows up when calculating correlation functions. This
rescaling also tells us that if we know how to solve the theory for, say, K = 1, we
know how to solve the theory for all K’s, as is expressed by Eq. (4.13). On the
other hand, the infinitesimal rescaling of φi = 2π − into φi = 2π + is dramatic
since it turns a large angle, 2π − , into a small one, , in the 2d–XY model. The
2d–XY model is thus certainly not scale invariant.
180
4. KOSTERLITZ-THOULESS TRANSITION
(a)
(b)
Figure 3. (a) An elementary plaquette supporting a
configuration of four planar spins of unit length that
would match the magnetic field induced by a thin
current-carrying wire threading the center of the plaquette in three-dimensional electrostatics if two-dimensional
space is embedded in three-dimensional space. (b) An
elementary plaquette supporting a configuration of four
planar spins of unit length that would match the electric
field induced by a thin charged wire threading the center of the plaquette in three-dimensional electrostatics if
two-dimensional space is embedded in three-dimensional
space. The representation of vortices in the continuum
given in the text corresponds to configurations of spins
that vary very little on the lattice scale but that behave
like the spin configuration in (a) along closed path that
are much longer than the perimeter of an elementary plaquette.
for any closed path that encloses x with x arbitrarily chosen in R2 . In
the 2d–XY model, only the spin
cos φi
∼ eiφi
(4.15)
sin φi
must be single valued. Hence, φi and φi + 2π are physically indistinguishable. There is thus no a priori reason to demand that, after the
continuum limit has been taken, the field
φ : R2 → R
(4.16)
is single valued.
An example of a multivalued field is the function
Θ : R2 \ {x1 , · · · , xM } → R,
x → Θ(x) =
M
X
i=1
mi arctan
(x − xi )2
(x − xi )1
.
(4.17)
4.2. CLASSICAL TWO-DIMENSIONAL XY MODEL
181
The condition that exp iΘ(x) be single valued demands that
mi ∈ Z,
i = 1, · · · , M.
(4.18)
The field Θ is smooth everywhere except at the isolated points x1 , · · · ,
xM . Around any xi , the open disc Ui can be chosen sufficiently small
so that counterclockwise integration about its boundary ∂Ui yields
I
dΘ = 2πmi .
(4.19)
∂Ui
Correspondingly, each singularity xi is said to carry vorticity or charge
mi . The vorticity mi counts how many times Θ winds about xi . Figure 3(a) depicts the configuration
π
cos Θ + π2
∼ ei(Θ+ 2 ) .
(4.20a)
π
sin Θ + 2
Figure 3(b) depicts the configuration
cos Θ
∼ eiΘ .
sin Θ
(4.20b)
The dimensionless energy stored by Θ in a finite region Ω ⊂ R2 of
linear size L that contains all the singularities of Θ is
Z
2
K
d2 x ∂µ Θ ,
SCb [Θ, K, L] :=
(4.21)
2
Ω
where the label Cb for the action stands for Coulomb. Observe that
this dimensionless energy is form invariant under the rescaling
x = κx0 ,
L = κL0 ,
0 < κ ∈ R,
(4.22)
i.e.,
SCb [Θ, K, L] = SCb [Θ, K, L0 ].
(4.23)
To proceed with the evaluation of the dimensionless energy stored
by Θ, observe that the field
e : R2 \ {x1 , · · · , xM } → R,
Θ
M
X
(4.24)
x − xi e
,
x → Θ(x) = −
mi ln ` i=1
where ` is some arbitrarily chosen length scale, is related to Θ by the
Cauchy-Riemann relation 2
e
∂1 Θ = +∂2 Θ,
2
e
∂2 Θ = −∂1 Θ.
(4.32)
Let
z = x + iy = |z|eiϕ ∈ C,
|z| :=
p
x2 + y 2 ≡ r,
arg(z) := arctan
y
≡ ϕ.
x
(4.25)
182
4. KOSTERLITZ-THOULESS TRANSITION
e is called the dual field to Θ. With the notation
The field Θ
e
∂eµ := µν ∂ν ,
µν = −νµ ,
12 = 1,
∂µ Θ = +∂eµ Θ,
e = −∂eµ Θ,
∂µ Θ
(4.33)
the property
∂eµ2 = µν µλ ∂ν ∂λ = ∂µ2 ,
and the two-dimensional Green function
x − y 1
2 −1
− ∂µ
(x, y) = − ln 2π
` (4.34)
(4.35)
of Laplace operator −∂µ2 ≡ −∆, the dimensionless energy stored by Θ
becomes
Z
2
K
SCb [Θ, K, L] =
d2 x ∂µ Θ
2
Ω
Z
(4.36)
2
K
2
e .
d x ∂µ Θ
=
2
Ω
If arg(z) and ln |z| are understood as real-valued functions on C \ {0}, then they
obey the Cauchy-Riemann conditions
(∂x arg)(z) =
(∂y arg)(z) =
1
1+
1
1+
y
(−) 2
y 2
x
x
(+)
y 2
x
=−
y
= −(∂y ln |z|),
r2
1
x
= + 2 = +(∂x ln |z|).
x
r
(4.26)
The complex-valued function log(z) := ln |z| + iarg(z) is thus analytic, for ln |z| and
arg(z) are a pair of conjugate single-valued harmonic functions
0 = (∂x2 + ∂y2 ) ln |z| = (∂x2 + ∂y2 )arg(z),
on the Riemann sheet 0 ≤ arg(z) < 2π of C \ {0}. By definition
I
d arg(z) = 2π
(4.27)
(4.28)
for any closed curve winding once counterclockwise around the origin. Observe
that, whereas
1
∂ 2 + ∂y2 ln |z| = δ(|z|),
(4.29)
2π x
the singularity of arg(z) at the origin is not manifest in
1
∂x2 + ∂y2 arg(z) = 0,
(4.30)
2π
it is manifest in
1
∂x ∂y − ∂y ∂x arg(z) = δ(|z|).
(4.31)
2π
Correspondingly, the vector field ∂arg(z) can be interpreted as the magnetic field
depicted in Fig. 3(a), while the vector field ∂ ln |z| can be interpreted as the electric
field depicted in Fig 3(b).
4.2. CLASSICAL TWO-DIMENSIONAL XY MODEL
183
By orienting the infinitesimal surface element d2 x according to d2 x →
dx1 ∧ dx2 = −dx2 ∧ dx1 and in combination with partial integration,
Z
h i K Z
K
e −∂µ2 Θ
e .
e ∂µ Θ
e +
dx1 ∧ dx2 ∂µ Θ
d2 x Θ
SCb [Θ, K, L] =
2
2
Ω
Ω
(4.37)
With the help of Eq. (4.33),
Z
h i K Z
K
e −∂µ2 Θ
e .
e −∂eµ Θ +
SCb [Θ, K, L] =
dx1 ∧ dx2 ∂µ Θ
d2 x Θ
2
2
Ω
Ω
(4.38)
Application of Stokes theorem then gives,
"
#
I
Z
M
X
K
e
e −∂µ Θ + K
SCb [Θ, K, L] =
dxµ Θ
d2 x Θ(x)
(−)2 2π
ml δ(x − xl )
2
2
l=1
∂Ω
Ω
(4.39)
Finally, theorem 20 from section 4.6.2 of Ref. [43] delivers
M
2
P
mi
L
i=1
2K
ln
SCb [Θ, K, L] = (2π)
2
2π
`
M
X
mk ml xk − xl 2K
+ (2π)
ln −
.
2 k,l=1
2π
`
(4.40)
As expected from the form invariance (4.23) of the dimensionless
energy stored by Θ under length rescaling, the dependence on ` of
the boundary contribution cancels the dependence on ` of the bulk
contribution on the last line of Eq. (4.40). The
P boundary term is the
dimensionless self-energy of a single charge
mi concentrated in a
i
radius ` about the origin and guarantees that the dimensionless energy
stored by Θ is strictly positive. The short-distance behavior of the
Green function is regulated by
x − y = ln a
ln (4.41)
` `
when x is within a lattice spacing away from y. As explained in sections
3.3.2 and 3.4, the choice for the ultraviolet regularization of the Green
function is arbitrary from the point of view of field theory. Within the
2d–XY model, the Green function is unambiguously defined all the way
to the lattice spacing, the characteristic length at which scale invariance
is lost and the Green function is not well approximated by the logarithm
anymore. The same is also true of the spin-wave approximation on the
lattice, i.e., the bilinear action on the penultimate line of Eq. (4.8).
184
4. KOSTERLITZ-THOULESS TRANSITION
The dimensionless energy stored in Θ is finite in the thermodynamic
(infrared) limit L → ∞ if and only if charge neutrality holds,
M
X
mi = 0.
(4.42)
i=1
For example, a single-vortex configuration carrying charge m stores the
energy
a
L
L
2
2
2
πm J ln
− πm J ln
= πm J ln
(4.43)
`
`
a
which seems insurmountable in the thermodynamic limit. In two dimensions, the fact that the two-dimensional Green function for Laplace
operator −∂µ2 ≡ −∆ grows logarithmically at long distances as a result of scale invariance means that one must be careful when balancing
energy with entropy. The entropy of a single vortex is proportional
to the logarithm of all the distinct ways of placing it in the underlying square lattice, the proportionality constant being the Boltzmann
constant which was set to unity, i.e.,
2
L
(4.44)
ln
.
a
Thus, the reduced free energy βF of a single vortex becomes

 > 0 if K > 2 2 ,
πm
L
βF = πm2 K − 2 ln
a  ≤ 0 if K ≤ 2 .
πm2
(4.45)
At sufficiently high temperatures, the free energy for a single vortex
is dominated by the entropy gain. At sufficiently low temperatures
(the lower |m|, the lower one must go down in temperature), the free
energy for a single vortex is dominated by the energy cost. By this
argument, the Kosterlitz-Thouless criterion, vortices become important
when K ∼ 2/π. [41]
Equation (4.45) suggests the following scenario that would reconcile
the prediction of the spin-wave approximation with the prediction of
the high-temperature expansion on the 2d–XY model.
At zero temperature, no vortices are allowed for energetic reasons.
At any finite temperature, the condition of charge neutrality must apply to vortices in the thermodynamic limit, i.e., vortices come in pairs
of opposite charges. A quasi-long-range ordered phase driven by spin
waves for sufficiently small temperatures can only survive the presence
of vortices if vortices form bound states due to their strong logarithmic
interaction at large distances, the simplest bound state being a dipole.
The lower the temperature the fewer and tighter the bound states. As
the size of a bound state becomes of the order of the lattice spacing,
the vortices making up the bound state annihilate. Far away from
4.2. CLASSICAL TWO-DIMENSIONAL XY MODEL
185
bound states of vortices, the local disturbance to the spin-wave texture
induced by vortices, as is depicted in Fig. 3, is negligible.
In the opposite limit of high temperatures, there is room for an
entropy driven transition by which single-vortices behave like a weakly
interacting gas for sufficiently high densities. Here, increasing the temperature increases the density of bound states until a critical density
is reached above which screening of the bare logarithmic interaction
takes place at long distances and vortex bound states unbind. Vortices
strongly disrupt ferromagnetic or quasi-long-range order on a microscopic scale as is depicted in Fig. 3 and, if they are driven by entropy to
unbind from tight-dipole pairs, they also strongly disrupt ferromagnetic
or quasi-long-range order on macroscopic scales. By this entropy-driven
mechanism, the quasi-long-range ordered phase would be washed out
and turned into a paramagnetic phase.
There are caveats to this scenario which can be simply stated by the
following interpretation of Eq. (4.45). There is only one dimensionless
parameter available, the ratio K of the Heisenberg exchange interaction
and the temperature, when describing the 2d–XY model or the 2d
O(2) NLσM augmented by vortices. The density of vortices cannot be
tuned independently of K, the density of vortices is a function of K.
The concept of screening of the bare logarithmic interaction between
vortices is nothing but interpreting K as a scale-dependent coupling
constant. A finite-temperature and entropy-driven phase transition
between a low-temperature phase – in which vortices are confined into
bound states – and a high-temperature phase – in which vortices are
deconfined – demands that the renormalized coupling constant Kren
renormalizes in the infrared limit to:
(1) infinity if the initial (bare) value Kbare corresponds to a temperature below the critical temperature.
(2) zero if the initial (bare) value Kbare corresponds to a temperature above the critical temperature.
If Kren renormalizes to zero whatever the value taken by Kbare , entropy
always dominates, ferromagnetic long-range order is destroyed at any
finite temperature, and the paramagnetic phase extends to all temperatures. If Kren renormalizes to infinity whatever the value taken by
Kbare , energy always dominates, quasi-long-range order extends to all
finite temperatures, and the paramagnetic phase only exist at T = ∞.
In the latter case, the 2d O(2) NLσM augmented by vortices does not
capture the physics of the 2d–XY model.
Needed is thus a field theory for the 2d O(2) NLσM augmented
by vortices on which a renormalization-group analysis for the dependence of the interaction between vortices on length rescaling can be
performed.
A welcome simplifying feature is that spin waves and vortices do not
interact with each other at the Gaussian order of the gradient expansion
186
4. KOSTERLITZ-THOULESS TRANSITION
(4.8). The dimensionless energy of a spin wave φ superposed to a vortex
configuration Θ is additive,
Z
Z
Z
2 K
2 K
2
K
2
2
d x ∂µ φ + ∂µ Θ =
d x ∂µ φ +
d2 x ∂µ Θ ,
2
2
2
(4.46)
as follows after partial integration on the cross-term
Z
Z
2
e
d x(∂µ φ)(∂µ Θ) = d2 x(∂µ φ)(∂eµ Θ),
(4.47)
e is
keeping in mind that the two-dimensional vector field ∂µ Θ = ∂eµ Θ
divergence free and that spin waves vanish at infinity. It is therefore
sufficient to study the vortices alone. 3
Vortices form the so-called two-dimensional Coulomb gas (2d–Cb–
gas). From now on the vorticities will be restricted to
mi = ±1.
(4.48)
As the energy of a single vortex of vorticity m ∈ Z is proportional to
m2 , this simplification is of no consequence with regard to the existence
of a critical value Kc at which quasi-long-range order could be destroyed
by the entropy-driven deconfinement of vortices.
The simplest “physical” probe as a diagnostic of a putative transition from the spin-wave to the paramagnetic phase is the two-point
spin-spin correlation function
+i[φ(x)+Θ(x)] −i[φ(y)+Θ(y)] +i[φ(x)−φ(y)] +i[Θ(x)−Θ(y)] e
e
=
e
×
e
Zsw ×ZCb
Zsw
ZCb
1
a 2πK +i[Θ(x)−Θ(y)] × e
.
= ZCb
x − y
(4.49)
Needed
is the partition
function for the 2d–Cb–gas
and an evaluation
of e+i[Θ(x)−Θ(y)] Z . The correlation function e+i[Θ(x)−Θ(y)] Z must
Cb
Cb
decay algebraically at sufficiently low temperatures and exponentially
fast at sufficiently high temperatures if the vortex sector can account
for the paramagnetic phase of the 2d–XY model.
3
One might wonder if this decoupling of spin waves and vortices is an artifact
of the Gaussian approximation to the gradient expansion. Villain has shown in
Ref. [44] that this decoupling is not an artifact in that he constructed a lattice
model, now called the Villain model, that shares with the XY model the compact
nature of angular degrees of freedom but is nevertheless Gaussian. The decoupling
between spin waves and vortices is a rigorous property of the Villain model, as
is shown in appendix D. The phase diagram of the Villain model is believed to
be identical to the one of the XY model in that the same phases of matter are
separated by the same phase transitions. Transition temperatures are not equal,
though, as they reflect different microscopic details. [45]
4.3. THE COULOMB-GAS REPRESENTATION OF THE CLASSICAL 2d–XY MODEL
187
4.3. The Coulomb-gas representation of the classical 2d–XY
model
We cannot rely solely on the naive continuum limit of the 2d–XY
model to define the 2d–Cb–gas. The continuum limit is scale invariant and vortices of opposite charges are not prevented from collapsing
towards each other under the attractive force
m m x − x2
F (x1 − x2 ) = + 1 2 1
(4.50a)
2π |x1 − x2 |2
induced by the two-body potential
x
1
(4.50b)
VCb;a (x) = − ln 2π
a
that solves Poisson equation
(−∂µ2 VCb;a )(x) = δ(x)
(4.50c)
with the boundary condition
VCb;a (a) = 0.
(4.50d)
A hardcore two-body repulsive potential between all vortices that vanishes for separations larger than the lattice spacing must be introduced
by hand. In the presence of this hardcore repulsive potential, vortices cannot occupy the same volume a2 . Another effect of the hardcore repulsive potential is to change the so-called vortex core energy
VCb;a (a) = 0 to the finite positive value Ecore (a). The vortex core energy is unambiguously defined in the 2d–XY model as the value of
the two-point function hcos(φi − φj )iZXY when i = j. The vortex core
energy could also be taken as the limit i = j of the two-point function
hcos(φi − φj )iZSW where ZSW is the lattice partition function defined
with the Gaussian link energy on the penultimate line of Eq. (4.8).
From the point of view of field theory, Ecore (a) is a high-energy cutoff that requires a lattice regularization for its determination and thus
depends on detailed knowledge of the microscopic theory.
The grand-canonical partition function for the 2d–Cb–gas in a finite
volume Ω ⊂ R2 of linear dimension L is
M
M
∞
∞
X
X
2
Y+ + Y− −
βCb
ZCb :=
exp +
M+ − M− VCb;a (L)
M
!
M
!
2
+
−
M+ =0 M− =0


M ≡M+ +M−
Z
X
× DM [x] exp −βCb
mk ml VCb;a (xk − xl ) .
Ω
1≤k<l
(4.51a)
Here,
Y ≡ Y+ = Y− = eβCb µCb ,
1
µCb = − Ecore;Cb (a)
2
(4.51b)
188
4. KOSTERLITZ-THOULESS TRANSITION
is the fugacity of unit-charge vortices, βCb is the 2d–Cb–gas inverse
temperature, and
d2 xM+ d2 xM+ +1
d2 xM+ +M−
d2 x 1
DM [x] := 2 · · ·
···
f (x1 , · · · , xM )
a
a2
a2
a2
d2 xM+ d2 xM+ +1
d2 xM+ +M−
d2 x1
,
≡ 2 ···
···
a2
a2
a2
{z
}
|a
6=
(4.51c)
is the infinitesimal volume element of phase space. The function f
vanishes whenever two of its arguments are within a lattice spacing.
Otherwise f is unity. The grand-canonical partition of the 2d–Cb–gas is
related to the vortex sector of the 2d–XY model once the identification
[see Eq. (4.40)]
(2π)2 K → βCb .
(4.52)
is made.
4.4. Equivalence between the Coulomb gas and Sine-Gordon
model
4.4.1. Definitions and statement of results. An equivalence
between the grand-canonical partition function of the 2d–Cb–gas (4.51)
and the canonical partition function of the two-dimensional Sine-Gordon
(2d–SG) model will be derived. This equivalence allows to make contact between the 2d–XY model and quantum systems in one-space and
one-time dimensions.
This equivalence can be proved at different levels. The strongest
equivalence consists in establishing a one-to-one correspondence between all correlation functions in the 2d–Cb–gas with all correlation
functions in the 2d–SG model. A weaker equivalence is the proof that
the partition function of the 2d–SG model can be rewritten as the
grand-canonical partition function of the 2d–Cb–gas.
The definition of the 2d–SG model is that of a real-valued scalar
field in two-dimensional Euclidean space that is self-interacting with a
cosine potential. Thus, the generating function for the 2d–SG model is

Z
ZSG [J] :=
Z
D[θ] exp −
d2 x

1
h
(∂µ θ)2 − cos θ + Jθ  .
2t
t
Ω
(4.53a)
The real and positive parameter t plays the role of a dimensionless
temperature. The real parameter h plays the role of a magnetic field.
It carries the dimension of inverse area. Differentiation with respect of
4.4. EQUIVALENCE BETWEEN THE COULOMB GAS AND SINE-GORDON MODEL
189
the source J with dimensions of inverse area generates all correlation
functions for the scalar field θ. The model is defined on a large domain
Ω of linear extent L in the Euclidean plane. We will always assume the
boundary condition [recall Eq. (4.39)]
Z
d2 x ∂µ θ(∂µ θ) = 0.
(4.53b)
Ω
The dependence on the lattice spacing a indicates the need to introduce
a short-distance cutoff to regularize the theory at short distances.
We show in section 4.4.2 how to obtain the grand-canonical partition function of the 2d–Cb–gas model through a formal power expansion in the reduced magnetic field of the canonical partition function
of the 2d–SG model. More precisely, we show that, with the inverse
x
1
VCb;L (x) := −
ln 2π
L
x
L
1
1
(4.54)
ln +
ln =−
2π
a
2π
a
By Eq. (4.50b) ≡ VCb;a (x) − VCb;a (L)
of the 2d–Laplace operator −∂µ2 ≡ −∆, i.e., the 2d–Cb potential for
a point charge in two dimensions that implements boundary condition
(4.53b), the 2d–SG partition function (4.53a) becomes


Z
Z
1
h
ZSG [J = 0] =
D[θ] exp − d2 x
(∂µ θ)2 − cos θ 
2t
t
∝
∞
X
∞
X
M+ =0 M− =0
Z
×
M
z+ +
Ω
M
z− −
M+ ! M− !

e+
βSG
(M+ −M− )2 VCb;a (L)
2
M =M+ +M−
DM [x] exp −βSG
Ω
X

mk ml VCb;a (xk − xl ) .
1≤k<l
(4.55a)
Here, the effective fugacities z± , inverse temperature βSG , and phase
space measure DM [x] are
h
1
z± :=
× eβSG µSG ,
µSG := − Ecore;SG (a),
βSG := t,
2t
2
d2 x
d2 xM
DM [x] := 2 1 · · ·
f (x1 , · · · , xM ),
M = M+ + M− ,
2
a
a
 0, if ∃k, l, such that 1 ≤ k < l ≤ M , and |xk − xl | ≤ a,
f (x1 , · · · , xM ) :=
|
{z
}

1, otherwise,
∈Ω×···×Ω
(4.55b)
190
4. KOSTERLITZ-THOULESS TRANSITION
respectively.
In other words, vortices have a hardcore radius a L and hardcore
energy Ecore;SG (a), respectively. The hardcore energy Ecore;SG (a) must
be determined microscopically, i.e., it cannot be derived within the 2d–
SG field theory. Rather, it plays here the role of a (phenomenological)
high-energy cutoff. By comparison with Eq. (4.51), we infer that the
2d–XY , Cb–gas, and SG models are equivalent once
t
h
2
−πK Ddia
(2π) K ↔ βCb ↔ t,
e
↔Y ↔
× e− 2 Ecore;SG (a) ,
2t
(4.56)
have been identified. Here, Ddia is defined in appendix D to be the
diagonal matrix element of the Green function for the Villain model on
the square lattice, see Eq. (D.24a).
4.4.2. Formal expansion in powers of the reduced magnetic
field. We perform the formal expansion
+ ht
R
d2 x cos θ
e
=
Z
∞
M Z
X 1
h
2
d x1 · · · d2 xM (cos θ)(x1 ) · · · (cos θ)(xM ) =
M
!
t
M =0
Ω
Ω
Ω
MZ
Z
∞
X
1
h
2
d x1 · · · d2 xM e+iθ(x1 ) + e−iθ(x1 ) · · · e+iθ(xM ) + e−iθ(xM ) .
M ! 2t
M =0
Ω
Ω
(4.57)
Insertion of this expansion in the generating function (4.53a) yields,
owing to the fact that one can freely rename integration variables,
M X
Z
Z
M ∞
X
h
1
M
2
d x1 · · · d2 xM −m
ZSG [J] =
m
M ! 2t
m=0
M =0
Ω
Ω
(4.58a)
* MP
+unnor
−m
m
Z
Z
P
d2 y 1 · · ·
×
Ω
i
d2 y m
e
k=1
θ(xk )−i
l=1
θ(y l )
,
J
Ω
where
h (· · · )
iunnor
J
Z
:=
−
D[θ] e
R
Ω
1
d2 x( 2t
(∂µ θ)2 +Jθ)
(· · · )
(4.58b)
denotes an unnormalized average. Consider the source term with dimension of inverse area given by
J (x) := J(x) −
M
−m
X
k=1
iδ(x − xk ) +
m
X
l=1
iδ(x − y l ).
(4.59)
4.4. EQUIVALENCE BETWEEN THE COULOMB GAS AND SINE-GORDON MODEL
191
Needed is the term by term evaluation of
M X
Z
Z
∞
M X
1
h
M
2
ZSG [J] =
d x1 · · · d2 xM −m
m
M
!
2t
m=0
M =0
Ω
Ω
Z
×
d2 y 1 · · ·
Ω
Z
−
d2 y m e
R
d2 xJ θ
unnor
(4.60)
.
Ω
J=0
Ω
Completion of the square gives
− 12 1 R 2 R 2
R 2
unnor 2 −1 (x−y)J (y)
− d xJ θ
+ 2 d x d yJ (x)(− 1t ∂µ
)
1
2
e Ω Ω
.
e Ω
= Det − ∂µ
t
J=0
(4.61)
Equation (4.61) suggests defining
(M,m)
Seff [J]
t
:= −
2
Z
2
Z
dx
Ω
d2 yJ (x) VCb;L (x − y) J (y).
(4.62)
Ω
(M,m)
Two comments are of order here. First, if Seff [J] is to be interpreted as a dimensionless action, it has to be positive definite. Second,
Eq. (4.62) is form invariant under rescaling of the coordinates. The latter observation allows to extract the explicit dependence of Eq. (4.62)
on L from
(M,m)
Seff [J]
t
=−
2
+ it
Z
2
dx
Ω
Ω
M
X
Z
mk
k=1
+
Z
d2 y J(x) VCb;L (x − y) J(y)
d2 x J(x) VCb;L (x − z k )
(4.63a)
Ω
M
X
t
m m V
(z − z l ),
2 k,l=1 k l Cb;L k
where z k = xk (z k = y k ) for k = 1, · · · , M −m (k = M −m+1, · · · , M )
and mk = +1 (mk = −1) for k = 1, · · · , M −m (k = M −m+1, · · · , M ),
by trading VCb;L for VCb;a , as is done on the last line of Eq. (4.54).
192
4. KOSTERLITZ-THOULESS TRANSITION
Hence,
(M,m)
Seff [J]
t
=−
2
Z
2
Z
dx
Ω
d2 y J(x) VCb;L (x − y) J(y)
Ω
Z
Z
M
X
2
− it(M − 2m)VCb;a (L) d x J(x)+ it
mk d2 x J(x)VCb;a (x − z k )
k=1
Ω
Ω
M
X
t
t
− (M − 2m)2 VCb;a (L) +
m m V (z − z l ).
2
2 k,l=1 k l Cb;a k
(4.63b)
Equation (4.63) shows that the relationships between correlation functions in the 2d–SG model and the 2d–Cb–gas are non-local. The SG
field θ is not related to the spin-wave field φ in a local way.
If the source J for the correlation functions in the 2d–SG model is
set to zero,
∞ X
M
h M −m
h m
X
2t
2t
ZSG [J = 0] ∝
(M
−
m)!
m!
M =0 m=0
+ 2t (M −2m)2 VCb;a (L)
Z
×e
d2 z 1 · · ·
Ω
Z
d2 z M e
− 2t
M
P
k,l=1
mk ml VCb;a (z k −z l )
.
Ω
(4.64)
Finally, by attaching the contribution to the Boltzmann weight from
the self-energy of the Coulomb gas to the dimensionless ratio h/(2t),
we obtain
h
iM −m h
im
− 2t VCb;a (a)
− 2t VCb;a (a)
h
h
∞
M
×
e
×
e
X X 2t
2t
ZSG [J = 0] ∝
(M − m)!
m!
M =0 m=0
+ 2t (M −2m)2 VCb;a (L)
Z
×e
2
Z
d z1 · · ·
Ω
2
−t
d zM e
M
P
1≤k<l
mk ml VCb;a (z k −z l )
Ω
(4.65)
Comparison with the partition function (4.51) of the 2d–Cb–gas yields
the 2d–Cb–gas representation of the 2d–SG–model, Eqs. (4.55a) and
(4.56), after a core energy has been added by hand.
4.4.3. Sine-Gordon representation of the spin-spin correlation function in the 2d–XY model. The two-point spin-spin correlation function in the 2d–XY model is approximated by Eq. (4.49) in
the 2d O(2) NLσM augmented by vortices. How should one represent
.
4.4. EQUIVALENCE BETWEEN THE COULOMB GAS AND SINE-GORDON MODEL
193
the 2d–Cb gas correlation function
+i[Θ(x)−Θ(y)] e
Z
(4.66)
Cb
within the 2d–SG field theory?
On the one hand, we perform the following manipulations. Let Γx,y
be a smooth path connecting x to y within the region Ω defined before
Eq. (4.21). The argument of the two-point correlation function (4.66)
can then be written as the following line integral
Z
dsµ ∂µ Θ (s)
i [Θ(y) − Θ(x)] = i
Γx,y
Z
By Eq. (4.33)
e (s)
dsµ ∂eµ Θ
= i
Γx,y
By Eq. (4.24)
= −i
X
mi
i
∂ν ln(L/`) = 0
= +2πi
s − xi ∂
dsµ µν
ln
∂sν ` Z
Γx,y
X
i
Z
mi
dsµ µν
∂
V
(s − xi ).
(4.67)
∂sν Cb;L
Γx,y
On the other hand, Eqs. (4.67) and (4.59) suggest choosing the
source
JΓx,y (s) =
JΓx,y (s)
{z
|
Z
:= α
Γx,y
duµ µν
−i
M
−m
X
M
−m
X
k=1
δ(s − xk ) −
!
δ(s − y l )
l=1
k=1
}
∂
δ(s − u) − i
∂uν
δ(s − xk ) −
m
X
m
X
!
δ(s − y l )
l=1
(4.68)
in the 2d–Cb–gas expansion of the SG model (4.60). Indeed,
Z
Z
M
X
∂
2
d s JΓx,y (s) θ(s) = α
duµ µν
θ (u) − i
mk θ(z k ),
∂uν
k=1
Γx,y
(4.69)
where the constant α will be fixed shortly and z k = xk (z k = y k ) for
k = 1, · · · , M − m (k = M − m + 1, · · · , M ) and mk = +1 (mk = −1)
for k = 1, · · · , M − m (k = M − m + 1, · · · , M ). To fix the constant α
for the Ansatz
Z
∂
duµ µν
JΓx,y (s) := α
δ(s − u)
(4.70)
∂uν
Γx,y
194
4. KOSTERLITZ-THOULESS TRANSITION
that we made in Eq. (4.68), observe that Eq. (4.63a) with the Ansatz
(4.68) as argument becomes
Z
Z
t
(M,m)
2
Seff [JΓx,y ] = −
d s d2 t JΓx,y (s)VCb;L (s − t)JΓx,y (t)
2
+ it
Ω
Ω
M
X
Z
mk
k=1
d2 s JΓx,y (s)VCb;L (s − z k )
Ω
M
X
t
m mV
(z − z l )
2 k,l=1 k l Cb;L k
Z
Z
t 2
duµ
dūµ̄ µν µ̄ν̄ ∂ν ∂ν̄ VCb;L (u − ū)
=− α
2
+
Γx,y
+ itα
M
X
Γx,y
Z
duµ µν ∂ν VCb;L (u − z k )
mk
k=1
Γx,y
M
t X
+
m mV
(z − z l ).
2 k,l=1 k l Cb;L k
(4.71)
The constant α can now be chosen by demanding that the penultimate
(underlined) line reduces to the right-hand side of Eq. (4.67), i.e,
2π
1
(4.72)
=
.
t
2πK
It is time to turn our attention to the term quadratic in JΓx,y . It
can be simplified with the help of
α=
µν µ̄ν̄ = δµµ̄ δν ν̄ − δµν̄ δν µ̄ .
(4.73)
It becomes
t
D(x − y) := −
2
2π
t
2 Z
Z
duµ
Γx,y
=
=
Eq. (4.56)
− 2t
R
2π 2
+ 2t
R
2π 2
t
t
duµ
Γx,y
Γx,y
R
Γx,y
Γx,y
duµ
dūµ̄ µν µ̄ν̄ ∂ν ∂ν̄ VCb;L (u − ū)
R
ū 1
dūµ̄ −δµµ̄ δ(u − ū) − − 2π
∂µ ∂µ̄ ln u−
L
dūµ δ(u − ū) −
Γx,y
2π x − y = constant +
ln t
a x − y 1
.
= constant +
ln 2πK a 2π
t
ln La − ln y−x
L
(4.74)
4.4. EQUIVALENCE BETWEEN THE COULOMB GAS AND SINE-GORDON MODEL
195
(a)
(b)
h
8⇡
(c)
Y
2
⇡
t
K
Y
1
T /Tc
Figure 4. (a) Stability analysis of the critical line
h = 0 in the 2d–SG model. Vertical arrows indicate
whether the magnetic field decreases or increases under
coarse graining. (b) Stability analysis of the critical line
Y = 0 in the 2d–Cb–gas model. Vertical arrows indicate
whether the fugacity decreases or increases under coarse
graining. (c) Same as in (b) but with the horizontal axis
K replaced by T /TKT = KKT /K, KKT = 2/π.
To recapitulate, we have found that the effective action (4.62) becomes
(M,m)
Seff [JΓx,y ]
M
t X
= D(x−y)−i [Θ(x) − Θ(y)]+
m mV
(z −z l ),
2 k,l=1 k l Cb;L k
(4.75)
when the source JΓx,y is chosen as in Eqs. (4.70) and (4.72), whereby:
• D(x−y) is independent of the M −m positive vortices located
at xk and the m negative vortices located at y l .
• Up to a constant, exp − D(x − y) is the spin-wave two-point
function,
1
a + 2πK
−D(x−y)
constant
e
=e
× = econstant × e+i[φ(x)−φ(y)] sw .
x−y
(4.76)
In other words, we have proved that the spin-spin two-point correlation
function (4.49) in the 2d–XY model within the continuum spin-wave
approximation augmented by vortices has the 2d-SG representation
* 2π R
+
∂
−
du
θ
(u)
µ
µν
t
∂u
+i[φ(x)+Θ(x)]−i[φ(y)+Θ(y)] ν
e
= e Γx,y
.
Z ×Z
sw
Cb
ZSG
(4.77)
As a corollary, the 2d–SG representation of the 2d–Cb–gas correlation
function (4.66) is
1 * − 2π R du ∂ θ(u) +
µ µν ∂uν
a − 2πK
t
+i[Θ(x)−Θ(y)] Γx,y
e
=
e
.
x − y ZCb
ZSG
(4.78)
196
4. KOSTERLITZ-THOULESS TRANSITION
4.4.4. Stability analysis of the line of fixed points in the
2d–Sine-Gordon model. Now that we have identified the correlation
function in the 2d–SG model that approximates the spin-spin two-point
correlation function in the 2d–XY model, we can deduce from the 2d–
SG model the stability of the spin-wave phase. The spin-wave phase
is obtained by switching off vortices. In the 2d–SG model, turning
off the magnetic field h amounts to removing all vortices from the
Cb gas sector of the XY model by tuning the vortex core energy to
infinity. With the magnetic field h turned off, the 2d–SG model reduces
to the free 2d scalar field theory. All correlation functions for the
exponentiated SG scalar field exp[iθ(x)] are algebraic when the reduced
temperature t is finite. The horizontal line h = 0 in the t − h plane is a
line of critical points. The magnetic field, as a perturbation to this line
of fixed points, has engineering dimensions 2 and scaling dimension
t
(4.79a)
∆h =
= πK,
4π
since
lim hcos θ(x) cos θ(y)iZ
∼ lim e+i[θ(x)−θ(y)] Z
h→0
SG
h→0
tVCb;a (x−y)
SG
= e
t
a 2π
= x − y
a 2πK
= .
(4.79b)
x − y
In section 3.3.1, we saw that a small perturbation to a critical fixed
point in d-dimensions is infrared irrelevant, marginal, or relevant if
its scaling dimension is larger, equal, or smaller than d, respectively.
Hence, the magnetic field in the 2d–SG model (or, equivalently, the
fugacity in the 2d–Cb–gas) is infrared:
• irrelevant when
2
(4.80)
t > 8π ⇐⇒ K > ≡ KKT .
π
• marginal when
2
(4.81)
t = 8π ⇐⇒ K = ≡ KKT .
π
• relevant when
2
(4.82)
t < 8π ⇐⇒ K < ≡ KKT .
π
infrared irrelevance of the cosine potential means that the algebraic
(spin-wave) phase is stable to a weak perturbing magnetic field (vortices
with large core energy). infrared relevance of the cosine potential means
that the algebraic (spin-wave) phase is unstable to a weak perturbing
magnetic field (vortices with large core energy). The criterion (4.82)
section 3.3.2
4.5. FUGACITY EXPANSION OF n-POINT FUNCTIONS IN THE SINE-GORDON MODEL
197
for the instability of the algebraic phase agrees with the KosterlitzThouless criterion (4.45). 4 This stability analysis of the line of fixed
point h = Y = 0 is summarized by Fig. 4. Needed is a better grasp
of the flow obeyed by the reduced temperature (reduced spin stiffness)
and by the magnetic field (fugacity).
4.5. Fugacity expansion of n-point functions in the
Sine-Gordon model
We start from the 2d–SG Lagrangian
LSG [θ] :=
1
h
(∂µ θ)2 − cos θ.
2t
t
(4.83)
Here, we remember the bookkeeping
K :=
t
,
4π 2
Y ∼
h
,
2t
(4.84)
to make contact with the 2d–Cb–gas representation of the 2d–XY
model with reduced spin stiffness K. We shall denote thermal averaging with angular brackets,
Z
1
h(· · · )i :=
D[θ] e−SSG [θ] (· · · ),
ZSG
Z
(4.85)
−SSG [θ]
ZSG := D[θ] e
,
R
where SSG [θ] := d2 x LSG is the SG action obtained from Eq. (4.83).
Let n be a positive integer, choose n points x1 , · · · , xn on the
Euclidean plane, and define
Fx1 ,··· ,xn := ei1 θ(x1 ) · · · ein θ(xn ) ,
1 , · · · , n = ±1, (4.86a)
Z
hFx1 ,··· ,xn iunnor := D[θ] e−SSG [θ] Fx1 ,··· ,xn .
(4.86b)
Thermal averaging of Eq. (4.86a) is obtained by dividing the unnormalized average in Eq. (4.86b) by the SG partition function,
hFx1 ,··· ,xn i =
hFx1 ,··· ,xn iunnor
ZSG
.
(4.87)
We are going to compute both hFx1 ,··· ,xn i and hFx1 ,··· ,xn iunnor through
a formal power expansion in h.
4
Vortices with charge m ∈ Z are induced by the cosine potential cos(mθ) in
the 2d–SG model.
198
4. KOSTERLITZ-THOULESS TRANSITION
To this end, the key identity that is needed is [recall the expansion
done in Eq. (4.58a)]
m X
Z
∞
m X
1
h
m
=
d2 y 1 · · · d2 y m
p
{z
}
|
m!
2t
m=0
p=0
6=
D
Eunnor
× eiθ(y1 ) · · · eiθ(yp ) e−iθ(yp+1 ) · · · e−iθ(ym ) Fx1 ,··· ,xn
.
unnor
hFx1 ,··· ,xn i
h=0
(4.88)
Integrations over coordinates are done with the hardcore constraint
that no two points ever coincide as indicated by the underbrace. Implementing the hardcore constraint is equivalent to a renormalization
of the magnetic field h by the core energy of the vortices in the 2d–Cb–
gas interpretation. The renormalized magnetic field is essentially the
2d–Cb–fugacity. Thermal averaging on the last line must be performed
with h = 0, in which case averaging over θ is Gaussian.
4.5.1. Fugacity Expansion of the two-point function. We
treat the case of the two-point function
Fx1 ,x2 := eiθ(x1 )−iθ(x2 ) ≡ F12 .
(4.89)
The expansion in powers of h/2t of its thermal average is
hF12 i ≡
∞
X
(n)
F12 (h/2t)n
n=0
∞
P
=:
(m)
f12 (h/2t)m
m=0
∞
P
1+
,
Z (n)
(4.90a)
(h/2t)n
n=1
(2n+1)
where F12
(2n+1)
= f12
(0)
(0)
(2)
F12
(2)
f12
= Z (2n+1) = 0,
F12 = f12 ,
(4)
=
(4)
−
(4.90b)
(0)
f12 Z (2) ,
(2)
(4.90c)
(0)
(0)
F12 = f12 − f12 Z (2) + f12 Z (4) + f12 Z (2) × Z (2) ,(4.90d)
4.5. FUGACITY EXPANSION OF n-POINT FUNCTIONS IN THE SINE-GORDON MODEL
199
and the generic term
(2n)
F12
(2n)
= f12
(2n−2) (2)
(0)
− f12
Z + · · · + f12 Z (2n)

(2n−4)
+ f12
(2n−6)
(2)
Z
× Z (2)} +2f12
| {z
2−times

(2)
Z
× Z (4)} + · · ·  − · · ·
| {z
2−times
(0)
(2)
+ (−)n f12 Z
· · × Z (2)} .
| × ·{z
n−times
(4.90e)
The coefficients in the power expansions in h/2t of the numerator and
denominator are
Z
D
1
(2n)
2
2
iθ(y 1 )+···+iθ(y n )−iθ(y n+1 )−···−iθ(y 2n )
f12 =
d
y
·
·
·
d
y
2n e
}
(n!)2 | 1 {z
6=
Eunnor
× e+iθ(x1 )−iθ(x2 )
h=0
(4.90f)
and
Z
(2n)
1
=
(n!)2
Z
unnor
d2 y 1 · · · d2 y 2n eiθ(y1 )+···+iθ(yn )−iθ(yn+1 )−···−iθ(y2n ) h=0 ,
{z
}
|
6=
(4.90g)
respectively.
4.5.2. Two-point function to lowest order in h/2t. We need
(the short distance cutoff is set to one: a = 1)
(0)
(0)
F12 = f12 ,
unnor
(0)
f12 = eiθ(x1 )−iθ(x2 ) h=0 =
(4.91a)
1
,
|x1 − x2 |2πK
(4.91b)
t
.
2π
(4.91c)
where we have used [recall Eq. (3.106d)]
− (∂µ2 )−1 (x1 , x2 ) = −
1
ln |x1 − x2 |,
2π
2πK =
Thus,
(0)
F12 = |x1 − x2 |−2πK .
(4.92)
200
4. KOSTERLITZ-THOULESS TRANSITION
4.5.3. Two-point function to second order in h/2t. We need
(the short distance cutoff is set to one: a = 1)
(2)
(2)
(0)
F12 = f12 − f12 Z (2) ,
(4.93a)
Z
(2)
f12 =
unnor
d2 y 1 d2 y 2 eiθ(y1 )−iθ(y2 )+iθ(x1 )−iθ(x2 ) h=0
| {z }
6=
Z
=
1
d y1d y2
| {z
} |x1 − x2 |2πK |y 1 − y 2 |2πK
2
2
6=
|y 1 − x1 ||y 2 − x2 |
|y 1 − x2 ||y 2 − x1 |
2πK
,
(4.93b)
Z (2) =
Z
unnor
d2 y 1 d2 y 2 eiθ(y1 )−iθ(y2 ) h=0
| {z }
6=
Z
=
1
d2 y 1 d2 y 2
,
| {z } |y 1 − y 2 |2πK
6=
(4.93c)
(0)
f12 Z (2) =
Z
1
|x1 − x2 |2πK
1
,
d2 y 1 d2 y 2
| {z } |y 1 − y 2 |2πK
6=
(4.93d)
where we have used [recall Eq. (3.106d)]
− (∂µ2 )−1 (x1 , x2 ) = −
1
ln |x1 − x2 |,
2π
2πK =
t
.
2π
(4.93e)
Thus,
(2)
F12
1
=
|x1 − x2 |2πK
Z
d2 y 1 d2 y 2
| {z }
6=
Kx1 x2 (y 1 , y 2 ; 2πK) − Kx1 x2 (y 1 , y 2 ; 0)
|y 1 − y 2 |2πK
,
(4.94a)
where
Kx1 x2 (y 1 , y 2 ; κ) :=
|y 1 − x1 ||y 2 − x2 |
|y 1 − x2 ||y 2 − x1 |
κ
.
(4.94b)
We now estimate the integrals over y 1 and y 2 . To this end, we
introduce the center of mass and relative coordinates
1
Y := (y 1 +y 2 ),
2
1
y 12 := y 1 −y 2 ⇐⇒ y 1 = Y + y 12 ,
2
1
y 2 = Y − y 12 .
2
(4.95)
4.5. FUGACITY EXPANSION OF n-POINT FUNCTIONS IN THE SINE-GORDON MODEL
201
The first step is to approximate the difference of two logarithms to
third order in the relative coordinates,
1
1
ln |y 1 − x1 | − ln |y 2 − x1 | = ln |Y − x1 + y 12 | − ln |Y − x1 − y 12 |
2
2
∂ ln |Y − x1 |
· y 12 + O(y 312 ),
= +
∂Y
(4.96a)
1
1
ln |y 2 − x2 | − ln |y 1 − x2 | = ln |Y − x2 − y 12 | − ln |Y − x2 + y 12 |
2
2
∂ ln |Y − x2 |
= −
· y 12 + O(y 312 ).
∂Y
(4.96b)
The second step is to expand the exponential of the difference of
logarithms in powers of the relative coordinates:
Kx1 x2 (y 1 , y 2 ; κ) :=
|y 1 − x1 ||y 2 − x2 |
|y 1 − x2 ||y 2 − x1 |
κ
∂ (ln |Y −x1 |−ln |Y −x2 |)
3 )
κ
·y
+O(y
12
12
∂Y
= e
= 1
∂ (ln |Y − x1 | − ln |Y − x2 |)
· y 12
∂Y
2
1 2 ∂ (ln |Y − x1 | − ln |Y − x2 |)
+
κ
· y 12
2
∂Y
+ O(y 312 ).
(4.97)
+ κ
The third step is to perform the integration over the measure d2 y 1 d2 y 2 =
d2 Y d2 y 12 , where the hardcore constraint is implemented by |y 12 | > a,
with the approximation (4.97) of the function Kx1 x2 (y 1 , y 2 ; κ). We will
assume that the domain of integration is invariant under y 12 → −y 12 .
The two terms independent of y 12 in the numerator of the integrand
cancel out. By assumption integration over the terms linear in y 12 in
the numerator of the integrand vanish. Finally, we are left with the
contribution from the term quadratic in y 12 in the numerator of the
202
4. KOSTERLITZ-THOULESS TRANSITION
integrand. We thus need, for any function f (|z|),
Z
d2 z(x · z)2 f (|z|) =
Z∞
Z2π
drr
0
0
Z∞
Z2π
=
drr
0
Z∞
=
dϕ(x1 r cos ϕ + x2 r sin ϕ)2 f (r)
dϕ(x21 r2 cos2 ϕ + x22 r2 sin2 ϕ + 2x1 x2 r2 cos ϕ sin ϕ) f (r)
0
drrπ x21 r2 + x22 r2 f (r)
0
= π|x|
2
Z∞
drr3 f (r).
(4.98)
0
Notice that this intermediary result can also be understood as follows.
The measure and the integrand are invariant under rotations of the
domain of integration. Hence,
Z
2
2
d z(x·z) f (|z|) = |x|
2
Z
2
2
d z(n̂·z) f (|z|) = |x|
2
Z∞
3
Z2π
dr r f (r)
0
dϕ cos2 ϕ
0
(4.99)
for any n̂, n̂ · n̂ = 1. We conclude that (after reinserting the short
distance cutoff a)
Z
Z
h
i
2
d y 1 d2 y 2 Kx1 x2 (y 1 , y 2 ; κa ) − Kx1 x2 (y 1 , y 2 ; κb ) f (|y 1 − y 2 |) ≈
π 2
κa − κ2b
2
max
yZ12
3
Y
Zmax
∂ (ln |Y − x1 | − ln |Y − x2 |) 2
.
d Y ∂Y
d|y 12 ||y 12 | f (|y 12 |)
a
2
0
(4.100)
With the help of partial integration with respect to the center-of-mass
coordinate Y and the Green function (4.93e),
Y
Zmax
2
∂
(ln
|Y
−
x
|
−
ln
|Y
−
x
|)
1
2
=
d2 Y ∂Y
0
Y
Zmax
d2 Y (ln |Y − x1 | − ln |Y − x2 |) (−2π) [δ (Y − x1 ) − δ (Y − x2 )] .
0
(4.101)
4.5. FUGACITY EXPANSION OF n-POINT FUNCTIONS IN THE SINE-GORDON MODEL
203
Insertion of Eq. (4.101) into Eq. (4.100) gives
Z
Z
h
i
2
d y 1 d2 y 2 Kx1 x2 (y 1 , y 2 ; κa ) − Kx1 x2 (y 1 , y 2 ; κb ) f (|y 1 − y 2 |) ≈
max
yZ12
x
−
x
4π 2 2
2
κa − κ2b ln 1
d|y 12 ||y 12 |3 f (|y 12 |),
2
a
a
(4.102)
which implies
(2)
F12
 ymax

Z12
y −2πK
x x −2πK
12 4
2
3 12 
8π K
d|y 12 | |y 12 | ln 12 .
≈
a
a
a
a
(4.103)
By collecting all terms up to and including second order in the
fugacity, one can write
x −2πx(1)
hF12 i ≈ 12 ,
(4.104a)
a
Z∞
3 2 2
x(1) := K − 4π K Y
dy y 3−2πK .
(4.104b)
1
Here, we have introduced the squared dimensionless fugacity
2 2
ah
2
.
(4.104c)
Y :=
2t
Before proceeding to an RG interpretation of this result, we observe
that when K is larger than 2/π, then the y integral is convergent and
the scaling exponent x(1) is a well-defined number. The assumption
that the density of vortices is small is then consistent. On the other
hand, when K is smaller or equal to 2/π, then the y integral is divergent
so that the scaling exponent x(1) is ill-defined. The assumption that
the density of vortices is small is not consistent. The bare logarithmic
interaction between vortices is screened so as to change the functional
form of the decay of the spin-spin correlation function.
Notice that if we rewrite
Z∞
dy y
1
3−2πK
Zel
=
dy y
1
3−2πK
Z∞
+
dy y 3−2πK
el
el(4−2πK) − 1
=
+ el(4−2πK)
4 − 2πK
Z∞
1
dy y 3−2πK ,(4.105)
204
4. KOSTERLITZ-THOULESS TRANSITION
then Eq. (4.104b) is form invariant, i.e.,
0 2
x(1) = K 0 − 4π 3 K 2 (Y )
Z∞
dy y 3−2πK ,
(4.106a)
1
where
el(4−2πK) − 1
4 − 2πK
3 2 2
K − 4π K Y l + O(l2 ),
a2 h0
2t
Y el(2−πK)
Y [1 + l(2 − πK) + O(l2 )].
K 0 := K − 4π 3 K 2 Y 2
=
Y 0 :=
=
=
(4.106b)
(4.106c)
If l is chosen to be arbitrary small, Eqs. (4.106b) and (4.106c) can be
rewritten as differential equations,
dK
= −4π 3 K 2 Y 2 ,
dl
(4.107a)
dY
= (2 − πK)Y.
(4.107b)
dl
These equations were first derived by Kosterlitz in Ref. [42]. They
are often called the Kosterlitz-Thouless RG equations in the literature.
Kosterlitz derived these equations by using the Coulomb gas representation of the 2d–XY model. Here, we used the Sine-Gordon representation of the 2d–XY model to derive the KT RG equations. [46]
Equation (4.107b) reproduces the Kosterlitz-Thouless criterion (4.45)
and the scaling analysis (4.82). However, there is more information to
be gained from Eq. (4.107a) that encodes the screening of the logarithmic interaction between a pair of vortices far apart as anticipated by
Berezinskii in Ref. [40]. Kosterlitz RG equations are left invariant by
Y → −Y.
(4.108)
This invariance reflects the condition of charge neutrality; vortices occur in pairs of opposite charges in the thermodynamic limit.
4.6. Kosterlitz renormalization-group equations
4.6.1. Kosterlitz RG equations in the vicinity of X = Y = 0.
In this section we want to analyze in details the Kosterlitz RG equations
(4.107a) and (4.107b) in the vicinity of K = 2/π and Y = 0. Define
the scaling variable
X := 2 − πK ⇐⇒ K =
(2 − X)
.
π
(4.109)
4.6. KOSTERLITZ RENORMALIZATION-GROUP EQUATIONS
205
Y
II
II
I
III
X
Figure 5. Kosterlitz RG flow in the vicinity of fixed
point X = Y = 0 in the half-plane Y > 0. The flow in
the half-plane Y < 0 is obtained by reflection symmetry
about the line Y = 0.
The physical interpretation of X is that it is proportional to the deviation in the temperature T of the 2d–XY model away from the
Kosterlitz-Thouless critical temperature (4.82), (recall that the Boltzmann constant is set to unity)
X ≡ π (KKT − K) ≡ πJ
1
TKT
1
−
T
πJ
=
TKT
(
T − TKT
+O
TKT
"
T − TKT
TKT
(4.110)
Equations (4.107a) and (4.107b) are rewritten
dX
= +4π 2 (2 − X)2 Y 2 ,
dl
(4.111a)
dY
= XY.
dl
(4.111b)
in the X − Y coupling plane.
Next, we expand Eqs. (4.111a) and (4.111b) in the vicinity of the
fixed point
0 = X,
0 = Y,
(4.112)
to the first non-trivial order. We thus find
dX
= +(4π)2 Y 2 + O(XY 2 ),
dl
(4.113a)
dY
= XY.
dl
(4.113b)
2 #)
.
206
4. KOSTERLITZ-THOULESS TRANSITION
These two equations are brought to the more symmetric form
dX
= +2Y 2 ≡ βx ,
(4.114a)
dl
dY
= 2XY ≡ βy ,
(4.114b)
dl
by another redefinition of the running coupling constants X and Y and
of the rescaling parameter l,
1
l → l/2.
(4.114c)
X → 4X,
Y → Y,
π
It is possible to find curves in the X − Y coupling plane that are
invariant under the Kosterlitz RG equations (4.114a) and (4.114b).
Define the family of hyperbolas parametrized by α ∈ R
Γα : R −→ R2 ,
X(l)
l −→
,
Y (l)
X 2 (l) − Y 2 (l) = α.
(4.115)
Under the Kosterlitz RG equations (4.114a) and (4.114b)
d 2
X (l) − Y 2 (l) = 2 X(l)βx − Y (l)βy
dl
= 4 (XY 2 )(l) − (Y XY )(l)
= 0,
∀l ∈ R.
(4.116)
In view of the invariance of the Kosterlitz RG equations (4.111a) and
(4.111b) under Y → −Y , we need to distinguish three cases.
(1) When α > 0, the hyperbola (4.115) is parametrized by (without loss of generality, Y ≥ 0)
√ 1 + s2
√
2s
X = (±) α
,
Y = α
,
0 ≤ s < 1.
(4.117)
2
1−s
1 − s2
Equation (4.114a) reads
dX
− 2Y 2
0 =
dl
√ 2s(1 − s2 ) + (1 + s2 )2s ds
4s2
= (±) α
−
2α
(1 − s2 )2
dl
(1 − s2 )2
√
√
4s
ds
= (±) α
−
(±)2
αs .
(4.118)
(1 − s2 )2
dl
The RG equation for the parameter s is
√
ds
= (±)2 α s,
(4.119a)
dl
with the solution
√ s(l) = s(l0 ) exp (±)2 α l .
(4.119b)
4.6. KOSTERLITZ RENORMALIZATION-GROUP EQUATIONS
207
(a) When α > 0 and X(l) > 0 (high-temperature
phase), one
√
must choose the positive root + α in Eq. (4.117) and the
solution to Eq. (4.119a) is
√ s(l) = s(l0 ) exp +2 α l .
(4.120)
This is the solution corresponding to an initial inverse
reduced temperature K(l0 ) below Kc . 2/π. The fugacity, initially very small, grows exponentially fast until
the self-consistency of the perturbative expansion is lost.
This behavior is the one expected when vortices are relevant perturbations to the spin-wave phase. From the
high-temperature expansion, this phase is believed to be
the paramagnetic phase.
(b) When α > 0 and X(l) < 0 (low temperature
phase), one
√
must choose the negative root − α in Eq. (4.117) and
the solution to Eq. (4.119a) is
√ (4.121)
s(l) = s(l0 ) exp −2 α l .
This is the solution corresponding to an initial inverse
reduced temperature K(l0 ) above Kc & 2/π. The fugacity, initially very small, decreases exponentially fast. The
self-consistency of the perturbative expansion improves
under the RG group flow. This behavior is the one expected when vortices are irrelevant perturbations to the
spin-wave phase.
(2) When α < 0, the hyperbola (4.115) is parametrized by (without loss of generality, Y ≥ 0)
X=
p
2s
|α|
,
1 − s2
Y =
p
|α|
1 + s2
,
1 − s2
−1 < s < 1.
(4.122)
Equation (4.114a) reads
dX
− 2Y 2
dl
p 2(1 − s2 ) + 4s2 ds
(1 + s2 )2
=
|α|
−
2|α|
(1 − s2 )2
dl
(1 − s2 )2
p (1 + s2 )
p
ds
2
= 2 |α|
− |α|(1 + s ) .
(1 − s2 )2
dl
0 =
The RG equation for the parameter s is
ds p
= |α| 1 + s2 ,
dl
with the solution
p
arctan[s(l)] − arctan[s(l0 )] = |α| (l − l0 ) .
(4.123)
(4.124a)
(4.124b)
208
4. KOSTERLITZ-THOULESS TRANSITION
For any α < 0, s(l) increases with l. The fugacity, initially
very small, increases under the RG group flow until the selfconsistency of the perturbative expansion is lost. This behavior is the one expected when vortices are relevant perturbations
to the spin-wave phase.
According to this analysis, the half-plane Y > 0 can be divided into
three regions separated by the half-lines
Y = +X,
X > 0,
and
Y = −X,
X < 0,
(4.125)
respectively (see Fig. 5). Region I is defined by X < 0 and |X| >
Y > 0. This is the regime in which the spin-wave phase is stable to
the thermal nucleation of vortices with a large core energy. In this
regime spin-spin correlation functions decay algebraically fast and the
interaction between vortices grows logarithmically for large separations.
Vortices can only appear in tight bound states at low temperatures.
Quasi-long-range order is associated to an infinitely large correlation
length. Region II is defined by Y > |X| > 0. Region III is defined
by X > Y > 0. In both regimes II and III, the spin-wave phase is
unstable to the thermal nucleation of vortices. Spin-spin correlation
functions decay exponentially fast and the interaction between vortices
is screened at long distances. Vortices are deconfined at long distances.
The difference between region II and region III is that vortices are also
ultraviolet relevant in region II whereas they are ultraviolet irrelevant
in region III. In region II, the field theory never reduces to a free scalar
field theory obtained by ignoring the cosine potential, be it at long
or short distances. In region III, the field theory is asymptotically
free. The field theory reduces to the free scalar field theory obtained
by ignoring the cosine potential at short distances. The property of
asymptotic freedom is of little use to the understanding of the 2d–XY
model however, since there is no justification for approximating the
2d–XY model by a field theory on length scales of the order of the
lattice spacing. The separatrix Y = |X| is a line of phase transitions.
These transitions are continuous but very weak as we demonstrate
by estimating how the correlation length diverges upon approaching
X = Y = 0 from the high-temperature regime X > 0.
4.6.2. Correlation length near X = Y = 0. The initial value
of the fugacity Y0 ≡ Y (l0 ) at X0 ≡ X(l0 ) is inferred from Eq. (4.56).
We assume that it belongs to region II. There is one hyperbola (4.115),
that goes through the coordinate (X0 , Y0 ) in region II of the X − Y
coupling plane as is depicted in Fig. 6 when Y0 > X0 . This hyperbola
is labeled by the value
α0 = X02 − Y02 < 0.
(4.126)
This
p hyperbola intersects the fugacity axis at the value Yintersection =
|α0 |. The Kosterlitz RG flow (4.114a) and (4.114b) takes (X0 , Y0 ) to
4.6. KOSTERLITZ RENORMALIZATION-GROUP EQUATIONS
209
Y
Yl
Y0
I
Xc X0
II
III
Xl
X
Figure 6. Blow up of Fig. 5 in the vicinity of the fixed
point X = Y = 0 in the half-plane Y > 0. The dotted
line represents the initial value Y0 ≡ Y (l0 ) of the fugacity
as a function of the initial value X0 ≡ X(l0 ). The case
considered here is when (X0 , Y0 ), depicted by a black dot,
is in region II and is very close to the separatrix Y = −X.
The intercept (white dot) between the separatrix Y =
−X and the dotted line defines the critical temperature
and fugacity, i.e., Xc and Yc , respectively.
the point (X(l), Y (l)) of region II as is depicted in Fig. 6. When l l0 ,
we should expect to be deep in the paramagnetic phase. The correlation
length ξ(l) defined by the asymptotic exponential decay length of the
spin-spin two-point function should be very small for (X(l), Y (l)) ∼
(1, 1), say of the order of the lattice spacing. The question we want
to answer is what is the value of the initial correlation length ξ0 for
(X0 , Y0 ) very close to X = Y = 0?
The answer to this question requires two steps. First, we observe
that the transformation law obeyed by the correlation length under the
RG flow is
ξ(l) = ξ(l0 ) e−(l−l0 )/2 .
(4.127)
[The argument l/2 comes from the redefinition of l in Eq. (4.114c).] By
assumption
ξ(l) ∼ a.
(4.128)
The second step consists in expressing l in terms of X0 and Y0 given
that X0 and Y0 are very close to the origin X = Y = 0. To this end,
one divides Eq. (4.114a) by Eq. (4.114b),
dX
Y
=
dY
X
⇐⇒ XdX = Y dY
=⇒ X 2 (l) − Y 2 (l) = α0 .
(4.129)
210
4. KOSTERLITZ-THOULESS TRANSITION
By Eq. (4.114a)
Zl
l − l0 =
dl0
l0
X(l)
Z
=
dX
.
2Y 2
(4.130)
X0
By Eq. (4.129)
l − l0
1
=
2
X(l)
Z
X0
Region II has α0 < 0
1
=
2
dX
− α0
X2
X(l)
Z
X2
X0
dX
+ |α0 |
"
1
= p
arctan
2 |α0 |
X(l)
p
|α0 |
!
− arctan
!#
X0
p (4.131).
|α0 |
By assumption, |α0 | is very small. More precisely, note by inspection
of Fig. 6 that (X0 , Y0 ) is very close to the separatrix Y = −X. We may
do the linearization
"
2 #
T
−
T
T
−
T
c
c
X0 = −Y0 + c2 Y0
+O
,
(4.132)
Tc
Tc
where c2 is some positive number that depends on details at the microscopic level. Hence,
|α0 | = Y02 − X02
2
T − Tc
2
2
≈ Y0 − −Y0 + c Y0
Tc
T − Tc
≈ 2c2 Y02
.
Tc
(4.133)
On the other hand, we have also assumed that X(l) > 0 is of order 1.
√
Thus, to a first approximation, we may replace X(l)/ α0 by +∞ and
√
X0 / α0 by −∞. In this way,
−1/2
π
π
T − Tc
l − l0 ≈ p
≈ √
.
(4.134)
Tc
2 2 c Y0
2 |α0 |
This is the desired relationship for l in terms of the initial conditions.
At last we obtain the correlation length [recall the rescaling done in
4.6. KOSTERLITZ RENORMALIZATION-GROUP EQUATIONS
211
Eq. (4.114c)]
ξ(l0 ) ≡ ξ0 ≈ a × exp
π
−1/2
√
,
t
4 2 c Y0
t :=
T − Tc
.
Tc
(4.135)
The correlation length in the paramagnetic phase diverges faster
than any power of t upon approaching the Kosterlitz-Thouless transition. It can be shown that the regime of validity of Eq. (4.135) demands that t < 10−2 for which values ξ0 is at least 108 lattice spacings.
Testing Eq. (4.135) experimentally demands very clean samples and
is hopeless numerically. It is known from the theory of critical phenomena that the free energy per unit volume can be decomposed into
a regular and a singular contribution at a critical point and that the
singular contribution is roughly given by ξ0−2 upon approaching the
critical point from the side of the disordered phase. It follows that
the free energy per unit volume is of the form exp(−|const| × t−1/2 )
and thus has an essential singularity at t = 0. All derivatives of the
free energy per unit volume are continuous functions of t through the
transition. Consequently, the Kosterlitz-Thouless phase transition lies
outside of the 19-th century classification of phase transitions in terms
of the order at which the derivative of the free energy is discontinuous. The Kosterlitz-Thouless transition suggests that a perhaps better
classification of phase transitions should simply be one distinguishing
continuous from non-continuous (first order) phase transitions.
A more elegant way of expressing l and thus the correlation length
ξ(l0 ) in terms of X0 and Y0 is to choose once more a new set of running
coupling constants. Let
X+ := Y + X,
Y := 12 (X+ + X− ),
⇐⇒
X− := Y − X,
X := 12 (X+ − X− ).
(4.136)
This choice is nothing but a rotation by π/4 of Figs. 5 and 6. The
separatrix (4.125) are
X− = 0,
X+ = 0.
(4.137)
The hyperbolas (4.115) are
X+ X− = −α.
(4.138)
212
4. KOSTERLITZ-THOULESS TRANSITION
The Kosterlitz RG equations (4.114a) and (4.114b) are
dX+
= 2 XY + Y 2
dl
1
=
X+2 − X−2 + X+2 + X−2 + 2X+ X−
2
= X+2 + X+ X−
= X+2 − α,
(4.139a)
dX−
= 2 XY − Y 2
dl
1
=
X+2 − X−2 − X+2 − X−2 − 2X+ X−
2
= −X−2 − X+ X−
= −X−2 + α.
(4.139b)
These can be rewritten as
dX+
= −dl,
α − X+2
dX−
= +dl,
α − X−2
(4.140a)
(4.140b)
which can be immediately integrated. For X+ (l), it is found that

X+ (l)
X+ (l0 )


+arctan √
− arctan √
,
if α < 0,


|α|
|α|





p
n h√
i
h√
io
α+X+ (l)
α+X+ (l0 )
1
|α|(l−l0 ) =
√
√
− 2 ln α−X (l) − ln α−X (l ) , if α > 0 and α > X+2 ,

+
+ 0





n h
h
√ i
√ io


 − 1 ln X+ (l)+√α − ln X+ (l0 )+√α , if α > 0 and α < X+2 ,
2
X (l)− α
X (l )− α
+
+
0
(4.141)
whereas

X− (l)
X− (l0 )


−arctan √
+ arctan √
,


|α|
|α|





p
n h√
i
h√
io
α+X− (l)
α+X− (l0 )
1
|α|(l−l0 ) =
√
√
+
ln
−
ln
,
2

α−X− (l)
α−X− (l0 )





n h
h
√ i
√ io


 + 1 ln X− (l)+√α − ln X− (l0 )+√α ,
2
X (l)− α
X (l )− α
−
−
0
if α < 0,
if α > 0 and α > X−2 ,
if α > 0 and α < X−2 ,
(4.142)
for X− (l). Since we are interested in region II, we use the solution
"
#
"
#
p
X+ (l)
X+ (l0 )
|α0 |(l − l0 ) = +arctan p
− arctan p
(4.143)
|α0 |
|α0 |
4.6. KOSTERLITZ RENORMALIZATION-GROUP EQUATIONS
213
Y
II
II
I
III
X
Figure 7. Kosterlitz RG flow in the vicinity of fixed
point X = Y = 0 in region I. Initial conditions (black
dot) correspond to a temperature below Tc (white dot).
for X+ (l), which, by assumption, is very close to the separatrix Y = X,
i.e.,
X+ (l) ∼ 2Y (l) ∼ 2X(l) ∼ 1 X+ (l0 ) > 0.
(4.144)
The limit in which
X+ (l0 )
p
=
|α0 |
is small gives
p
π
|α0 |(l−l0 ) ∼ ,
2
s
X+ (l0 ) X (l ) −
π
l−l0 ∼ p
,
2 |α0 |
(4.145)
0
l − l0
π
∼ p
, (4.146)
2
4 |α0 |
so that Eqs. (4.127) and (4.128) become
ξ(l0 ) ∼ a e(l−l0 )/2 ∼ a exp
π
p
4 |α0 |
!
.
(4.147)
4.6.3. Universal jump of the spin stiffness. Below the critical
temperature Tc (white dot in Fig. 7), the RG trajectory takes the initial coupling constants (black dot in Fig. 7) to the Gaussian fixed line
Y = 0 at some particular value of X(l = ∞) > X0 or, equivalently,
K(l = ∞) < K0 . A finite value of K(l = ∞) means that screening
effects of the bare logarithmic vortex interaction are only partially effective in making spin-spin correlation functions decay faster with large
separations. The exponent
1
l→∞ 2πK(l)
η(K0 ) = lim
(4.148)
that characterizes the algebraic decay of the spin-spin two-point function in region I is non-universal and monotonically decreasing (increasing) function of K (i.e., T ). When K0 = Kc (i.e., T = Tc ), we may use
214
4. KOSTERLITZ-THOULESS TRANSITION
the fact that liml→∞ K(l) = KKT = (2/π) so that the value
1
1
=
l→∞ 2π × (2/π)
4
η(Kc ) = lim
(4.149)
is, however, universal.
The quantity K(l = ∞) is called the spin stiffness as it measures
the sensitivity to changes in the boundary conditions. To illustrate
this, choose a rectangular geometry with linear dimensions L × L0 in
the x̂ and ŷ directions. Periodic boundary conditions are imposed in
the ŷ direction. Twisted boundary conditions are imposed in the x̂
direction, i.e., the spin-wave field φ obeys
φ(L, y) = φ(0, y) + α
φ(x, L0 ) = φ(x, 0),
α ∈ R.
(4.150)
The change in the dimensionless energy (4.7) induced by changing the
boundary condition from periodic to α–twisted is
1 α 2
LL0 .
(4.151)
∆Ssw = K
2
L
To see this, observe that if φ obeys α–twisted boundary conditions,
then φe defined by
e y) + α x
φ(x, y) = φ(x,
(4.152)
L
must obey periodic boundary conditions. The spin-wave stiffness Υsw
is defined by
2
∆Ssw
K
L
×
= .
(4.153)
Υsw :=
0
α
LL
2
It can be shown that in the presence of vortices
 1
 2 K(l = ∞), if K0 > Kc ,
1
K ,
if K0 = Kc ,
Υsw → Υsw+Cb =
 2 KT
0,
if K0 < Kc .
(4.154)
The stiffness Υsw+Cb thus exhibits a universal jump coming from the
paramagnetic phase. As a model of 2d superfluidity for thin films of
He4 , the 2d–XY model predicts a jump of the superfluidity density ρs
at the superfluid transition,
2
~2 ρs (Tc )
=
,
m2 kB Tc
π
(4.155)
where m is the mass of the helium atom. [47]
4.7. Problems
4.7.1. The classical two-dimensional random phase XY model.
4.7. PROBLEMS
215
Introduction. We define the two-dimensional random phase XY
model on the square lattice Λ by the partition function


2π
YZ
ZXY [A] := 
dφi  e−SXY [A]
(4.156a)
i∈Λ 0
with the classical action
X
SXY [A] := K
1 − cos φi − φj − Aij .
(4.156b)
hiji
The product of the inverse temperature β (the Boltzmann constant
kB = 1) with the ferromagnetic exchange coupling J > 0 is K ≡
β J ≥ 0, hiji is any directed nearest-neighbor pair of lattice sites from
Λ, 0 ≤ φi < 2π is an angle, and the real-valued random numbers
{Aij ≡ −Aji } obey the distribution law (probability distribution)
Y e−A2ij /(2 gA )
p
P [A] :=
.
2π
g
A
hiji
(4.156c)
The choice for this probability distribution is motivated by simplicity,
for it only depends on one (dimensionless) coupling gA and is amenable
to (Gaussian) integrations. In the limit gA = 0, the statistical ensemble
consist of one element {Aij = 0}. This is the clean limit. In the
opposite limit gA = ∞, all choices from {Aij ∈ R} are equally likely.
This is the gauge glass limit.
The sample ZXY [A = 0] from the statistical ensemble {ZXY [A]}
of partition functions supports a fully saturated ferromagnetic ordered
state at vanishing temperature 1/K = 0 and a quasi-long-range-ordered
phase for 0 < 1/K < π/2 up to the KT transition temperature 1/K =
π/2 above which thermal fluctuations select a paramagnetic phase.
Consider any sample ZXY [A] from the statistical ensemble {ZXY [A]}
of partition functions such that the flux
Φ := Aij + Ajk + Akl + Ali
(4.157)
through any elementary square plaquette from Λ, here labeled counterclockwise by the vertices i, j, k, and l as shown in Fig. 2, equals
π mod 2π. Such a partition function ZXY [A] realizes a classical twodimensional frustrated XY magnet whose ground state at vanishing
temperature 1/K = 0 is not the fully saturated ferromagnetic ordered
state.
(The number of plaquettes in Λ is taken to be even so that
P
Φ
= 0 mod 2π.)
The question to be addressed is which of this two cases is typical
of the statistical ensemble {ZXY [A]} as a function of K and gA ?
216
4. KOSTERLITZ-THOULESS TRANSITION
The answer to this question is encoded by the probability distribution


Z Y
−A2ij /(2 gA )
e
 δ(Z − ZXY [A]).
PXY [Z] :=  dAij p
(4.158)
2π
g
A
hiji
However, the probability distribution (4.158) has no more than a symbolic value given that the functional dependence of the partition function ZXY [A] on {Aij } is not known in closed form.
The effect of a random phase (disorder) on the phase diagram from
Fig. 4 and RG flows from Fig. 5 of the classical two-dimensional XY
model must be studied with the help of less ambitious means than
by computing the probability distribution (4.158). Our strategy is
going to be to compute the disorder average of some two-point correlation function to the first non-trivial order in the fugacity expansion
of section 4.5, where we are now attaching the magnitude of the vorticity whose fugacity is included in the RG analysis. Hence, Y1 is the
(bare) fugacity of vortices carrying the vorticities ±1. In this way,
the stability analysis of the quasi-long-range-ordered phase captured
by Fig. 4(c) will become the shaded area bounded by the parabola
and the horizontal axis in Fig. 8(a). According to this calculation,
there is a re-entrant phase transition from a quasi-long-range-ordered
phase to a paramagnetic phase for any fixed but not too strong disorder strength gA upon lowering the reduced temperature 1/K. [48]
What we will not do here is to show that this perturbative stability
analysis is misleading. [49, 50, 51] Indeed, it can be shown that the
regime of stability to the second order in the fugacity expansion shrinks
as is shown in Fig. 8(b). [52] The breakdown of this perturbative approach has two possible interpretations. [52] It could either signal that
the quasi-long-range-ordered phase is unstable to any gA , or that this
phase remains critical for sufficiently small gA , but with critical exponents showing a non-analytic dependence on the disorder strength
gA > 0. It is believed that the latter scenario holds and that the quasilong-range-ordered phase at Y1 = 0 is stable in a region of the plane
Y1 = 0 approximately delimited by the segment parallel to the 1/K
axis that joins (1/K,
gA ) = (0,π/8) to the maximum of the parabola
(1)
at (1/K, gA ) = π/4, gA (π/4) [the dashed segment in Fig. 8(a)] and
(1)
continues with the branch gA (1/K) for π/4 < 1/K < π/2 of the
parabola. [53, 54]
The problem of a single plaquette. Exercise 1.1: Define the plaquette Hamiltonian
X
H [A] := −J
cos φi − φj − Aij ,
J > 0,
(4.159a)
hiji∈
4.7. PROBLEMS
(a)
(b)
gA
⇡
8
Y1
(1)
⇡
2
gA
⇡
8
gA
1
K
217
Y1
(1)
(2)
gA gA
⇡
2
1
⇡ K
Figure 8. (a) Stability analysis of the critical plane
Y1 = 0 in the classical two-dimensional random phase
XY model to first-order in perturbation theory in powers of the bare charge-one fugacity Y1 . The coupling constant 1/K is the reduced temperature, i.e., the temperature in units of the spin exchange coupling. The coupling
constant gA is the variance of the random phases. It is
dimensionless and measures the strength of the disorder.
(1)
The parabolic boundary gA (1/K) is the value of the
disorder strength beyond which Y1 becomes relevant in a
one loop RG analysis. The additional dark-shaded area
below the horizontal dashed line results from a stability
analysis that is non-perturbative in Y1 . (b) If the stability analysis of (a) is extended to second-order in perturbation theory in powers of the bare charge-one fugacity, then the regime of stability of the quasi-long-rangeordered phase with the charge-one and charge-two fugacities renormalizing to zero is the shaded intersection be(1)
tween the area below the parabolic boundary gA (1/K)
(2)
and the area below the parabolic boundary gA (1/K).
where denotes the plaquette shown in in Fig. 2, with hiji any directed
nearest-neighbor pair of sites from , and the flux
X
Φ =
Aij
(4.159b)
hiji∈
is given.
(a) What transformation law of Aij with hiji ∈ combined with
the transformation law
φi → φi + χ i ,
0 ≤ χi < 2π,
i ∈ ,
(4.160)
leaves Eq. (4.159) invariant? Does this transformation law
leave the probability distribution (4.156c) invariant for 0 <
gA ≤ ∞?
218
4. KOSTERLITZ-THOULESS TRANSITION
(b) On how many of the angles φi with i ∈ does H [A] depend?
(c) Show that the four angles φi , φj , φk , and φl from Fig. 2 must
obey
  

Aij − Ali
φi
+2 −1 0 −1
−1 +2 −1 0   φj  Ajk − Aij 
  


 0 −1 +2 −1 φk  = Akl − Ajk 
−1 0 −1 +2
φl
Ali − Akl

(4.161)
if they are to minimize H [A].
(d) Solve Eq. (4.161) when the flux (4.159b) vanishes or equals π
and comment on the nature of the ground states.
Factorization into the spin wave and vortex sectors. We assume the
naive continuum limit for the classical two-dimensional random phase
XY model (4.156) by which
Z
SXY [A] → K
d2 x
2
1
∂µ φ + ∂µ Θ + Aµ
2
(4.162a)
and
P [A] → e
− 2 g1
A
R
d2 x A2µ
.
(4.162b)
The summation convention over the repeated index µ = 1, 2 is implied.
We also adopt the notation of Eq. (4.46) by which φ is vortex free while
Θ is not.
Exercise 2.1:
(a) It is shown in appendix D that the factorization into a spinwave and vortex sector holds for the Villain model in the presence of the random phases Aij ∈ R. Show that the same holds
for the continuum limit (4.162) by making use of the decomposition
Aµ =: ∂˜µ θ + ∂µ η,
∂˜µ := µν ∂ν ,
(4.163)
both on the action (4.162a) and the probability distribution (4.162b).
(b) Express the random magnetic field
b := µν ∂µ Aν
in terms of θ and η defined by Eq. (4.163).
(4.164)
4.7. PROBLEMS
(c) Compute the disorder averages
R
D[A] P [A] Aµ (x) Aν (y)
Aµ (x) Aν (y) := R
,
D[A] P [A]
R
D[A] P [A] θ(x) θ(y)
θ(x) θ(y) := R
,
D[A] P [A]
R
D[A] P [A] η(x) η(y)
,
η(x) η(y) := R
D[A] P [A]
R
D[A] P [A] b(x) b(y)
.
b(x) b(y) := R
D[A] P [A]
219
µ, ν = 1,(4.165)
2,
(4.166)
(4.167)
(4.168)
(d) Show that the spin-wave sector can be represented by the (random) partition function with the (random) Gaussian action
Z
2
K
Ssw [φ; η] := d2 x
∂µ φ − ∂µ η .
(4.169)
2
The gauge transformation φ → φ + η decouples the spin wave
φ from the longitudinal disorder η.
(e) Show that the vortex sector can be represented by the Coulomb
(Cb) gas with the action
X
X
xk − xl 2
SCb [Θ, θ] := Ec
(mk −nk ) −π K
(mk −nk ) (ml −nl ) ln `
k
k6=l
(4.170)
where Ec is the core energy of vortices, mk ∈ Z is the vorticity
of a “thermal” vortex at xk present in Θ, while nk ∈ R is the
vorticity of a “quenched” vortex at xk present in θ.
(f) Explain why the vorticity of the random magnetic field is not
quantized while that of Θ is. The transverse disorder θ cannot
be gauged away.
(g) Show that the Cb gas with the action (4.170) can be represented by the SG theory with the action
Z
1
h1
i
2
2
2
SSG [χ; θ] := d x
(∂ χ) −
cos χ +
χ (∂µ θ)
(4.171a)
2t µ
t
2π
and the identifications
t
h
K = 2,
Y1 ∼ 1 .
(4.171b)
4π
2t
The index 1 of h1 or Y1 refers to the vorticity. The bare fugacity hm or Ym of thermal vortices with vorticity ±m ∈ Z
larger in magnitude than unity is set to zero. Nevertheless,
they are always generated under RG. Their irrelevances in the
spin-wave phase and at the KT transition justify safely ignoring them when discussing the clean case at gA = 0. They
cannot be neglected as soon as gA > 0.
220
4. KOSTERLITZ-THOULESS TRANSITION
Moments of the two-point function in the spin-wave phase. Exercise 3.1: A prerequisite for the phase diagram shown in Fig. 8 is that
the spin-wave phase defined by the condition Y1 = 0 remains critical
for any gA . Show that, in the spin-wave sector,
e+iφ(x1 ) e−iφ(x2
) q
q
2π K
a
∝ e+iq η(x1 ) e−iq η(x2 )
x1 − x2 q+gA K q2
2π K
a
= (4.172)
x1 − x2 for any positive q and with a the UV cutoff. Angular bracketsR denote
2
2
thermal averaging with the normalized measure ∝ D[φ] e−(K/2) d y (∂µ φ−∂µ η) .
The overline denotes
disorder averaging with the normalized measure
R
−(1/2gA ) d2 y (∂µ η)2
∝ D[η] e
. The effect of disorder is thus, on average,
to increase the value of the critical exponent that characterizes the
algebraic decay of the q-th moment of the two-point function in the
spin-wave phase.
Moments of the two-point function in the SG theory. Define the
thermal two-point function
R
D
E
D[χ] e−SSG [χ,θ] eiχ(x1 )−iχ(x2 )
Fx1 ,x2 [θ] :=
.
(4.173)
ZSG [θ]
D
E
We have seen in section 4.5 that by expanding Fx1 ,x2 [θ = 0] in
powers of a very small fugacity h1 /2t, all coefficients of the expansion
in the fugacity are ill-defined unless a short distance cutoff a is imposed. The arbitrariness in the choice of the short distance cutoff was
used to derive RG equations obeyed by the fugacity and the reduced
temperature. The RG equations were integrated to determine whether
the initial assumption of a very small fugacity is consistent. The irrelevance, marginality, and relevance of the fugacity then determines
the spin-wave phase, KT transition, and disordered phase of the XY
model, respectively, i.e., Figs. 4 and 5.
The transformation
it
χ→χ+
θ
(4.174)
2π
plays an essential role in what follows.
D
E
Exercise 4.1: Show that the fugacity expansion of Fx1 ,x2 [θ]
depends on correlation functions calculated for vanishing “magnetic
field” h1 (fugacity h1 /2t) such as
Z
d2 y 1 · · · d2 y 2n hei[χ(x1 )−χ(x2 )+χ(y1 )+···−χ(y2n )] ih1 =0 ,
(4.175a)
4.7. PROBLEMS
221
on the one hand, but also such as
Z
n
i[χ(x
)−χ(x
)]
i[χ(y
)−χ(y
)]
2
2
1
2 i
1
2
he
d y1d y2 e
h1 =0
,
(4.175b)
h1 =0
on the other hand. Here, the overlines denote
disorder averaging with
R
−(1/2gA ) d2 y (∂µ θ)2
the normalized measure ∝ D[θ] e
.
Exercise 4.2: Show that
qt
a 2π − q t θ(y ) + q t θ(y )
+iχ(y ) −iχ(y ) q
e 2π 1 e 2π 2
1 e
2
e
∝ h1 =0
y1 − y2 a 2π K q(1−gA K q)
= (4.176)
y −y 1
2
for any positive q and with a the UV cutoff. Compare this result
with Eq. (4.172). Contrast the cases gA = 0 and gA > 0 and give an
interpretation.
Exercise 4.3: The parabola
1
2 1
1
(1)
:=
1−
(4.177)
gA
K
K
π K
is obtained by requiring that the scaling exponent on the right-hand
side of Eq. (4.176) for the first positive integer moment q = 1 be “marginal”, i.e., equals 4. What happens if the scaling exponent is smaller
than 4? Hint: See the discussion below Eq. (4.104).
Exercise 4.4: The parabola
1
1
2 1
(q)
gA
:=
1−
(4.178)
K
Kq
π Kq
for any q > 0 is obtained by requiring that the scaling exponent on the
right-hand side of Eq. (4.176) be “marginal”, i.e., equals 4.
Screening of quenched vortices by thermal vortices. The ground
state of the Cb gas defined in Eq. (4.170) is the state that nucleates
integer-valued vortices out of the thermal field Θ in order to minimize
the Cb energy contained in the quenched vortices out of the random
field θ. Screening is not perfect because the vorticity of the quenched
vortices is not quantized.
Exercise 5.1: Show that screening by thermal vortices mk of
quenched vortices nk is least efficient if nk = ±1/2.
Exercise 5.2: Show that the probability distribution for the field
θ with the restriction that its vortices are quantized in units of 1/2 can
be interpreted as the classical two-dimensional XY model on a square
lattice with the reduced temperature gA .
Exercise 5.3: Show that gA = π/8 is the KT “transition temperature” if the quenched vortices are quantized in units of 1/2.
Exercise 5.4: Argue that the segment (Y1 , 1/K, gA ) = (0, 0, gA ) in
Fig. 8 with 0 ≤ gA ≤ π/8 must be critical.
Part 2
Fermions
CHAPTER 5
Non-interacting fermions
Outline
The physics of non-interacting fermions is reviewed.
5.1. Introduction
This chapter is a review devoted to the second quantization of
fermions and to the thermodynamic and transport properties of the
non-interacting electron gas.
When the dispersion of the non-interacting electron gas is assumed
to be the non-relativistic parabolic spectrum of electrons in vacuum,
we say that we are dealing with the non-interacting jellium model.
The non-interacting jellium model treats electrons in a metal as if
they were in vacuum, except for a homogeneous, inert, and positive
background charge that restores charge neutrality and represents the
crudest approximation to the ions of a metal. This background charge
plays no role in this chapter and will thus be omitted entirely. However,
this background charge plays an important role when the Coulomb
interaction between electrons in the jellium model is accounted for, as
we shall see in the next chapter.
After a quick summary of second quantization for fermions (section
5.2), the notions of the Fermi sea and the Fermi surface will be reviewed (section 5.3). We shall see that thermodynamic properties are
controlled by the Fermi surface at sufficiently low temperatures. The
same is also true of transport properties.
The sections on the time-ordered Green functions for the noninteracting jellium model (section 5.4), the Grassmann coherent states
(appendix E.1), fermionic path integrals (appendix E.2), Jordan-Wigner
fermions (appendix E.3), the electronic correlation energy (appendix E.4),
and the fluctuation-dissipation theorem (appendix E.5) are included for
completeness.
5.2. Second quantization for fermions
The Hilbert space for a many-electron system is constructed by
taking the direct sum of all antisymmetric (exterior) tensor products
of a single-electron Hilbert space. This construction is called second
quantization for electrons and is the natural quantum counterpart of
the grand-canonical ensemble in classical statistical mechanics. We will
225
226
5. NON-INTERACTING FERMIONS
present the formalism of second quantization for fermions by taking the
fermions to be spinless in order to simplify notation. This economy also
makes sense whenever the electronic spin is a mere bystander that plays
no consequential role. Furthermore, there are collective excitations in
condensed matter systems that, to a good approximation, behave like
spinless electrons (see appendix E.3).
Assume that the single-particle Hamiltonian (from now on, ~ = 1
unless specified)
∆
H=−
+ U (r)
(5.1a)
2m
with appropriate boundary conditions has the countable basis of eigenfunctions
Z
X
∗
Hψn (r) = εn ψn (r),
dd r ψm
(r)ψn (r) = δm,n ,
ψn∗ (r)ψn (r 0 ) = δ(r−r 0 ),
n
V
(5.1b)
in the single-particle Hilbert space H of square integrable and twice
differentiable functions on Rd . We also assume that the single-particle
potential U (r) is bounded from below, i.e., there exists a single-particle
and non-degenerate 1 ground-state energy, say ε0 . Hence, the energy
eigenvalue index n can be chosen to run over the non-negative integers,
n = 0, 1, 2, · · · . The evolution in time of any solution of Schrödinger
equation
(1)
i∂t Ψ(r, t) = HΨ(r, t),
can be written as
X
Cn ψn (r) e−iεn t ,
Ψ(r, t) =
n
Ψ(r, t = 0) given,
Z
Cn =
(5.2a)
dd r ψn∗ (r)Ψ(r, t = 0).
V
(5.2b)
The formalism of second quantization starts with the following two
postulates.
(1) There exists a set of pairs of adjoint operators ĉ†n (creation
operator) and ĉn (annihilation operator) labeled by the energy
eigenvalue index n and obeying the fermionic algebra 2
{ĉm , ĉ†n } = δm,n ,
{ĉm , ĉn } = {ĉ†m , ĉ†n } = 0,
m, n = 0, 1, 2, · · · .
(5.3)
(2) There exists a non-degenerate vacuum state |0i that is annihilated by all annihilation operators,
ĉn |0i = 0,
1
n = 0, 1, 2, · · · .
(5.4)
By hypothesis fermions are spinless and there is no Kramer degeneracy associated to the spin-1/2 degrees of freedom of real electrons.
2 The conventions for the commutator and anticommutator of any two “objects”
A and B are [A, B] := AB − BA and {A, B} := AB + BA, respectively.
5.2. SECOND QUANTIZATION FOR FERMIONS
227
With these postulates in hand, we define the Heisenberg representation
for the operator-valued field (in short, quantum field),
X
(5.5a)
ψ̂ † (r, t) :=
ĉ†n ψn∗ (r) e+iεn t ,
n
together with its adjoint
ψ̂(r, t) :=
X
ĉn ψn (r) e−iεn t .
(5.5b)
n
The fermionic algebra (5.3) endows the quantum fields ψ̂ † (r, t) and
ψ̂(r, t) with the equal-time algebra 3
{ψ̂(r, t), ψ̂ † (r 0 , t)} = δ(r−r 0 ),
{ψ̂(r, t), ψ̂(r 0 , t)} = {ψ̂ † (r, t), ψ̂ † (r 0 , t)} = 0.
(5.9)
†
The quantum fields ψ̂ (r, t) and ψ̂(r, t) act on the “big” many-particle
space
(N
)
∞
M
^
F :=
H(1) .
(5.10a)
N =0
Here, each
VN
H
(1)
is spanned by states of the form
Y † mi
ĉi
|m0 , · · · , mi−1 , mi , mi+1 , · · · i :=
|0i,
mi = 0, 1,
i
(5.10b)
with the condition on mi = 0, 1 that
X
mi = N.
(5.10c)
i
V
The algebra obeyed by the ĉ’s and their adjoints ensures that N H(1) is
the N -th antisymmetric power of H(1) , i.e., that the state |m0 , · · · , mi−1 , mi , mi+1 , · · · i
made of N identical particles of which mi have energy εi changes by a
sign under exchange of any two of the N particles. Hence, the “big”
many-particle Hilbert space (5.10a) is the sum over the subspaces of
wave functions for N identical particles that are antisymmetric under
3
Alternatively, if we start from the classical Lagrangian density
|∇ψ|2 (r, t)
− |ψ|2 (r, t)U (r),
2m
we can elevate the field ψ(r, t) and its momentum conjugate
L := (ψ ∗ i∂t ψ)(r, t) −
π(r, t) :=
δL
= iψ ∗ (r, t)
δ(∂t ψ)(r, t)
(5.6)
(5.7)
to the status of quantum fields ψ̂(r, t) and π̂(r, t) = iψ̂ † (r, t) obeying the equal-time
fermionic algebra
{ψ̂(r, t), π̂(r 0 , t)} = iδ(r−r 0 ),
{ψ̂(r, t), ψ̂(r 0 , t)} = {π̂(r, t), π̂(r 0 , t)} = 0. (5.8)
228
5. NON-INTERACTING FERMIONS
any odd permutation of the particles labels. 4 This “big” many-particle
Hilbert space is called the fermion Fock space in physics.
The rule to change the representation of operators from the Schrödinger
picture to the second quantized language is best illustrated by the following examples.
Example 1: The second-quantized representation Ĥ of the singleparticle Hamiltonian (5.1a) is
Z
Ĥ := dd r ψ̂ † (r, t) H ψ̂(r, t)
(5.11)
V
=
X
εn ĉ†n ĉn .
n
As it should be it is explicitly time independent.
Example 2: The second-quantized total particle-number operator
Q̂ is
Z
Q̂ := dd r ψ̂ † (r, t) 1 ψ̂(r, t)
(5.12)
V
=
X
ĉ†n ĉn .
n
It is explicitly time independent, as follows from the continuity equation
0 = (∂t ρ)(r, t) + (∇ · J )(r, t),
ρ(r, t) := |Ψ(r, t)|2 ,
1
J (r, t) :=
[Ψ∗ (r, t) (∇Ψ) (r, t) − (∇Ψ∗ ) (r, t)Ψ(r, t)] ,
2mi
(5.13)
obeyed by Schrödinger equation (5.2a). The number operator Q̂ is the
infinitesimal generator of global gauge transformations by which all N particle states in the fermion Fock space are multiplied by the same
phase factor. Thus, for any q ∈ R, a global gauge transformation on
the Fock space space is implemented by the operation
|m0 , · · · , mi−1 , mi , mi+1 , · · · i → e+iq Q̂ |m0 , · · · , mi−1 , mi , mi+1 , · · · i
(5.14)
on states, or, equivalently, 5
ĉ†n → e+iqQ̂ ĉ†n e−iqQ̂ = e+iq ĉ†n ,
4
5
(5.17)
An odd permutation is made of an odd product of pairwise exchanges.
We made use of
[ĉ† ĉ, ĉ] = ĉ† ĉĉ − ĉĉ† ĉ = ĉ† ĉĉ + ĉ† ĉĉ − ĉ† ĉĉ − ĉĉ† ĉ = ĉ† {ĉ, ĉ} − {ĉ† , ĉ}ĉ = −ĉ, (5.15)
and, similarly,
[ĉ† ĉ, ĉ† ] = +ĉ† .
(5.16)
5.2. SECOND QUANTIZATION FOR FERMIONS
229
and
ĉn → e+iqQ̂ ĉn e−iqQ̂ = e−iq ĉn ,
(5.18)
for all pairs of creation and annihilation operators, respectively. Equation (5.17) teaches us that any creation operator carries the particle
number +1. Equation (5.18) teaches us that any annihilation operator
carries the particle number −1.
Example 3: The second-quantized local particle-number density
operator ρ̂ and the particle-number current density operator Ĵ are
ρ̂(r, t) = ψ̂ † (r, t)1ψ̂(r, t),
(5.19a)
and
i
1 h †
†
Ĵ (r, t) :=
ψ̂ (r, t) ∇ψ̂ (r, t) − ∇ψ̂ (r, t)ψ̂(r, t) , (5.19b)
2mi
respectively. The continuity equation
0 = (∂t ρ̂)(r, t) + (∇ · Ĵ )(r, t)
(5.19c)
that follows from evaluating the commutator between ρ̂ and Ĥ is
obeyed as an operator equation.
The operators Ĥ, Q̂, ρ̂, and Ĵ all act on the Fock space F. They
are thus distinct from their single-particle counterparts H, Q, ρ, and J
(1)
whose actions are restricted to the Hilbert space
V1 H(1) . By construction,
the action of Ĥ, Q̂, ρ̂ and Ĵ on the subspace
H of F, say, coincide
(1)
with the action of H, Q, ρ, and J on H .
Example 1: A single-particle wave function is recovered by defining
the single-particle state
|mi := ĉ†m |0i
(5.20)
h0|ψ̂(r, t)|mi = ψm (r) e−iεm t .
(5.21)
and calculating the overlap
Example 2: Let |Φ0 i be the state defined by filling the N lowest
energy eigenstates of H,
|Φ0 i :=
N
Y
j=1
ĉ†j |0i.
(5.22)
230
5. NON-INTERACTING FERMIONS
This state is called the Fermi sea. The overlap
−iε t
−iε t
−iε t + ψ1 (r 1 ) e 1 ψ2 (r 1 ) e 2 · · · ψN (r 1 ) e N * N
Y
ψ1 (r 2 ) e−iε1 t ψ2 (r 2 ) e−iε2 t · · · ψN (r 2 ) e−iεN t 0 ψ̂(r j , t) Φ0 = ..
..
..
.
.
·
·
·
.
j=1
ψ (r ) e−iε1 t ψ (r ) e−iε2 t · · · ψ (r ) e−iεN t 1 N
2 N
N
N
ψ1 (r 1 ) ψ2 (r 1 ) · · · ψN (r 1 ) !
N
X
ψ1 (r 2 ) ψ2 (r 2 ) · · · ψN (r 2 ) ,
= exp −i
εj t ..
..
..
.
.
·
·
·
.
j=1
ψ (r ) ψ (r ) · · · ψ (r )
1
N
2
N
N
N
(5.23)
is the Slater determinant representation of the Fermi sea. N -particle
states that can be expressed by a single N × N Slater determinant
are said to be decomposable. Decomposable states form a very small
subset of the totality of N -particle states. The so-called Hartree-Fock
approximation to the quantum many-body problem seeks the best trial
function among decomposable states.
5.3. The non-interacting jellium model
The non-interacting jellium model describes non-interacting electrons with the mass m and the electrical charge −e (the electric charge
e is chosen positive by convention) moving freely in a box of linear size
L. Mathematically, the non-interacting jellium model in the volume
V = L3 , at temperature T = (kB β)−1 , and chemical potential µ is
defined by the grand-canonical partition function
Z(L3 , β, µ) := TrF e−β (Ĥ−µN̂ ) ,
(5.24a)
with the Hamiltonian and number operators
Ĥ :=
XX
σ
εσ,k ĉ†σ,k ĉσ,k ,
N̂ :=
XX
σ
k
acting on the Fock space
(
Y
m
F := span
ĉ†ι ι |0iσ =↑, ↓,
ι≡(σ,k)
ĉ†σ,k ĉσ,k ,
~2 k2
,
2m
(5.24b)
εσ,k :=
k
L
k ∈ Z3 ,
2π
mι = 0, 1,
)
ĉι |0i = 0,
{ĉι , ĉ†ι0 }
= δι,ι0 ,
{ĉ†ι , ĉ†ι0 }
= {ĉι , ĉι0 } = 0 .
(5.24c)
The choice of periodic boundary conditions does not affect bulk properties in the thermodynamic limit L → ∞.
5.3. THE NON-INTERACTING JELLIUM MODEL
231
What distinguishes the non-interacting jellium model from other
non-interacting electron models is the non-relativistic parabolic dispersion and the unboundness of the allowed momenta. In the presence
of a weak single-particle periodic perturbation of the jellium model,
momenta can be restricted to the first Brillouin zone, i.e., momenta
are bounded from above and below in magnitude, although the dispersion remains unbounded from above. In contrast to the jellium model,
tight-binding electronic models have a kinetic energy that is bounded
from below and from above. Correspondingly, the tight-binding singleparticle dispersion is periodic in the extended zone scheme with the
periodicity set by the first Brillouin zone.
In this section, we are going to derive the thermodynamic properties
of the non-interacting jellium model in the absence of a magnetic field.
We will then review the Sommerfeld semi-classical theory of transport
for non-interacting electrons. We close with the effects of a magnetic
field in the form of Pauli paramagnetism and of Landau diamagnetism.
5.3.1. Thermodynamics without magnetic field. Evaluation
of the grand-canonical partition function (5.24) is performed in two
steps when interactions are absent. First,
−β
3
Z(L , β, µ) = TrF e

= TrF 
P
(ει −µ)ĉ†ι ĉι
ι=(σ,k)

Y
†
e−β(ει −µ)ĉι ĉι 
ι=(σ,k)
=
Y
†
TrFι e−β(ει −µ)ĉι ĉι .
(5.25a)
ι=(σ,k)
Here, owing to the lack of interactions, we have interchanged the trace
and the product whereby the two-dimensional single-particle Fock space
(
)
† mι
Fι := span
ĉι
|0imι = 0, 1
(5.25b)
is introduced. Second, we can now perform the trace over each Fock
space Fι labeled by the single-particle quantum number ι independently,
Y X
Z(L3 , β, µ) =
e−β(ει −µ)mι
ι=(σ,k) mι =0,1
=
Y
1 + e−β(ει −µ) ,
(5.26)
ι=(σ,k)
2π
n,
L
where σ =↑, ↓, k =
and n ∈ Z3 . In terms of the Fermi-Dirac
distribution
1
eβ(ει −µ)
fFD (ει ) := β(ε −µ)
⇐⇒ 1 − fFD (ει ) := β(ε −µ)
,
(5.27a)
e ι
+1
e ι
+1
232
5. NON-INTERACTING FERMIONS
Eq. (5.26) becomes
Z(L3 , β, µ) =
Y
ι=(σ,k)
1
.
1 − fFD (ει )
(5.27b)
The internal energy U of the non-interacting jellium model is
TrF e−β (Ĥ−µN̂ ) Ĥ
U (L3 , β, µ) :=
TrF e−β (Ĥ−µN̂ )
∂ ln Z(L3 , β, µ)
∂ ln Z(L3 , β, µ)
= −
+ β −1 µ
∂β
∂µ
X e−β(ει −µ) ε
ι
=
1 + e−β(ει −µ)
ι=(σ,k)
X
=
fFD (ει ) ει .
(5.28a)
ι=(σ,k)
The grand-canonical potential F of the non-interacting jellium model
is
F (L3 , β, µ) := −β −1 ln Z(L3 , β, µ)
X
= −β −1
ln 1 + e−β(ει −µ)
ι=(σ,k)
= +β
−1
X
ln 1 − fFD (ει ) .
(5.28b)
ι=(σ,k)
The entropy S of the non-interacting jellium model is
∂F (L3 , β, µ)
∂T
∂F (L3 , β, µ)
= − kB
∂β −1
i
X h
= − kB
fFD (ει ) ln fFD (ει ) + 1 − fFD (ει ) ln 1 − fFD (ει ) .
S(L3 , β, µ) := −
ι=(σ,k)
(5.28c)
The pressure P of the non-interacting jellium model is obtained in two
steps. First, differentiation yields
∂F (L3 , β, µ)
∂L3
∂ X
ln 1 + e−β(ει −µ)
= +β −1 3
∂L
P (L3 , β, µ) := −
ι=(σ,k)
= +β −1
X
ι=(σ,k)
e−β(ει −µ)
∂ε
(−)β ι3 . (5.28d)
−β(ε
−µ)
ι
1+e
∂L
5.3. THE NON-INTERACTING JELLIUM MODEL
∂ει
∂L3
Second, for a quadratic dispersion,
so that
∂(L3 )−2/3
∂L3
∝
233
= −(2/3)(L3 )−2/3−1 ,
X e−β(ει −µ) ε
2
ι
P (L3 , β, µ) = + L−3
3
1 + e−β(ει −µ)
ι=(σ,k)
X
2
fFD (ει ) ει
= + L−3
3
ι=(σ,k)
Eq. (5.28a)
=
2 −3
L × U (L3 , β, µ).
3
(5.28e)
The average number of electrons is
Ne (L3 , β, µ) :=
−β (Ĥ−µN̂ )
TrF e
N̂
TrF e−β (Ĥ−µN̂ )
∂ ln Z(L3 , β, µ)
= β −1
∂µ
−β(ε
X
ι −µ)
e
=
1 + e−β(ει −µ)
ι=(σ,k)
X
=
fFD (ει ),
(5.28f)
ι=(σ,k)
while the average occupation number of the single-particle level ι =
(σ, k) is
hĉ†ι ĉι iL3 ,β,µ :=
TrF e−β (Ĥ−µN̂ ) ĉ†ι ĉι
TrF e−β (Ĥ−µN̂ )
= fFD (ει ).
(5.29)
We now take the thermodynamic limit L → ∞. In this limit,
the single-particle spectrum becomes continuous. Correspondingly, the
density of states per unit energy and per unit volume (a distribution)
ν(ε, L3 ) := L−3
X
ι=(σ,k)
δ (ε − ει )
(5.30)
234
5. NON-INTERACTING FERMIONS
6
becomes the continuous function of the single-particle energy ε,
ν(ε) =
d3 k
~2 k2
δ ε−
(2π)3
2m
XZ
σ=↑,↓
Z+∞
= 2 × 4π
dk k 2
~2 k 2
,
δ ε−
8π 3
2m
(5.32)
0
as 4π√ is the area of the unit sphere. With the help of ω :=
p
k = 2mω
, dk = dω 2~m2 ω there then follows that
~
Z+∞
2m
dω ω 1/2 δ (ε − ω)
~2
0
r
1 m 2mε
Θ(ε).
= 2× 2 2
2π ~
~2
1 m
ν(ε) = 2 × 2 2
2π ~
~2 k2
,
2m
r
(5.33)
Here, we have introduced the Heaviside step function
Θ(x) :=

 1,
if x > 0,
0,
if x < 0.

(5.34)
In the thermodynamic limit L → ∞, the internal energy per unit
volume u, the grand-canonical potential per unit volume f , the entropy
per unit volume s, the pressure p, and the average number of electrons
6
Dimensional analysis gives the estimate
ν(ε) ∝ |k(ε)|d × ε−1 ∝ ε(d/n)−1
in d dimensions and with the dispersion ε(k) ∝ |k|n .
(5.31)
5.3. THE NON-INTERACTING JELLIUM MODEL
235
kz
kF
kF
kx
ky
kF
Figure 1. The Fermi sea of the non-interacting jellium
model is a sphere in momentum or wave number space.
The Fermi surface is the surface of the sphere.
per unit volume ne are given by
lim L−3 U (L3 , β, µ)
Z
=
dε ν(ε) fFD (ε) ε,
u(β, µ) :=
L→∞
(5.35a)
R
lim L−3 F (L3 , β, µ)
Z
−1
= β
dε ν(ε) ln 1 − fFD (ε) ,
f (β, µ) :=
L→∞
(5.35b)
R
lim L−3 S(L3 , β, µ)
(5.35c)
Z
h
i
= −kB dε ν(ε) fFD (ε) ln fFD (ε) + 1 − fFD (ε) ln 1 − fFD (ε) ,
s(β, µ) :=
L→∞
R
p(β, µ) :=
lim P (L3 , β, µ)
L→∞
2
u(β, µ),
3
ne (β, µ) := lim L−3 Ne (L3 , β, µ)
L→∞
Z
=
dε ν(ε) fFD (ε),
=
(5.35d)
(5.35e)
R
respectively. Equations (5.35a-5.35e) hold for any non-interacting Fermi
system once the thermodynamic limit of the single-particle density of
states is known.
In the thermodynamic limit L → ∞, the ground state of the noninteracting jellium model is the Fermi sea with the Fermi wave vector
kF . The Fermi sea shown in Fig. 1 is the sphere with the radius
kF
(5.36a)
4πkF3 /3
(5.36b)
and the volume
236
5. NON-INTERACTING FERMIONS
obtained by filling all single-particle levels ει where ι = (σ, k) with the
wave vectors satisfying
0 ≤ |k| ≤ kF .
(5.36c)
Since there is a total of
4πkF3 /3
kF3
=
(5.37)
(2π)3
6π 2
single-particle wave vectors available per unit volume, the Fermi wave
vector is given by
1/3
k3
ne = 2 × F2 ⇐⇒ kF = 3π 2 ne
(5.38)
6π
when the number of electrons per unit volume is given by ne . The
Fermi energy εF is the largest single-particle energy that is occupied in
the Fermi sea,
~2 kF2
εF = ει ,
ι = (σ, kF ) ⇐⇒ εF = ει =
.
(5.39)
2m
The Fermi wave vector (or the Fermi energy) defines the Fermi surface.
Single-particle states above the Fermi surface are unoccupied, while
they are occupied below it in the ground state of the non-interacting
jellium model. The Fermi energy of the non-interacting jellium model
takes the form
~2 kF2
e2 aB kF2
e2
εF =
=
=
(kF aB )2 = Ry × (kF aB )2
(5.40)
2m
2
2aB
when expressed in term of the Bohr radius
~2
aB :=
(5.41)
me2
and the ground-state binding energy
e2
Ry :=
(5.42)
2aB
of the hydrogen atom, i.e., 13.6 eV. As good metals have
kF aB ≈ 1
(5.43)
of the order unity, their Fermi energy have the magnitude of a typical
atomic binding energy. The Fermi wave vector also defines the Fermi
velocity
~k
vF := F ,
(5.44)
m
which is three orders of magnitude smaller than the velocity of light
for good metals. Neglecting relativistic effects to describe electrons in
good metals is therefore justified to a first approximation. For copper,
[kF ] = 13.6 nm−1 ,
[λF ] = 0.46 nm,
[εF ] = 7.03 eV,
[vF ] = 0.005 c.
(5.45)
5.3. THE NON-INTERACTING JELLIUM MODEL
fFD
237
kB T
1
µ
"
Figure 2. The Fermi-Dirac function fFD (ε) :=
−1
eβ(ε−µ) + 1
is an analytic function of the energy ε
at any finite temperature. At zero temperature, it is discontinuous at the chemical potential µ. The unit step
at β = ∞ when ε = µ turns into a continuous and
monotonic decrease over the energy range β −1 = kB T
at any finite temperature. The sharp Fermi surface at
zero temperature is smeared over the temperature range
β −1 = kB T as depicted by the shaded box.
At zero temperature, the Fermi-Dirac distribution (5.27a) shown in
Fig. 2 is the step function
lim fFD (ε) = Θ (µ − ε) ,
(5.46)
β→∞
whose derivative with respect to energy is the delta function
lim
β→∞
dfFD
dε
(ε) = −δ (µ − ε) .
(5.47)
This suggests a Taylor expansion about the chemical potential µ of the
function
Zε
0
0
dε g(ε ) ⇐⇒
h(ε) :=
dh
dε
(ε) := g(ε)
(5.48)
−∞
that appears in the integral
Z
Z
dε g(ε) fFD (ε) =
R
R
df
dε h(ε) − FD
dε
(ε)
(5.49)
238
5. NON-INTERACTING FERMIONS
provided g vanishes as → −∞ and diverges no faster than polynomially for → +∞. The so-called Sommerfeld expansion
Z
Z
dfFD
dε h(ε) −
dε g(ε) fFD (ε) =
(ε)
dε
R
R
"
#
Z
∞
X
(ε − µ)m dm h
dfFD
=
dε h(µ) +
(µ) −
(ε)
m!
dεm
dε
m=1
R
= h(µ) +
∞ 2m X
d h
m=1
Zµ
=
dε g(ε) +
−∞
dε2m
Z
(µ)
(ε − µ)2m
dε
(2m)!
df
− FD
dε
(ε)
R
∞
X
am
m=1
d2m−1 g
dε2m−1
(µ) (kB T )2m
follows. To reach the third equality, we used the fact that
dfFD
β/4
−
(ε) =
2
dε
cosh β2 (ε − µ)
(5.50)
(5.51)
is an even function of ε − µ at any temperature. To reach the last
equality, we re-expressed h in terms of g and used the dimensionless
integration variable
x := β(ε − µ)
(5.52)
to write
Z
Z
(ε − µ)2m
dfFD
1
d
1
2m
2m
dε
−
(ε) = (kB T ) ×
dx x
−
(2m)!
dε
(2m)!
dx ex + 1
R
R
= (kB T )2m × 2
∞
X
(−1)i+1
i2m
i=1
|
= am (kB T )2m ,
{z
≡am
}
m = 1, 2, · · · .
(5.53)
The coefficients am introduced in the last equality are, up
of
P to a factor
n−1 −s
2, the values taken by the Dirichlet eta function η(s) := ∞
(−1)
n
.
n=1
If g varies significantly on the energy scale of µ, i.e.,
2m−1 g(µ)
d
g
(µ) ≈ 2m−1 ,
m = 1, 2, · · · ,
(5.54)
2m−1
dε
µ
then the ratio of two successive
terms in the Sommerfeld expansion is
of the order O (kB T /µ)2 so that, up to order four in the Sommerfeld
5.3. THE NON-INTERACTING JELLIUM MODEL
239
expansion,
Zµ
Z
dε g(ε) fFD (ε) =
dε g(ε) +
7π 4 000
π2 0
g (µ) (kB T )2 +
g (µ) (kB T )4 .
6
360
−∞
R
(5.55)
The Sommerfeld expansion (5.55) applied to the internal energy density (5.35a) and the average occupation number density (5.35e) yields
Zµ
u(T, µ) =
dε ν(ε) ε +
−∞
Zµ
ne (T, µ) =
π2
(kB T )2 [µν 0 (µ) + ν(µ)] + · · · ,
6
(5.56)
2
dε ν(ε) +
π
(kB T )2 ν 0 (µ) + · · · ,
6
−∞
respectively.
We now assume that
µ = εF + O (kB T )2 ,
(5.57)
an assumption whose consistency we shall shortly verify. Under this
assumption and owing to the vanishing of the density of states for
negative energies,
ZεF
dε ν(ε) ε + εF (µ − εF ) ν(εF ) +
u(T, µ) =
π2
[εF ν 0 (εF ) + ν(εF )] (kB T )2 + · · · ,
6
0
ZεF
dε ν(ε) + (µ − εF ) ν(εF ) +
ne (T, µ) =
π2 0
ν (εF ) (kB T )2 + · · · .
6
0
(5.58)
We also assume that the electronic density is temperature independent
ZεF
dε ν(ε) ≡ ne ,
ne (T, µ) = ne (T = 0, µ) =
∀T, µ.
(5.59)
0
This implies that the chemical potential µ is a function of temperature
and of the electronic density ne given by
µ = εF −
π 2 ν 0 (εF )
(kB T )2 + · · · ,
6 ν(εF )
(5.60)
while the internal energy density reduces to
ZεF
u(T ) =
0
π2
dε ν(ε) ε + ν(εF ) (kB T )2 + · · · .
6
(5.61)
240
5. NON-INTERACTING FERMIONS
This result could have been guessed from the following argument.
The difference between the internal energy density at finite and at zero
temperature is the product of three factors. First, there is the support
kB T of the Fermi-Dirac distribution over which it varies significantly at
the non-vanishing temperature T . Second, there is the non-vanishing
density of states at the Fermi energy ν(εF ). Finally, there is the characteristic excitation energy kB T measured relative to the Fermi energy,
i.e.,
ZεF
u(T ) −
dε ν(ε) ε ∝ ν(εF ) (kB T )2 .
(5.62)
0
At last, the specific heat of the non-interacting jellium model at
fixed electronic concentration is
∂u(T ) Cv (T ) :=
∂T ne
=
π2
ν(εF )kB2 T + · · · .
3
(5.63)
This result holds for any non-vanishing single-particle density of states
at the Fermi energy. For the non-interacting jellium model
π 2 kB ne kB T
(5.64)
Cv (T ) =
+ ··· .
2
εF
For comparison, a classical ideal gas has the constant volume specific
heat
3kB ne
.
(5.65)
2
The Fermi-Dirac statistics suppresses the classical result by the multiplicative factor
π 2 kB T
.
3 εF
(5.66)
The prediction of a linear specific heat for a non-interacting Fermi
gas is a simple test of how important electronic interactions are in a
metal. It is customary to call the linear coefficient of the temperature
dependence of the specific heat the γ coefficient and to plot
Cv
= γ + AT 2
(5.67)
T
linearly, i.e., as a function of T 2 . For good metals, the linear dependence on temperature of the specific heat becomes comparable to the
cubic dependence at a few degrees Kelvin.
5.3. THE NON-INTERACTING JELLIUM MODEL
241
v
B
Ey
ev ^ B
+ + + + + + +
- - - - - - -
j
Ex
b
z
b
y
b
x
Figure 3. The set up for Hall’s experiment is the following. The electric charge e is chosen positive by conb pointing along the posvention. A dc electric field Ex x
itive x-Cartesian axis is applied on a metallic wire. It
b along the
induces an electronic steady-state current jx x
positive x-Cartesian axis. A dc magnetic field of magnitude B is pointing along the positive z-Cartesian axis.
x |) B
b∧z
b = − e |vcx | B y
b is balx
The Lorentz force (−e) (−|v
c
b that
anced by the force induced by the electric field Ey y
points along the negative y-Cartesian axis. The latter
force is induced by the electric charge that have accumulated on the boundaries along the y-Cartesian axis.
Here, we are assuming overall charge neutrality and a
steady state. For positive charge carriers, v points along
the positive x-Cartesian axis and thus induces an elecb pointing along the positive y-Cartesian
trical field Ey y
axis. Changing the sign of the charge carrier leaves
jx = ne (∓e)(∓|vx |) unchanged but reverses the sign of
E
Ey and thus of the the Hall coefficient RH := j yB .
x
5.3.2. Sommerfeld semi-classical theory of transport. The
semi-classical theory of transport in metals by Sommerfeld is a quantum extension of the classical kinetic theory of transport by Drude. We
thus review first the classical theory of transport in metals by Drude.
The Drude model of electrical transport in metals assumes that
electricity is carried by small (point-like) hard spheres quantized in
the units of e with e > 0 the electric charge that undergo elastic and
instantaneous scattering events with a probability per unit time 1/τ
while they move freely (ballistically) between the collisions. Assuming
isotropy in space, let
j = ne (−e) v
(5.68)
be the electric current per unit area and per unit time carried by an
electronic density ne of electrons moving at the average velocity
v=
(−e) E τ
m
(5.69)
242
5. NON-INTERACTING FERMIONS
induced by a dc (static) electric field E between the collisions with
probability per unit time 1/τ . The linear relation
ne e2 τ
,
m
defines the Drude conductivity σD and the Drude resistivity
−1
ne e 2 τ
E = ρD j,
ρD =
.
m
j = σD E,
σD =
(5.70)
(5.71)
b
Isotropy of space is broken by a dc (static) magnetic field B = B z
perpendicular to a rectangular metallic sample as is shown in Fig. 3.
One defines the magnetoresistance
E
ρ(B) := x
(5.72a)
jx
and the Hall coefficient
Ey
(5.72b)
RH :=
jx B
induced by solving the steady-state equation
(m v)
eB
b −
(m v) ∧ z
(5.72c)
0 = − eE +
mc
τ
with
b + Ey y
b,
b.
E := Ex x
v := vx x
(5.72d)
Multiplication by
σ
n eτ
− D =− e
(5.73)
e
m
of the steady-state equation and the introduction of the cyclotron frequency
eB
ωc :=
(5.74)
mc
yields
1
ω τ
Ex
1
+ωc τ
jx
σD
.
=
⇐⇒ ρ(B) = ρD ,
RH = − c = −
Ey
−ωc τ
1
0
σD B
ne e c
(5.75)
The Drude magnetoresistance is independent of the applied magnetic
field. The Drude Hall coefficient depends only on the electronic density and on the sign of the charge carrier. Measurements of the Drude
conductivity and of the Hall coefficient allow to extract τ and ne . For
good metals ne is of the order 1022 per cubic centimeter and τ is of order 10−14 second at room temperature (although strongly temperature
dependent). Drude’s mean free path
`D := veqp τ
with the characteristic (equipartition) velocity
1
3
2
m veqp
:= kB T
2
2
(5.76a)
(5.76b)
5.3. THE NON-INTERACTING JELLIUM MODEL
b -axis
Temperature gradient along the x
x
vx ⌧
x + vx ⌧
b
z
b
y
243
b
x
Figure 4. The Drude theory for the thermal current
assumes a directionally isotropic distribution of velocities after an elastic and instantaneous scattering event.
Two such collisions are depicted by a star of arrows representing the distribution of velocities after scattering.
We assume that the direction of the temperature gradient (black arrow) from high to low temperatures is from
left to right. This is depicted with the vectors emerging from a scattering event at x − vx τ longer than the
vectors emerging from a scattering event at x + vx τ . At
a midpoint between the left and right scattering events,
the electrons moving from left to right are more energetic
than the electrons moving from right to left. This yields
a net thermal current to theright that can be modeledby
ne
jx = n2e ×vx ×E
T (x−v
x τ ) − 2 ×vx ×E T (x+vx τ ) ≈
ne × vx2 τ × ∂E
− dT
where ne is the electronic density,
∂T
dx
vx is the velocity at x, E T (x ∓ vx τ ) is the thermal energy at the last scattering event. Equation (5.77)
follows
1 2
∂E
2
with the
identifications
v
→
v
,
n
×
→
C
x
e
v , and
3
∂T
dT
− dx → (−∇T ).
yields a mean free path of the order of the Ångström at room temperature that is one order of magnitude too small.
The Drude model of thermal transport assumes that the thermal
current per unit area and per unit time j D is given by (see Fig. 4)
j D := −κD ∇T
(5.77a)
with
1 2
1
3
3
v τ Cv ,
m v 2 = kB T,
Cv = ne kB .
(5.77b)
3
2
2
2
Drude thus predicts the universal ratio
2
κD
3 kB
=
(5.78)
σD T
2 e
in agreement with the empirical law of Wiedemann and Franz.
Drude constructed his theory of transport in metals by assuming
point-like charge carriers (the electrons) that are in local thermodynamic equilibrium and whose probability distribution of velocities is
κD =
244
5. NON-INTERACTING FERMIONS
the Maxwell-Boltzmann distribution
3/2
1
mβ
2
fB (v) := ne
e− 2 m v β .
2π
(5.79)
Thus
fB (v) d3 v
(5.80)
is the number of electrons per unit volume with velocities in the range
d3 v about v. Sommerfeld’s theory of transport in metals simply replaces the Maxwell-Boltzmann distribution (5.79) by the Fermi-Dirac
distribution
1
1 m 3
fFD (v) := 2 ×
.
(5.81)
1
2
(2π)3 ~
e( 2 m v −µ)β + 1
This approximation is justified if positions and momenta of electrons
can be specified as accurately as necessary without violating the uncertainty principle. Since the typical momentum of an electron in a metal
is
~ kF ,
(5.82)
we must demand that the momentum uncertainty ∆p satisfies
∆p ~ kF .
(5.83)
As the uncertainty in the electronic position ∆x is given by
∆x ∼
~
,
∆p
(5.84)
∆x 1
.
kF
(5.85)
1
aB
(5.86)
it follows that
However, for a good metal
kF ∼
so that
∆x aB .
(5.87)
We conclude that a classical description of electrons requires that the
uncertainty in their position be much larger than the Bohr radius. A
classical description of transport in metals is prohibited if electrons are
localized in space within atomic distances. Two characteristic length
scales enter the Sommerfeld’s or Drude’s theory of transport in metals.
First, there is the characteristic range λ of variations in space of the
external probes applied to a metal in order to induce transport, say
an electromagnetic field or a temperature gradient. One must demand
that
λ kF−1
(5.88)
5.3. THE NON-INTERACTING JELLIUM MODEL
245
for a semi-classical treatment à la Sommerfeld to hold. Second, there
is the mean free path `S which must therefore also satisfy
`S kF−1
(5.89)
for a semi-classical treatment à la Sommerfeld to hold.
The replacement by the Fermi-Dirac distribution (5.81) of the MaxwellBoltzmann distribution (5.79) only affects transport coefficients that
depend on the equilibrium velocity distribution. If one assumes that
the rate 1/τ at which elastic scattering occurs between electrons is
independent of the electron energy, then the dc conductivity, magnetoresistance, and Hall coefficient agree in the Sommerfeld and Drude
models. On the other hand, the Drude mean free path (5.76) is changed
to
`S = vF τ
(5.90)
which can be larger than `D by two order of magnitude at room temperature. Similarly, the thermal velocity
r
q
∼ kB T /m =
(5.91)
(kB T ) /εF × (εF /m)
in the thermal conductivity (5.77) must be replaced by the Fermi velocity
q
∼ εF /m
(5.92)
while the Drude specific heat
∼ ne kB
must be replaced by the smaller specific heat
kB T
∼
× ne kB .
εF
(5.93)
(5.94)
The enhancement factor εF / (kB T ) induced by the use of the Fermi
velocity cancels the reduction factor (kB T )/εF induced by the use of
the Fermi gas specific heat. The empirical law of Wiedemann and
Franz (5.78) is thus also satisfied in the model of Sommerfeld albeit
with the universal coefficient
2
κS
π 2 kB
=
.
(5.95)
σD T
3
e
5.3.3. Pauli paramagnetism. So far we have assumed that the
single-particle dispersion ει does not depend on the electronic spin.
We are now going to treat a simple model in which the single-particle
energy dispersion becomes spin dependent by accounting for a Zeeman
term, but neglecting the orbital response to the presence of an external
magnetic field B with the magnitude B = |B|.
246
5. NON-INTERACTING FERMIONS
To this end, we recall that the Zeeman energy for a magnetic moment µ (not to be confused with the chemical potential) in the presence
of a uniform magnetic field B is
− µ · B.
(5.96)
The magnetic moment of an electron with the spin operator S is
gµ
µ = − B S ≈ −µB σ
(5.97)
~
owing to the negative charge of the electron and the electron g-factor
being approximately 2. Hence, we work with the grand-canonical partition function
Z(L3 , β, µ, B) := Tr e−β (Ĥ−µN̂ ) ,
F
Ĥ :=
XX
σ=±1
εσ,k ĉ†σ,k ĉσ,k ,
N̂ :=
XX
σ=±1
k
(
† mι
Y
F := span
ĉι
ι≡(σ,k)
|0iσ = −1, +1,
ĉ†σ,k ĉσ,k ,
εσ,k
k
L
k ∈ Z3 ,
2π
~2 k2
:=
+ σ µB B,
2m
mι = 0, 1,
)
ĉι |0i = 0,
{ĉι , ĉ†ι0 }
= δι,ι0 ,
{ĉ†ι , ĉ†ι0 }
= {ĉι , ĉι0 } = 0 ,
(5.98a)
where have introduced the Bohr magneton (the electric charge e is
chosen positive by convention)
e~
.
2mc
The Bohr magneton has the units of a magnetic moment.
We want to compute the statistical average
h
i
gµ
TrF e−β (Ĥ−µN̂ ) − ~ B S
i.
h
M P (L3 , β, µ, B) := L−3
TrF e−β (Ĥ−µN̂ )
µB :=
(5.98b)
(5.99)
Because the uniform magnetic field B only breaks the SU (2) spinrotation symmetry down to the subgroup U (1) of rotations about the
quantization axis in spin space, only the component MP (L3 , β, µ, B)
of the magnetization per unit volume (5.99) along the quantization
axis that is selected by the applied magnetic field B is non-vanishing.
Hence, we are after
ln Z(L3 , β, µ, B)
MP (L , β, µ, B) ≡:= +L β
∂B
and the corresponding spin susceptibility
3
−3 −1 ∂
χP (L3 , β, µ, B) :=
∂MP (L3 , β, µ, B)
∂B
(5.100a)
(5.100b)
5.3. THE NON-INTERACTING JELLIUM MODEL
247
in the thermodynamic limit L → ∞ holding the electronic density ne
fixed.
Each electron with spin parallel to B contributes
− L−3 × µB
(5.101)
to the magnetization density. Each electron with spin antiparallel to
B contributes
+ L−3 × µB
(5.102)
to the magnetization density. If
ne± (β, µ, B)
(5.103)
denotes the density of electrons with spin parallel (+) and antiparallel
(−) to B in the thermodynamic limit, then the magnetization density
is
MP (β, µ, B) = −µB ne+ (β, µ, B) − ne− (β, µ, B)
(5.104)
in the thermodynamic limit. Of course, the constraint
ne = ne+ (β, µ, B) + ne− (β, µ, B)
(5.105)
must hold for all β, µ and B. This constraint fixes the dependence of
the chemical potential on β and B. For ease of notation, we drop the
arguments of ne± , M , and χ from now on.
When B = 0, the density of states per unit energy, per unit volume,
and per spin ν± (ε) obeys
1
ν± (ε) = ν(ε)
2
(5.106)
with ν(ε) defined in Eq. (5.33). When B 6= 0,
1
ν± (ε) = ν(ε ∓ µB B),
2
(5.107)
for an electron with spin down relative to the quantization axis B/B
in spin space lowers its energy by µB B. Hence,
Z
ne± = dε ν± (ε) fFD (ε).
(5.108)
R
We shall assume that
µ B B εF ,
(5.109)
248
5. NON-INTERACTING FERMIONS
a reasonable assumption since a B of 104 Gauss gives µB B of order
10−4 × εF . We then do the Taylor expansions
1
ν± (ε) = ν(ε ∓ µB B)
2
1
1
= ν(ε) ∓ µB Bν 0 (ε) + · · · ,
2Z
2
dε ν± (ε) fFD (ε)
ne± =
R
Z
1
dε ν(ε)fFD (ε) ∓ µB B dε ν 0 (ε) fFD (ε) + · · · ,
(5.110)
2
R
R
MP = − µB ne+ − ne−
Z
2
= + µB B dε ν 0 (ε) fFD (ε) + · · ·
1
=
2
Z
R
=+
µ2B B
Z
0
dfFD (ε)
dε ν(ε) −
+ ··· ,
dε
R
subject to the constraint that
Z
ne = dε ν(ε) fFD (ε) + · · · .
(5.111)
R
We can then use Eq. (5.60) to solve for the chemical potential
µ = εF + · · · .
(5.112)
At zero temperature
MP = µ2B ν(εF ) B,
χP = µ2B ν(εF ),
(5.113)
with corrections of the order (kB T /εF )2 at finite temperature. This
result, known as the Pauli paramagnetism, is a dramatic manifestation
of the Pauli principle. It should be contrasted to Curie’s law
(gL µB )2 J(J + 1)
gL µB B
χ P = ni
+O
(5.114)
3
kB T
kB T
for non-interacting ions with density ni , total angular momentum quantum number J, and Landé factor gL .
5.3.4. Landau levels in a magnetic field. We take the jellium
model in the presence of the magnetic field
 
 
0
0



 ≡ ∇ ∧ A.
0
Bx
B=
=∇ ∧
(5.115a)
B
0
5.3. THE NON-INTERACTING JELLIUM MODEL
249
The relevant single-particle Hamiltonian is the Pauli Hamiltonian for
an electron carrying the negative charge −e. It is
2
~
(−e)
∇−
A σ0 − (−µB ) σ3 B
i
c
#
"
(5.115b)
2
~
e
1
2 2
2 2
−~ ∂x +
∂ + B x − ~ ∂z σ0 + µB σ3 B.
=
2m
i y c
1
H=
2m
The eigenvalue problem
eikz z eiky y
Ψ(r) := √ × √ ×φ(x)×ξσ ,
L
L
HΨ(r) = εΨ(r),
0 ≤ x, y, z ≤ L,
(5.116)
L
L
with ξσ ∈ C2 a two-component spinor and 2π
kz = mz ∈ Z, 2π
ky = my ∈
Z, reduces, for any given 0 ≤ y ≤ L, to solving the one-dimensional
Harmonic oscillator for the wave function φ. The corresponding orthonormal eigenfunctions and energy eigenvalues are
√ −1/2
φn,ky (x) = 2 n! π`c
×Hn
n
x + ky `2c
2 2
2
×e−(x+ky `c ) /(2`c ) ,
`
0 ≤ x ≤ L,
(5.117a)
and
εn,kz ,σ
~2 kz2
1
=
+ ~ωc n +
+ µB B σ,
2m
2
(5.117b)
with
|e B|
ωc :=
,
mc
s
~c
,
|e B|
(5.117c)
respectively. (The functions Hn are the Hermite polynomials.) Energy
eigenvalues do not depend on ky = 2πmy /L. Energy levels are thus
degenerate. The degeneracy of the energy level with quantum numbers
n, kz , and σ is
n = 0, 1, 2, · · · ,
σ = ±,
`c :=
L2
2π`2c
(5.118)
as follows from the constraint on ky `2c ,
0≤
2πmy 2
` ≤ L,
L c
my ∈ Z
⇐⇒
0 ≤ my ≤
L2
,
2π`2c
my ∈ Z.
(5.119)
250
5. NON-INTERACTING FERMIONS
In the thermodynamic limit L → ∞, the density of states per unit
energy, per unit volume, per spin, and in the n-th Landau level is
X
L2
ν(ε, σ, n) := lim L−3 ×
×
δ ε − εn,kz ,σ
2
L→∞
2π`c
kz
Z
~2 kz2
1
dkz
1
δ ε−
− ~ωc n +
− µB B σ
×
=
2π`2c
2π
2m
2
R
3/2
(2m) ωc Θ ε − ~ωc n + 12 − µB Bσ
q
.
(5.120)
=
8 π 2 ~2
ε − ~ω n + 1 − µ B σ
c
2
B
For a fixed n = 0, 1, 2, · · · and a fixed σ = ±, this density of states
has a square root singularity that is typical of a free one-dimensional
electron gas. The smooth density of state (5.33) is strongly affected by
a magnetic field through the square root singularities. The positions
of these singularities depend on the magnetic field.
The grand-canonical partition function of the jellium model perturbed by a static and spatially uniform magnetic field pointing along
the z Cartesian axis is given by Eq. (5.26) with the identifications
(n = 0, 1, 2, · · · )
~2 kz2
1
2π
ι → (n, kz , σ), ει →
+ ~ωc n +
k ∈ Z, σ = ±.
+ µB B σ,
2m
2
L z
(5.121)
The magnetization per unit volume M can be calculated in closed form
with the help of the Poisson formula. It is [55]


πεF m
π
r
∞
sin
−
4
µ B
1 πk T
εF X 1
2 B .
√
M = χP B 1 − + B
3
µB B µB B m=1 m sinh π kB T m
µB B
(5.122)
The susceptibility
∂M
(5.123)
∂B
reduces to the sum of the Pauli (paramagnetic) susceptibility
χ :=
χP = µ2B ν(εF )
(5.124)
and the Landau (diamagnetic) susceptibility
1
χL = − χP
3
(5.125)
kB T
1.
µB B
(5.126)
in the limit
5.4. TIME-ORDERED GREEN FUNCTIONS
251
kz
B
kF
ky
kx
Figure 5. The extremal area among all the discs obtained by intersecting the Fermi sea with planes perpendicular to the applied magnetic field is that of the equatorial plane.
In the opposite limit
kB T
1.
(5.127)
µB B
of very low temperatures, the dependence of χ on 1/B oscillates with
the dominant period ∆(1/B) given by
πεF
∆(1/B) = 2π,
(5.128)
µB
i.e.,
2µB
∆(1/B) =
εF
e~
2m
= 2×
× 2 2
2 m c ~ kF
2πe 1
=
,
(5.129)
~c A(kF )
where
A(kF ) = πkF2
(5.130)
is the extremal area among all the discs obtained by intersecting the
Fermi sea with planes perpendicular to the magnetic field. This oscillatory behavior of the uniform and static magnetic susceptibility for
the jellium model was explained by Landau in 1930 within the noninteracting jellium model shortly after qualitatively similar oscillations
were measured in metals by de Haas and van Alphen the same year.
This is the so-called de-Haas-van-Alphen effect. Onsager showed in
1952 how to generalize Landau’s analysis to the nearly free electron
model.
5.4. Time-ordered Green functions
5.4.1. Definitions. Before specializing to the case of the noninteracting jellium model, we consider the generic case of a conserved
252
5. NON-INTERACTING FERMIONS
many-body Hamiltonian
Ĥµ ≡ Ĥ − µ N̂
(5.131)
acting on a Z2 -graded Fock space F. The Fock space
(
hh
ii
hh
ii hh
ii
Y n
F := span
â†ι ι |0i âι , â†ι0 = δι,ι0 , âι , âι0 = â†ι , â†ι0 = 0, âι |0i = 0,
ι
)
deg (âι ) = 0 ⇒ nι = 0, 1, 2, · · · ,
deg (âι ) = 1 ⇒ nι = 0, 1
(5.132a)
is Z2 -graded because any pair âι , â†ι carries, through its degree
(5.132b)
deg â†ι ≡ deg âι = 0, 1,
the bosonic or fermionic algebra
hh
ii
†
âι , âι0 := âι â†ι0 − (−1)deg(âι )deg(âι0 ) â†ι0 âι = δι,ι0 ,
ii
hh
â†ι , â†ι0 := â†ι â†ι0 − (−1)deg(âι )deg(âι0 ) â†ι0 â†ι = 0,
hh
ii
âι , âι0 := âι âι0 − (−1)deg(âι )deg(âι0 ) âι0 âι = 0,
(5.132c)
whenever
deg (âι ) = 0
(5.132d)
deg (âι ) = 1,
(5.132e)
or
respectively. We shall assume that the many-body Hamiltonian Ĥµ
has a Taylor expansion in powers of the operators â’s generating the
Z2 -graded Fock space (5.132) in such a way that it can be decomposed
into the sum of two non-commuting and conserved Hermitean operators
Ĥ0,µ and Ĥ1 ,
h
i
Ĥµ = Ĥ0,µ + Ĥ1 ,
Ĥ0,µ , Ĥ1 6= 0
(5.133)
whereby Ĥ0,µ is the quadratic form
X
Ĥ0,µ =
(ει − µ) â†ι âι ,
(5.134)
ι
while Ĥ1 is of higher order in the â’s. We work in the grand-canonical
ensemble with the grand-canonical partition function
Z(β, µ) := TrF e−β Ĥµ .
(5.135)
Let 0 ≤ λ ≤ 1 be a dimensionless coupling that allows us to treat
the interaction Ĥ1 adiabatically, i.e., we define
Ĥµ (λ) := Ĥ0,µ + λĤ1 .
(5.136)
5.4. TIME-ORDERED GREEN FUNCTIONS
253
We shall use the notation
Ĥµ ≡ Ĥµ (λ = 1)
(5.137)
so that Ĥµ (λ) interpolates between Ĥ0,µ and Ĥµ as λ varies between 0
and 1.
The grand-canonical potential in the grand-canonical ensemble is
defined by
F (β, µ; λ) := U (β, µ; λ) − T S(β, µ; λ)
1
≡ − ln TrF e−β Ĥµ (λ)
(5.138)
β
1
≡ − ln Z(β, µ; λ).
β
The thermal expectation value of the interaction is
−β Ĥµ (λ)
D E
Ĥ1
∂F (β, µ; λ) TrF e
(5.139)
=
≡ Ĥ1
.
∂λ
β,µ;λ
TrF e−β Ĥµ (λ)
The change in the grand-canonical potential induced by switching on
the interaction adiabatically is
Z1
Z1
E
∂F (β, µ; λ)
dλ D
=
λĤ1
.
F (β, µ; 1) − F (β, µ; 0) = dλ
∂λ
λ
β,µ;λ
0
0
(5.140)
It turns out (see appendix E.4) that the grand-canonical expectation
value
D
E
λĤ1
(5.141)
β,µ;λ
can be related to the so-called time-ordered single-particle Green function. One important physical meaning of the time-ordered singleparticle Green function is thus that it encodes the correlation energy (5.140). With this motivation in mind, we define time-ordered
Green functions in the grand-canonical ensemble as a conclusion to
this chapter.
5.4.2. Time-ordered Green functions in imaginary time.
Let  be any operator acting on the Fock space (5.132) on which the
grand-canonical partition function (5.135). is defined. Examples of
fermionic operators for the non-interacting jellium model are
Z
Z
1
1
†
3
+ik·r †
ĉσ,k = √
d re
ψ̂σ (r),
ĉσ,k = √
d3 r e−ik·r ψ̂σ (r),
V
V
V
1 X −ik·r †
ψ̂σ† (r) = √
e
ĉσ,k ,
V k
V
1 X +ik·r
ψ̂σ (r) = √
e
ĉσ,k .
V k
(5.142)
254
5. NON-INTERACTING FERMIONS
The symmetric convention for the normalization by the volume V = L3
is here chosen so that
√ the ĉ’s are dimensionless while the ψ̂’s have
the dimensions of 1/ V . Examples of bosonic operators for the noninteracting jellium model are
Z
XX †
ρ̂q ≡
ĉk,σ ĉk+q,σ = d3 r e−iq·r ρ̂(r),
σ=↑,↓
ρ̂(r) ≡
k
X
σ=↑,↓
V
ψ̂σ† (r)ψ̂σ (r)
1 X +iq·r
=
e
ρ̂q .
V q
(5.143)
The asymmetric convention for the normalization by the volume is here
chosen so that the ρ̂’s are dimensionless in momentum space while they
have the dimensions of 1/V in position space.
Operator Â, as any operator acting on the Fock space including the
kinetic energy operator Ĥ0,µ or the interaction Ĥ1 , is explicitly time
independent. Let τ ∈ R be a real parameter with dimension of time
that we call imaginary time. We endow the operator  with the explicit
dependence on imaginary time
ÂH (τ, τ0 ) := e+(τ −τ0 )Ĥµ Â(τ0 ) e−(τ −τ0 )Ĥµ ,
Â(τ0 ) ≡ Â.
(5.144)
The index H stands for the Heisenberg picture. Alternatively, we endow
the operator  with the explicit dependence on imaginary time
ÂI (τ, τ0 ) := e+(τ −τ0 )Ĥ0,µ Â(τ0 ) e−(τ −τ0 )Ĥ0,µ ,
Â(τ0 ) ≡ Â.
(5.145)
The index I stands for the interacting picture.
The equations of motion obeyed by  in the Heisenberg and interacting pictures follow from taking imaginary time τ − τ0 to be infinitesimal. They are
h
i
∂τ ÂH (τ, τ0 ) = Ĥµ , ÂH (τ, τ0 ) ,
ÂH (τ0 ) = Â,
(5.146)
in the Heisenberg picture and
h
i
∂τ ÂI (τ, τ0 ) = Ĥ0,µ , ÂI (τ, τ0 ) ,
ÂI (τ0 ) = Â,
(5.147)
in the interacting picture.
In the Schrödinger picture, states at the imaginary time τ0 are related to states at the imaginary time τ by multiplication of the former
state from the left with the imaginary-time evolution operator
ÛS (τ, τ0 ) = e−(τ −τ0 )Ĥµ ,
(5.148a)
as their imaginary-time evolution is governed by the imaginary-time
Schrödinger equation
∂τ ÛS (τ, τ0 ) = −Ĥµ ÛS (τ, τ0 ) ⇐⇒ ∂τ ΨS (τ ) = −Ĥµ ΨS (τ ),
ΨS (τ0 ) given.
(5.148b)
5.4. TIME-ORDERED GREEN FUNCTIONS
255
In the interacting picture, states at imaginary time τ0 are related to
states at the imaginary time τ by multiplication of the former state
from the left with the operator


Zτ


ÛI (τ, τ0 ) ≡ Tτ exp − dτ 0 Ĥ1I (τ 0 , τ0 )
τ0
:= 1 +
∞
X
n=1
(−1)n
Zτ2
Zτ
dτn · · ·
τ0
dτ1 Ĥ1I (τn , τ0 ) · · · Ĥ1I (τ2 , τ0 )Ĥ1I (τ1 , τ0 ),
τ0
(5.149a)
as their imaginary-time evolution is governed by the imaginary-time
first-order differential equation
∂τ ÛI (τ, τ0 ) = −Ĥ1I (τ, τ0 ) ÛI (τ, τ0 ) ⇐⇒
∂τ ΨI (τ ) = −Ĥ1I (τ, τ0 ) ΨI (τ ),
ΨI (τ0 ) given.
(5.149b)
The operation of imaginary-time ordering used in Eqs (5.148a) and (5.149a)
is defined for any pair of operators
Tτ Â(τ1 , τ0 ) B̂(τ2 , τ0 ) := Â(τ1 , τ0 ) B̂(τ2 , τ0 )Θ (τ1 − τ2 )
≡
± B̂(τ2 , τ0 ) Â(τ1 , τ0 )Θ (τ2 − τ1 )

when τ1 > τ2 ,
 Â(τ1 , τ0 ) B̂(τ2 , τ0 ),

(±)B̂(τ2 , τ0 ) Â(τ1 , τ0 ),
when τ2 > τ1 ,
(5.150)
irrespective of how the imaginary-time evolution is implemented. The
sign + holds for a pair of bosonic operators or for a mixed pair of
bosonic and fermionic operators. The sign − holds for a pair of fermionic
operators.
Because the interaction does not commute with the kinetic energy,
Ĥ1I (τ, τ0 ) = e+(τ −τ0 )Ĥ0,µ Ĥ1I (τ0 ) e−(τ −τ0 )Ĥ0,µ
(5.151)
depends explicitly on imaginary time in the Schrödinger-like equation (5.149b). Hence, the integration over imaginary time cannot be
performed explicitly in Eq. (5.149a). Neither ÛS (τ, τ0 ) nor ÛI (τ, τ0 ) are
unitary, but they share the composition law
ÛS (τ, τ 0 ) ÛS (τ 0 , τ0 ) = ÛS (τ, τ0 ) =⇒ ÛS−1 (τ, τ 0 ) = ÛS (τ 0 , τ ),
(5.152)
and
ÛI (τ, τ 0 ) ÛI (τ 0 , τ0 ) = ÛI (τ, τ0 ) =⇒ ÛI−1 (τ, τ 0 ) = ÛI (τ 0 , τ ),
(5.153)
256
5. NON-INTERACTING FERMIONS
for all triplets (τ, τ 0 , τ0 ), respectively. Either ÛS (τ, τ0 ) or ÛI (τ, τ0 ) become unitary under the analytical continuation
τ ∈ R → +it,
t ∈ R.
(5.154)
The relation between the imaginary-time evolution in the Schrödinger
and interaction pictures is
ÛI (τ1 , τ2 ) = e+(τ1 −τ0 )Ĥ0,µ ÛS (τ1 , τ2 ) e−(τ2 −τ0 )Ĥ0,µ ⇐⇒
ΨS (τ ) = e−(τ −τ0 )Ĥ0,µ ΨI (τ ),
(5.155)
where τ0 is the time at which ΨS (τ0 ) = ΨH (τ0 ) = ΨI (τ0 ).
Let  and B̂ be any pair of operators with the degrees deg(Â)
and deg(B̂), respectively, acting on the Fock space (5.132) on which
the grand-canonical partition function (5.135) is defined. The timeordered correlation function in imaginary time between  and B̂ is the
expectation value
h
i
−β Ĥµ
TrF e
Tτ ÂH (τ1 , τ0 ) B̂H (τ2 , τ0 )
Cβ,µ;Â,B̂ (τ1 , τ2 ) := −
Tr e−β Ĥµ
(5.156)
E
D F
.
≡ − Tτ ÂH (τ1 , τ0 ) B̂H (τ2 , τ0 )
β,µ
The sign on the right-hand side is convention and it is implicitly assumed that this correlation function does not depend on τ0 . In fact,
we are going to prove that:
(i) Translation invariance in imaginary time holds for the correlation function (5.156) as
Cβ,µ;Â,B̂ (τ1 , τ2 ) = Cβ,µ;Â,B̂ (τ1 − τ2 ).
(5.157a)
(ii) The correlation function (5.156) decays exponentially fast with
|τ1 − τ2 | only if
|τ1 − τ2 | < β.
(5.157b)
It grows exponentially fast with |τ1 − τ2 | otherwise.
(iii) If −β < τ1 − τ2 < 0, it then follows that
Cβ,µ;Â,B̂ (τ1 + β, τ2 ) = ±Cβ,µ;Â,B̂ (τ1 , τ2 ),
(5.157c)
where periodicity holds if  and B̂ commute while antiperiodicity holds if  and B̂ anticommute under the operation of
imaginary-time ordering.
5.4. TIME-ORDERED GREEN FUNCTIONS
257
(iv) If  and B̂ are bosonic, then
Cβ,µ;ÂB̂ (τ ) =
1 X −i$l τ
e
Cβ,µ;ÂB̂,i$ ⇐⇒
l
β l∈Z
Zβ −
Cβ,µ;ÂB̂,i$ =
l
dτ e+i$l τ Cβ,µ;ÂB̂ (τ )
0+
(5.157d)
with the bosonic Matsubara frequency $l = 2lπ/β. If  and
B̂ are fermionic, then
Cβ,µ;ÂB̂ (τ ) =
1 X −iωn τ
e
Cβ,µ;ÂB̂,iω ⇐⇒
n
β n∈Z
Zβ −
Cβ,µ;ÂB̂,iω =
n
dτ e+iωn τ Cβ,µ;ÂB̂ (τ )
0+
(5.157e)
with the fermionic Matsubara frequency ωn = (2n + 1)π/β. The
asymmetric convention for the normalization by β is the same
as in Eq. (5.143).
Proof. Cyclicity of the trace with the definition (5.144) implies
h
Cβ,µ;Â,B̂ (τ1 , τ2 ) = − Θ(τ1 − τ2 )
∓ Θ(τ2 − τ1 )
−β Ĥµ
TrF e
e
+(τ1 −τ2 )Ĥµ
−(τ1 −τ2 )Ĥµ
 e
B̂
i
TrF e−β Ĥµ
i
h
TrF e−β Ĥµ e+(τ2 −τ1 )Ĥµ B̂ e−(τ2 −τ1 )Ĥµ Â
TrF e−β Ĥµ
(5.158)
from which (i) and (iii) follow. Insertion of a complete basis of eigenstates of Ĥµ in Eq. (5.158), where, without loss of generality, the
many-body ground state energy is taken to be positive, implies that
the support of Cβ,µ;Â,B̂ (τ ) for which it is a decaying function of τ is
Eq. (5.157b). The Fourier transforms (5.157d) and (5.157e) follow from
the periodicity (iii).
258
5. NON-INTERACTING FERMIONS
The correlation function (5.156) cannot be evaluated exactly in
practice. For a systematic perturbation theory, a better suited representation of Eq. (5.156) is
"
#
TrF e−β Ĥ0,µ ÛI (τ0 + β, τ0 ) Tτ ÂI (τ1 , τ0 ) B̂I (τ2 , τ0 )
Cβ,µ;Â,B̂ (τ1 −τ2 ) = −
i.
h
TrF e−β Ĥ0,µ ÛI (τ0 + β, τ0 )
(5.159)
Proof. Equation (5.159) follows from Eq. (5.156) with the help of
Eq. (5.155). First, we observe that
e−β Ĥµ = US (τ0 + β, τ0 )
(5.160)
= e−β Ĥ0,µ UI (τ0 + β, τ0 ).
Second, we observe that
ÂH (τ1 , τ0 ) B̂H (τ2 , τ0 ) = e+(τ1 −τ0 )Ĥµ Â(τ0 ) e−(τ1 −τ2 )Ĥµ B̂(τ0 ) e−(τ2 −τ0 )Ĥµ
= e+(τ1 −τ0 )Ĥµ Â(τ0 ) ÛS (τ1 , τ2 ) B̂(τ0 ) e−(τ2 −τ0 )Ĥµ
Eq. (5.145)
= e+(τ1 −τ0 )Ĥµ e−(τ1 −τ0 )Ĥ0,µ ÂI (τ1 , τ0 ) e+(τ1 −τ0 )Ĥ0,µ ÛS (τ1 , τ2 )
× e−(τ2 −τ0 )Ĥ0,µ B̂I (τ2 , τ0 ) e+(τ2 −τ0 )Ĥ0,µ e−(τ2 −τ0 )Ĥµ
= ÛI (τ0 , τ1 ) ÂI (τ1 , τ0 ) ÛI (τ1 , τ2 ) B̂I (τ2 , τ0 ) ÛI (τ2 , τ0 ) .
(5.161)
Third, we recall that bosonic (fermionic) operators behave like complex
(Grassmann) numbers under the operation of τ -ordering. Hence, we
may move the bosonic operator ÛI (τ0 , τ1 ) to the right of ÂI (τ1 , τ0 ), while
we may move the bosonic operator ÛI (τ2 , τ0 ) to the left of B̂I (τ2 , τ0 ) in
Tτ ÛI (τ0 , τ1 ) ÂI (τ1 , τ0 ) ÛI (τ1 , τ2 ) B̂I (τ2 , τ0 ) ÛI (τ2 , τ0 ) .
(5.162)
Equation (5.159) then follows from the identity ÛI (τ0 , τ1 ) ÛI (τ1 , τ2 ) ÛI (τ2 , τ0 ) =
1.
Another useful tool to evaluate the correlation function (5.156) is
the equation of motion
D
E
−∂τ1 Cβ,µ;Â,B̂ (τ1 − τ2 ) = δ (τ1 − τ2 ) ÂH (τ1 , τ0 )B̂H (τ1 , τ0 ) ∓ B̂H (τ1 , τ0 )ÂH (τ1 , τ0 )
β,µ
D nh
i
oE
+ Tτ Ĥµ , ÂH (τ1 , τ0 ) B̂H (τ2 , τ0 )
β,µ
(5.163)
5.4. TIME-ORDERED GREEN FUNCTIONS
259
that is obeyed forh any unequal imaginary
times τ1 and τ2 . In general,
i
the commutator Ĥµ , ÂH (τ1 , τ0 ) is not proportional to ÂH (τ1 , τ0 ) so
that this equation does not close on its own. In fact, a closed set of
equations of motion is generically infinite.
The definition (5.156) readily generalizes to the 2n-point time-ordered
correlation function in imaginary time between operators Â1 , · · · , Ân
and B̂1 , · · · , B̂n acting on the Fock space (5.132) on which the grandcanonical partition function (5.135) is defined. It is
Cβ,µ;Â
1 ,··· ,Ân |B̂1 ,··· ,B̂n
(τ1 , · · · , τn |τ10 , · · · , τn0 ) :=
D E
(−1)n Tτ ÂH (τ1 , τ0 ) × · · · × ÂH (τn , τ0 ) × B̂H (τ10 , τ0 ) × · · · × B̂H (τn0 , τ0 )
(5.164)
Next, we are going to compute explicitly 2 and 4 points timeordered Green functions for the non-interacting jellium model, whereby
we shall make the identifications
~2 k2
†
 → ĉσ,k ,
B̂ → ĉσ,k ,
ει → εσ,k ≡
.
(5.165)
2m
5.4.3. Time-ordered Green functions in real time. Imaginary time τ and real time t are related by the analytical continuation
τ = it,
t ∈ R.
(5.166)
As before, let  and B̂ be any pair of operator with the degrees deg(Â)
and deg(B̂), respectively, acting on the Fock space (5.132) on which the
grand-canonical partition function (5.135) is defined. The time-ordered
correlation function in real time between  and B̂ is the expectation
value
h
i
TrF e−β Ĥµ Tt ÂH (t1 , t0 ) B̂H (t2 , t0 )
Cβ,µ;Â,B̂ (t1 , t2 ) = −
Tr e−β Ĥµ
(5.167a)
D F
E
≡ − Tt ÂH (t1 , t0 ) B̂H (t2 , t0 )
,
β,µ
where
ÂH (t, t0 ) := e+i(t−t0 )Ĥµ Â(t0 )e−i(t−t0 )Ĥµ
(5.167b)
and
Tt Â(t1 , t0 )B̂(t2 , t0 ) := Â(t1 , t0 )B̂(t2 , t0 )Θ (t1 − t2 )
≡
+ (−1)deg(Â)deg(B̂) B̂(t2 , t0 )Â(t1 , t0 )Θ (t2 − t1 )

when t1 > t2 ,
 Â(t1 , t0 )B̂(t2 , t0 ),

(−1)deg(Â)deg(B̂) B̂(t2 , t0 )Â(t1 , t0 ),
when t2 > t1 .
(5.167c)
.
β,µ
260
5. NON-INTERACTING FERMIONS
The sign on the right-hand side of Eq. (5.167a) is here convention, as
is the imaginary factor on the left-hand side of Eq. (5.167a).
5.4.4. Application to the non-interacting jellium model.
The grand-canonical partition function for the non-interacting jellium
model is defined in Eq. (5.24). We shall use the more compact notation
Ĥµ := Ĥ − µN̂ ,
ξσ,k ≡ εσ,k − µ.
(5.168)
5.4.4.1. Momentum-space representation. The imaginary-time-ordered
single-particle Green function in momentum space is defined by Eq. (5.156)
with the identifications
 → ĉσ1 ,k1 ,
B̂ → ĉ†σ2 ,k2 .
(5.169)
For any |τ1 − τ2 | < β, it is given by
E
D †
(τ1 − τ2 ) = − Tτ ĉH σ1 ,k1 (τ1 , τ0 )ĉH σ2 ,k2 (τ2 , τ0 )
Cβ,µ;ĉ
,ĉ†
σ1 ,k1
σ2 ,k2
β,µ
h
= − Θ(τ1 − τ2 )
−β Ĥµ +(τ1 −τ2 )Ĥµ
+(τ2 −τ1 )Ĥµ †
e
e
ĉσ ,k e
ĉσ ,k
1 1
2 2
TrF
−β Ĥµ
TrF e
h
+ Θ(τ2 − τ1 )
−β Ĥµ +(τ2 −τ1 )Ĥµ †
+(τ1 −τ2 )Ĥµ
e
ĉσ ,k e
ĉσ ,k
2 2
1 1
−β Ĥµ
TrF e
TrF e
≈ δσ1 ,σ2 δk1 ,k2 Gβ,µ (τ1 − τ2 , k1 ),
(5.170a)
where
n
h
i
o
Gβ,µ (τ, k) = − Θ(+τ ) 1 − feFD (ξk ) e−τ ξk − Θ(−τ )feFD (ξk ) e−τ ξk
(5.170b)
and [compare with the definition (5.27a) of the Fermi-Dirac distribution
that depends explicitly on the chemical potential]
feFD (ξk ) := fFD (εk ) =
1
.
(5.170c)
+1
e
It is extended to |τ1 − τ2 | > β by antiperiodicity. To reach the last
line, we made a small error that vanishes in the thermodynamic limit
by which the volume V = L3 of the system goes to infinity, while the
average number of electrons per unit volume,
ne := β −1
β ξk
∂ ln Z(V, β, µ)
,
∂µ
(5.171)
is held fixed. This is so because we need to introduce once the resolution
of the identity in terms of the exact many-body energy eigenstates
|ιi(Ne ) ,
(Ne )
Eι
i
the energy, Ne the electron number,
(5.172)
of Ĥµ between the creation and annihilation operators to go from the
second equality to the third equality of Eq. (5.170a). This brings the
i
5.4. TIME-ORDERED GREEN FUNCTIONS
261
exponentials
(N )
(Ne +1)
−Eι e
−(τ1 −τ2 ) Eι,σ,k
e
≈ e−(τ1 −τ2 )ξk ,
(5.173)
where the many-body energy eigenstate |ι, σ, ki(Ne +1) has one additional occupied single-particle level compared to the many-body energy eigenstate |ιi(Ne ) , to be thermal averaged when τ1 > τ2 , and the
exponentials
(N −1)
e
+(τ1 −τ2 ) Eι,σ,k
e
(Ne )
−Eι
≈ e−(τ1 −τ2 )ξk ,
(5.174)
where the many-body energy eigenstate |ι, σ, ki(Ne −1) has one less occupied single-particle level compared to the many-body energy eigenstate
|ιi(Ne ) , to be thermal averaged when τ2 > τ1 .
Owing to the fact that
h
i
lim 1 − feFD (ξk ) = Θ(+ξk ),
lim feFD (ξk ) = Θ(−ξk ), (5.175)
β→∞
β→∞
Eq. (5.170b) tells us that, at zero temperature, the imaginary-timeordered Green function is non-vanishing at positive (negative) time if
and only if the single-particle level ξk is unoccupied (occupied), i.e.,
lim Gβ,µ (τ, k) = − Θ(+τ ) Θ(+ξk ) e−τ ξk − Θ(−τ ) Θ(−ξk ) e−τ ξk .
β→∞
(5.176)
Owing to the antiperiodic dependence (5.157e), for any fermionic
Matsubara frequency ωn = (2n + 1)π/β,
Zβ −
Gβ,µ (ωn , k) :=
dτ e+iωn τ Gβ,µ (τ, k)
0+
Z0−
=
dτ e+iωn τ Gβ,µ (τ, k)
(5.177)
−β +
T →0
=
1
.
iωn − ξk
The real-time-ordered single-particle Green function in momentum
space follows from Eq. (5.170) with the analytical continuation (5.166),
i.e., it is
D E
†
Cβ,µ;ĉ
(t1 − t2 ) = − Tt ĉH σ1 ,k1 (t1 , t0 )ĉH σ2 ,k2 (t2 , t0 )
,ĉ†
σ1 ,k1
β,µ
σ2 ,k2
h
= − Θ(t1 − t2 )
TrF
i
−β Ĥµ
TrF e
h
+ Θ(t2 − t1 )
−β Ĥµ +i(t1 −t2 )Ĥµ
+i(t2 −t1 )Ĥµ †
e
e
ĉσ ,k e
ĉσ ,k
1 1
2 2
−β Ĥµ +i(t2 −t1 )Ĥµ †
+i(t1 −t2 )Ĥµ
e
ĉσ ,k e
ĉσ ,k
2 2
1 1
TrF e
−β Ĥµ
TrF e
≈ δσ1 ,σ2 δk1 ,k2 Gβ,µ (t1 − t2 , k1 )
(5.178a)
i
262
5. NON-INTERACTING FERMIONS
where
n
h
i
o
Gβ,µ (t, k) = − Θ(+t) 1 − feFD (ξk ) e−itξk − Θ(−t) feFD (ξk ) e−itξk .
(5.178b)
At zero temperature,
lim Gβ,µ (t, k) = − Θ(+t)Θ(+ξk ) e−itξk − Θ(−t) Θ(−ξk ) e−itξk .
β→∞
(5.179)
In real-frequency space,
Z
Gβ,µ (ω, k) := dt e+iωt Gβ,µ (t, k)
R
=−
feFD (ξk )
1 − feFD (ξk )
+
ω − ξk + i0+ ω − ξk − i0+
!
(5.180)
with the zero-temperature limit
lim Gβ,µ (ω, k) =
β→∞
−1
.
ω − ξk + i 0+ sgn(ξk )
(5.181)
5.4.4.2. Position-space representation. By combining Eqs. (5.142),
(5.170), and (5.178), we obtain the position-space representation
1 X X −ik1 ·r1 +ik2 ·r2
Cβ,µ;ψ̂ (r ),ψ̂† (r ) (τ1 − τ2 ) =
e
Cβ,µ;ĉ
(τ1 − τ2 )
†
σ1 1
σ2 2
σ1 ,k1 ,ĉσ2 ,k2
V k k
1
≈ δσ1 ,σ2
2
1 X −ik·(r1 −r2 )
e
Gβ,µ (τ1 − τ2 , k)
V k
≡ δσ1 ,σ2 Gβ,µ (τ1 − τ2 , r 1 − r 2 )
(5.182)
and
Cβ,µ;ψ̂
†
σ1(r 1 ),ψ̂σ2(r 2 )
(t1 − t2 ) =
1 X X −ik1 ·r1 +ik2 ·r2
(t1 − t2 )
Cβ,µ;ĉ
e
†
σ1 ,k1 ,ĉσ2 ,k2
V k k
1
≈ δσ1 ,σ2
2
1 X −ik·(r1 −r2 )
e
Gβ,µ (t1 − t2 , k)
V k
≡ δσ1 ,σ2 Gβ,µ (t1 − t2 , r 1 − r 2 )
(5.183)
of the single-particle Green function in imaginary and real times, respectively.
At equal points in space, it is useful to introduce the density of
states per spin
1 X
δ(ξk − ξ),
(5.184)
νe(ξ) := ν(ε) =
V k
5.4. TIME-ORDERED GREEN FUNCTIONS
263
in terms of which
Z
n
h
i
o
dξ νe(ξ) Θ(+τ ) 1 − feFD (ξ) e−τ ξ − Θ(−τ )feFD (ξ) e−τ ξ
Gβ,µ (τ, r = 0) = −
(5.185)
and
Z
Gβ,µ (t, r = 0) = −
n
h
i
o
−itξ
−itξ
e
e
dξ νe(ξ) Θ(+t) 1 − fFD (ξ) e
− Θ(−t)fFD (ξ) e
,
(5.186)
respectively. For the parabolic spectrum of the jellium model,
 p m
, for d = 1,

2π 2 ε
m
,
for d = 2,
ν(ε) =
ε := ξ + µ,
(5.187)
√
 2π
m 2m ε
, for d = 3,
2π 2
so that the density of states per spin can be taken to be the constant
νF around the Fermi energy at very low temperatures.
At zero temperature and assuming a constant density of states per
spin, the imaginary-time single-particle Green function at equal points
is, up to a proportionality constant, the Laplace transform of the sign
function, i.e.,
Z
Gβ=∞,µ (τ, r = 0) ≈ − νF dξ Θ(+τ )Θ(+ξ)e−τ ξ − Θ(−τ )Θ(−ξ)e−τ ξ
=−
νF
.
τ
(5.188)
Analytical continuation to real time gives
Z
Gβ=∞,µ (t, r = 0) ≈ − νF dξ Θ(+t) Θ(+ξ) e−itξ − Θ(−t) Θ(−ξ) e−itξ
=+
iνF
.
t − i0+ sgn(t)
(5.189)
The approximation by which the density of states per spin is assumed to
be constant becomes exact in the limits τ → ±∞ (t → ±∞). In other
words, Eqs. (5.188) and 5.189) become exact in the limits for which the
integrals on the right-hand sides are dominated by the contributions
around the Fermi energy ξ = 0. The algebraic decay on the righthand sides of Eqs. (5.188) and 5.189) is caused by the discontinuity at
the Fermi energy of the Fermi-Dirac distribution at zero temperature.
If the density of states per spin tames the discontinuity at the Fermi
energy of the Fermi-Dirac distribution at zero temperature, say because
it vanishes in a power law fashion at the Fermi energy, νe(ξ) ∼ |ξ|g with
g > 0, the long-time correlation probed by the single-particle Green
264
5. NON-INTERACTING FERMIONS
function at equal points decay faster, e.g.,
Gβ=∞,µ (t, r = 0) ∼ +iΓ(g + 1) e−iπ(1+g)/2
sgn(t)
.
|t|1+g
(5.190)
Tunneling experiments give access to the asymptotic time dependence
of the single-particle Green function at equal points in space. Thus,
they could signal whenever perturbations to the non-interacting limit
are sufficiently strong to change the exponent g from the value g = 0
to g > 0.
5.4.4.3. At equal times. We now combine Eqs. (5.183) and (5.178b)
to study
1 X −ik·r
e
Gβ,µ (t, k)
V k
h
i
o
1 X −ik·r n
−itξk
−itξk
e
e
e
Θ(+t) 1 − fFD (ξk ) e
− Θ(−t)fFD (ξk )e
=−
V k
(5.191)
Gβ,µ (t, r) :=
at equal times, i.e., in the limit t → −0+ (without loss of generality).
In this limit, the equal-time single-particle Green function in positionspace is the Fourier transform of the Fermi-Dirac distribution,
Gβ,µ (t = −0+ , r) =
1 X −ik·r e
e
fFD (ξk ).
V k
(5.192)
At zero temperature and in the thermodynamic limit, Eq. (5.192)
reduces to the Fourier transform over the Heaviside step function
+
dd k −ik·r
e
Θ(−ξk )
(2π)d
2 2
Z
dd k −ik·r
~ kF ~2 k2
=
e
Θ
−
(2π)d
2m
2m
Z
dd k −ik·r
=
e
Θ (kF − |k|) .
(2π)d
Z
lim Gβ,µ (t = −0 , r) =
β→∞
(5.193)
In d = 1,
lim Gβ,µ (t = −0+ , r) =
+kF
Z
β→∞
dk −ikr
e
2π
−kF
=
sin kF r
.
πr
(5.194)
5.4. TIME-ORDERED GREEN FUNCTIONS
265
For d > 1,
Z
+
lim Gβ,µ (t = −0 , r) =
β→∞
dd k −ik·r
e
Θ (kF − |k|)
(2π)d
1
= d
|r|
Z
kF |r|
Z
dωd
(2π)d
dp pd−1 e−ip cos θ1
0
1
= d
|r|
Z
2πδd,2 +(1−δd,2 )π
kF |r|
b
dΩ
d−1
(2π)d
Z
d−1
Z
dp p
0
dθ1 sind−2 θ1 e−ip cos θ1 ,
0
(5.195a)
where 0 ≤ θn < π for n = 1, · · · , d − 2 and 0 ≤ θd−1 < 2π with
dωd := sind−2 θ1 dθ1 sind−3 θ2 dθ2 · · · sind−1−i θi dθi · · · sin θd−2 dθd−2 dθd−1 ,
b
dΩ
sind−3 θ2 dθ2 · · · sind−1−i θi dθi · · · sin θd−2 dθd−2 dθd−1 .
d−1 :=
(5.195b)
(i) Example d = 2,
1/(2π)2
lim Gβ,µ (t = −0 , r) =
β→∞
|r|2
+
kF |r|
Z2π
Z
dp p
0
Eq. (13.6.22) from Ref. [56]
1/(2π)2
=
|r|2
dθ e−ip cos θ
0
kF |r|
Z
dp p
X
Z2π
Jn (p)
n∈Z
0
π
dθ e−in( 2 −θ)
0
kF |r|
1/(2π)
=
|r|2
Z
dp p J0 (p)
0
kF |r|
8.472.3 from Ref. [57]
1/(2π)
=
|r|2
Z
d
dp
p J1 (p)
dp
0
k
= F J1 (kF |r|).
2π|r|
(5.196a)
Hence [see Eq. (4.4.5) from Ref. [56]],
lim
|r| → ∞
β→∞
k
Gβ,µ (t = −0+ , r) ∼ + F
2π|r|
s
2
π
sin kF |r| −
πkF |r|
4
(5.196b)
if d = 2.
266
5. NON-INTERACTING FERMIONS
(ii) Example d = 3,
kF |r|
1/(2π)3
lim Gβ,µ (t = −0+ , r) =
β→∞
|r|3
Z
dp p2
0
Z2π
Zπ
dϕ
0
dθ sin θ e−ip cos θ
0
kF |r|
1/(2π)2
=
|r|3
Z
Z+1
dp p
dx e−ip x
2
−1
0
e−iπ/2 /(2π)2
=
|r|3
kF |r|
Z
dp p e+ip − e−ip
0
=e
kF
R|r|
−iπ
/(2π)2
|r|3
0
−iπ
=
dp
n
o
p e+ip + e−ip − e+ip + e−ip
d
dp
2h
e
/(2π)
+ikF |r|
−ikF |r|
k
|r|
e
+
e
F
|r|3
i
+ i e+ikF |r| − e−ikF |r| .
(5.197a)
Hence,
lim
|r| → ∞
β→∞
Gβ,µ (t = −0+ , r) ∼
kF
+ikF |r|−iπ
−ikF |r|+iπ
e
+
e
(2π)2 |r|2
(5.197b)
if d = 3. Examples (5.194), (5.196b), and (5.197b) illustrate the powerlaw decay of the equal-time single-particle Green function for large
separations in the non-interacting jellium model. This decay is slower
the lower the dimensionality. This slow decay reflects the discontinuity
of the Fermi-Dirac distribution at zero temperature. More generally,
it can be shown that, for any d-dimensional simply-connected closed
Fermi surface with a strictly positive-definite curvature tensor,
lim
|r| → ∞
β→∞
Gβ,µ (t = −0+ , r) ∼ A+ (b
r)
e−ikF (−br)·r−iπ(d+1)/4
|r|(d+1)/2
(5.198)
−ikF (+b
r )·r+iπ(d+1)/4
+ A− (b
r)
e
|r|(d+1)/2
b≡
where A± (b
r ) are dimensionful [same dimension as |kF (±b
r )|(d−1)/2 ], r
r/|r|, and, given a coordinate system in momentum space with the
center of gravity of the Fermi surface as origin, kF (±b
r ) = ±kF (b
r ) are
b are tangent
the two Fermi points for which the hyperplanes normal to r
in these points to the Fermi surface, see Fig. 6.
5.5. PROBLEMS
267
r
kF (b
r)
Figure 6. The Fermi vector kF (b
r ) is constructed as follows. Any fixed vector r in real space is determined by
b and magnitude |r|. The dashed line repits direction r
b. The vector kF (b
resents a hyperplane orthogonal to r
r)
is defined as the point where the hyperplane [(d − 1)dimensional] touches the [(d − 1)-dimensional] Fermi surface.
5.5. Problems
5.5.1. Equal-time non-interacting two-point Green function for a Fermi gas.
Introduction. We introduced in Eq. (5.192) the equal-time Green
function in position space Gβ,µ (t = −0+ , r). At zero temperature (β →
∞) and in the thermodynamic limit, we found
+
Z
Gβ,µ (t = −0 , r) =
dd k −ik·r
e
Θ (−ξk ) .
(2π)d
(5.199)
As usual, the function Θ is the Heaviside function step function. We
then proceeded to compute the asymptotic behavior for a spherical
Fermi surface in d = 1, 2, 3 dimensions and noted that the power-law
decay is slower the lower the dimension d. We are going to prove
Eq. (5.198).
Exercise 1.1: For any d-dimensional simply-connected closed Fermi
surface with a strictly positive-definite curvature tensor:
(a) Convince yourself that Gβ,µ (−0+ , r) can be written as
Z+∞
0
b) e−ik |r| ,
Gβ,µ (−0 , r) =
dk 0 N (k 0 , r
+
−∞
(5.200a)
268
5. NON-INTERACTING FERMIONS
where
Z
dd k
0
0
b
b
N (k , r ) =
δ
(k
−
k
·
r
)
Θ
−
ξ
k
(2π)d
α
b − k 0 Θ(+k 0 ) |+kF (+b
b − k0|
≈ Ã− (b
r ) Θ + kF (+b
r) · r
r) · r
α
b + k0| ,
b + k 0 Θ(−k 0 ) |−kF (−b
r) · r
+ Ã+ (b
r ) Θ − kF (−b
r) · r
(5.200b)
b ≡ r/|r| and, given a coorwhere ñ (b
r ) are dimensionful, r
dinate system in momentum space with the center of gravity of the Fermi surface as origin, kF (±b
r ) are the two Fermi
b are tangent in
points for which the hyperplanes normal to r
these points to the Fermi surface, see Fig. 6. For a given
b, this approximation is good for either k 0 ≈ kF (+b
b or
r
r) · r
b. Determine the exponent α as a function of
k 0 ≈ kF (−b
r) · r
the dimension d. What are the coefficients Ã+ and Ã− in the
case of a spherical Fermi surface?
(b) Do the k 0 integral in Eq. (5.200) to derive Eq. (5.198). Why
is the approximation in Eq. (5.200b) valid in the limit of large
distance |r|?
(c) Compare Eq. (5.198) with the results (5.194), (5.196b), and
(5.197b) for the spherical Fermi surface in d = 1, d = 2, and
d = 3, respectively.
(d) What happens if the Fermi surface has a flat piece?
5.5.2. Application of the Kubo formula to the Hall conductivity in the integer quantum Hall effect.
Introduction. The spectrum of a two-dimensional gas of free electrons is strongly reorganized when the electrons are subject to a perpendicular magnetic field. The parabolic dispersion, whose density of
states is constant as a function of energy, turns into a sequence of
flat bands with an equidistant separation in energy, the so-called Landau levels. Each of these Landau levels comprises an extensive number of degenerate single-particle states. Whenever an integer number
ñ = 1, 2, · · · of Landau levels is completely filled with electrons and
the next-higher Landau level is empty, the single-Slater-determinant
ground state is incompressible and insulating as far as longitudinal
charge transport is concerned. However, this incompressible state features a non-vanishing and quantized transverse conductivity σH =
ñ e2 /h. This is the integer quantum Hall effect (IQHE). [58]
We are going to derive these features of the IQHE. We will see
that the non-vanishing Hall conductivity is intimately related to the
fact that the electrons experience non-commutative quantum geometry.
This means that the two in-plane components of the electron’s position
operator do not commute, when projected to the degrees of freedom
5.5. PROBLEMS
269
from one Landau level. This is in sharp contrast to the case of free
electrons, whose position operator components commute.
Diagonalizing the Landau Hamiltonian. Non-interacting electrons
confined to two-dimensional position space under a perpendicular uniform magnetic field B = B e3 of magnitude B > 0 are governed by the
Hamiltonian
1 2
Ĥ :=
π̂ ,
(5.201a)
2m
where the gauge-invariant momentum (−e < 0)
π̂ := p̂ − (−e) A(r̂)
(5.201b)
is given in terms of the vector potential
∂1 A2 (r) − ∂2 A1 (r) = B
(5.201c)
and the two components of the position operator r̂ satisfy canonical
commutation relations with the two components of the momentum
operator p̂
[r̂i , p̂i ] = iδij ,
i, j = 1, 2,
(5.201d)
in units where the speed of light and Planck’s constant are unity. However, the components of π̂ are not the generators of (magnetic) translations. These are instead given by the so-called guiding center momenta
K̂ := π̂ −
1
e ∧ r̂,
`2 3
(5.202a)
where
` := √
1
eB
(5.202b)
is the magnetic length.
Exercise 1.1: Show that
π̂ = im [Ĥ, r̂].
(5.203)
Exercise 1.2: Show that the components of the guiding center
momenta (5.202a) and the components of the gauge-invariant momenta (5.201b) satisfy the commutation relations
ij
ij
[π̂i , π̂j ] = −i 2 ,
[K̂i , K̂j ] = +i 2 ,
[K̂i , π̂j ] = 0,
i, j = 1, 2,
`
`
(5.204)
Due to the different sign of their commutators, π̂ and K̂ are sometimes
referred to as the left-handed and right-handed degree of freedom of
the Landau level electrons, respectively.
The corresponding position operators for the guiding center are
X̂ := r̂ + `2 e3 ∧ π̂,
= + `2 e3 ∧ K̂,
(5.205a)
270
5. NON-INTERACTING FERMIONS
and also satisfy the right-handed algebra
[X̂i , X̂j ] = +iij `2 ,
i, j = 1, 2.
(5.205b)
In order to diagonalize the Hamiltonian (5.201a), it is convenient
to introduce the ladder operators
`
↠:= √ (π̂1 + iπ̂2 ) ,
2
`
â := √ (π̂1 − iπ̂2 ) ,
2
(5.206a)
1 b̂ := √
X̂1 + iX̂2 .
2`
(5.206b)
and
1 b̂ := √
X̂1 − iX̂2 ,
2`
†
Exercise 1.3: Show that the ladder operators satisfy the bosonic
algebra
[â, ↠] = 1,
[b̂, b̂† ] = 1,
(5.206c)
with all other commutators vanishing.
Exercise 1.4: After expressing the Hamiltonian (5.201a) in terms
of the operators ↠, â, b̂† , and b̂, show that it has the discrete spectrum
of Landau levels indexed by n = 0, 1, 2, · · · ,
1
eB
ε n = ωc n +
,
ωc :=
,
(5.207)
2
m
and that a basis for the eigenstates of each Landau level n is given by
1
↠n b̂† m |0i,
|n, mi := √
n! m!
â |0i = b̂ |0i = 0,
(5.208)
where m = 0, 1, 2, · · · .
Non-commutative geometry and Hall conductivity. The projector
on the states in the n-th Landau level can be represented as
X
|n, mihn, m|.
(5.209)
P̂n :=
m
Exercise 2.1: Show that the guiding center position X̂ is nothing
but the projection of the position operator r̂ onto any given Landau
level, i.e.,
X̂ = P̂n r̂ P̂n .
(5.210)
In this sense, the position operators projected to any given Landau
level furnish a non-commutative geometry. The commutation relations (5.205b) say that their components X̂1 and X̂2 are canonically
conjugate variables, in the same way as the momentum and position
operators of a free electron are canonically conjugate.
This non-commutative geometry is at the heart of both the IQHE
and the FQHE. For example, it is intimately related to the quantized
5.5. PROBLEMS
271
Hall conductivity σH . The Kubo formula for the contribution of the
n-th Landau level (n = 0, 1, 2, · · · ) to the Hall conductivity is
e2 ~ 1 X X hn, m| π̂1 P̂n0 π̂2 |n, mi − (1 ↔ 2)
(n)
σH := − 2
, (5.211)
im A n0 6=n m
(εn − εn0 )2
P
where A = 2π m `2 is the area of the Hall droplet, and we reinstated
~.
Exercise 2.2: Use Eqs. (5.201b) and (5.203) to show that the Hall
conductivity (5.211) is given by
h
i
E
ie2 X D
(n)
n, m X̂1 , X̂2 n, m
σH = −
A~ m
(5.212)
e2
= ,
h
where the commutation relations (5.205b) were used to obtain the last
line. The role of the non-commutative position-operator algebra is
apparent in the penultimate line. If the components of the position
operator were to commute, as they do for free electrons, the Hall conductivity is bound to vanish. If the lowest ñ Landau levels are filled,
each of them contributes the same Hall conductivity (5.212) and the
total Hall conductivity is
X e2
σH =
h
n≤ñ
(5.213)
e2
= ñ .
h
5.5.3. The Hall conductivity and gauge invariance.
Introduction. Shortly after the discovery of the integer quantum
Hall effect (IQHE), [58] Laughlin produced a beautiful argument, Laughlin flux insertion argument, [59] that explains under what conditions
the Hall conductivity in two-dimensional position space must necessarily take universal (i.e., independent of the shape of the Hall bar,
independent of the precise value of the applied magnetic field on the
Hall bar, robust to changes in the mobility of the Hall bar, etc.) and
rational values in units of e2 /h, where −e < 0 is the electron charge.
We shall adapt his argument to the situation when assumptions L1,
L2, and L3 that shortly follow hold. We denote the many-body Hamiltonian for identical electrons by Ĥ. The necessary (but not sufficient)
conditions for the Hall conductivity at vanishing temperature to take
rational values (not only integer values as in the original argument of
Laughlin) in units of e2 /h are the following.
L1: The total number (charge) operator commutes with Ĥ
and this symmetry is not broken spontaneously by the ground
state. This condition implies that the Hall conductivity of a
272
5. NON-INTERACTING FERMIONS
two-dimensional superconductor need not take rational values
in units of e2 /h, see section 7.9.4.
L2: If two-dimensional Euclidean space is compactified so
as to be topologically equivalent (homeomorphic) to the twosphere S 2 , then the energy spectrum of Ĥ displays a gap between its ground state and all other energy eigenstates. A
Fermi liquid fails to satisfy this condition. A band insulator
meets this requirement, as does an integer number of filled
Landau levels.
L3: Galilean invariance is broken by Ĥ. [See Eq. (7.149) in
footnote 7 of section 7.7.1 for the definition of a Galilean transformation.] This condition implies that the Hall conductivity
of a two-dimensional gas of non-interacting electrons free to
propagate in the two-dimensional Euclidean space perpendicular to a uniform and static magnetic field need not take rational values in units of e2 /h. Galilean symmetry is always
broken in the laboratory, say by the ionic periodic potential
hosting the electrons or by crystalline defects. To appreciate
condition L3, we need Kohn theorem. [60]
Kohn theorem. Exercise 1.1: Consider Ne spinless fermions, each
carrying the electron charge −e < 0 and the mass m, in the presence
of static and uniform magnetic and electric fields B = B ez and E =
E ex , respectively. Assume that any two spinless fermions separated
by the distance r interact through the two-body translation-invariant
potential V (r).
(a) Write down the classical Lagrangian for these Ne spinless fermions.
(b) Derive the classical equations of motion for these Ne spinless
fermions.
(c) Go to the center-of-mass and relative coordinates for these
Ne spinless fermions and show that the electric field decouples
from the equations of motion for the relative coordinates, while
the equations of motion for the center of mass do not depend
on the two-body interaction potential.
(d) Show that the center of mass is drifting with the drift velocity
v = c E ∧ B/B 2 .
(e) Use the drift velocity of the center of mass and the density
ne of electron per unit area to show that the classical electric
current per unit time and per unit length is given by
0
0
−σH
E
j=
=
,
(5.214)
E
+σ
0
0
ne e c B
H
i.e., the Hall conductance is
σH =
ne e c
.
B
(5.215)
5.5. PROBLEMS
273
It is independent of the two-body interaction and depends continuously on ne . This is a manifestation of the classical version
of Kohn theorem. [60]
Exercise 1.2: The decoupling of the center-of-mass coordinate
from the two-body interaction at the classical level survives quantization, i.e., the many-body quantum Hamiltonian Ĥ is the sum of the
Hamiltonians Ĥcm and Ĥrc that depend solely on the center-of-mass
and relative coordinates, respectively. The quantum dynamics obeyed
by the center-of-mass position operator R̂ and the center-of-mass momentum operator P̂ is governed by
i
1 h 2
e
2
Ĥcm =
(5.216)
P̂ + (P̂y + B R̂x ) + e E R̂x
2M x
c
in the Landau gauge A = (0, B Rx , 0)T . The total mass is here denoted
by M .
(a) Solve for the eigenstates and eigenvalues of Ĥcm . Hint: Take
advantage of section 5.3.4.
(b) Define the center-of-mass (drift) velocity operator
V̂µ :=
i
[Ĥ , R̂ ],
~ cm µ
µ = x, y.
(5.217)
Compute the expectation value of the center-of-mass (drift) velocity operator in any eigenstate of Hamiltonian (5.216) to deduce that the classical result (5.215) also holds at the quantum
level. Explain why this conclusion could have been reached
without any calculation. We have established a quantum manifestation of Kohn theorem. [60]
(c) Fill ñ Landau levels and show that Eq. (5.213) follows. What
is the Hall conductivity if a Landau level is partially filled?
(This is why condition L3 is needed.)
(d) In his paper, [60] Kohn was only considering the case of a uniform magnetic field, i.e., E = 0. He showed that for any exact
eigenstate |Eι i that is not the ground state of the many-body
Hamiltonian Ĥ, there exists a pair of exact eigenstates with
the energies |Eι ± ~ ωc i, where ωc is the cyclotron frequency.
Construct this pair of eigenstates out of |Eι i and the center-ofmass momentum operator. What is the lowest excited state
of this kind? Hint: Derive the equations of motion obeyed
by the components of the center-of-mass momentum operator
orthogonal to the uniform applied magnetic field.
Laughlin flux insertion argument. We assume that spinless fermions
are confined to a ring embedded in three-dimensional Euclidean space
spanned by the basis vector eµ with µ = x, y, z ≡ 1, 2, 3. We take the
ring to be coplanar with e1 and e2 . In the context of the IQHE, the
274
5. NON-INTERACTING FERMIONS
many-body interactions between electrons can be neglected, while onebody interactions such as the confining potentials at the edges, impurity potentials, and a uniform magnetic field B = B e3 are present. [61]
Here, we assume the presence of generic one-body and many-body interactions that meet conditions L1, L2, and L3. The quantum dynamics in the ring considered as a closed system is governed by the
Hamiltonian Ĥ(r, R), where r is the inner radius of the ring and R is
the outer radius.
We may model the circles with radius r and R to be the inner
and outer edges (boundaries) of the ring, respectively. The interior
(bulk) of the ring is then the set of rings with radius strictly larger
than r and strictly smaller than R. The limit r → 0 and R → ∞ is
always understood as excluding the origin of R2 . Topologically, a ring
is homeomorphic to the punctured plane R2 \{0}. Any two points from
the punctured plane can be connected by a smooth path. The set of all
closed path of the punctured plane decomposes into a set of equivalence
classes. Two closed path are equivalent if they wind around the origin
the same number of times. The algebraic structure of the Abelian
group Z can be attached to this set of equivalence classes through the
so-called fundamental homotopy group π1 (U (1)) = Z, see footnote 5 in
section F.3.2. The experimental realization of this geometry is called
a Corbino disk.
By assumption L1, charge is a good quantum number in the ring,
i.e., Ĥ(r, R) is Hermitean with a global U (1) symmetry that is not
spontaneously broken.
By assumption L2, all excited eigenstates of Ĥ(r, R) are separated
from the ground state of the ring by an energy gap ∆(r, R) that remains
non-vanishing in the limit r → 0 and R → ∞ with all the points at
R = ∞ identified, i.e.,
lim ∆(r, R) > 0
r→0
R→∞
on the punctured two-sphere.
(5.218)
Exercise 2.1: Argue that this implies that all excited states whose
wave functions have support in the bulk (interior) of the ring are gapped
if open boundary conditions are imposed on a ring of finite width.
Condition L3 is satisfied for any finite value of R. The boundary
condition at the origin implementing the removal of the origin in R2 \
{0} implies that condition L3 is also met in the limit r → 0 and R → ∞,
irrespectively of the boundary conditions at infinity.
Exercise 2.2: Argue that excited states with support on the boundaries of the ring (inner or outer edges) show a gap bounded from below
by a term of order 1/R and that assumption L2 does not prevent this
gap from disappearing in the limit r → 0 and R → ∞ with open
5.5. PROBLEMS
275
boundary conditions for which finite size effects disappear, i.e.,
lim ∆(r, R) = 0
r→0
R→∞
(5.219)
is permissible with open boundary conditions at infinity.
Imagine attaching to the axis of symmetry of the ring an infinitely
long and infinitesimally thin solenoid. A slowly varying magnetic flux
present in the solenoid induces a vector potential tangential to any
circle coplanar to the ring with a time-dependent amplitude.
Exercise 2.3: Show that this time-dependent vector potential does
not generate a magnetic field in the ring, but it does generate a tangential electric field that can transfer electric charge from the inner
to the outer edges of the ring, or vice versa, through the off-diagonal
components of the conductivity tensor, i.e., the Hall conductivity.
Exercise 2.4: Argue that assumption L2 prevents dissipation, i.e.,
the conductivity tensor must be off-diagonal in polar coordinates. Hint:
Invoke adiabatic continuity.
To discuss the possible values that the Hall conductivity can take,
we recall that the Hall conductivity of a Hall bar as shown in Fig. 3 is
the linear response between an external applied electric field and the
charge current it induces in the circuit to which it weakly couples. To
reproduce the Hall setup shown in Fig. 3 for the Corbino geometry, we
imagine connecting the inner and outer edges of the ring to conducting
wires connected to a voltmeter so that the difference in chemical potential between the inner and outer edges is the electrostatic potential VH .
We also imagine cycling adiabatically the magnetic flux in the solenoid
from the value 0 to the value of q times the unit of flux quantum h c/e.
More precisely, we prepare the ring in a ground state of Ĥ(r, R). We
then adiabatically couple the ring to the solenoid and the conducting
wires during the adiabatic pumping of the flux in the solenoid. This
coupling is removed adiabatically after q pumping cycles by which the
magnetic flux q times the unit of flux quantum has been transferred to
the ring. The question we want to address is what is the final state of
the ring after q pumping cycles.
We make the Ansatz
p e2
,
(5.220)
σH =
q h
with p < q mutually coprime integers, for the Hall conductivity.
Exercise 2.5: Explain why a charge equal to p times the electron
charge is transferred from one edge to the other by q pumping cycles.
The energy that has been pumped through the solenoid into the
ring is removed once the charge transferred from one edge to the other
edge is brought back to its original edge through the conducting wires.
In doing so the initial state has been recovered. Hence, the rational
276
5. NON-INTERACTING FERMIONS
Ansatz (5.220) for the Hall conductivity does not contradict conditions
L1, L2, and L3.
Instead of the rational Ansatz (5.220) for the Hall conductivity, we
make the irrational Ansatz
e2
(5.221)
h
with 0 < ξ an irrational number. We are going to show that this Ansatz
contradicts assumption L2. To this end, we need the following theorem
from number theory.
Theorem (Hurwitz): For any irrational number ξ, there are infinitely many pairs of integers p and q such that
ξ − p < √ 1 .
(5.222)
q
5 q2
σH = ξ
Hence, it is always possible to choose a pair p and q of integers such
that
p
q ξ−
× e VH < ∆(r, R)
(5.223)
q
where ∆(r, R) is the gap in the ring. As was the case for the rational
Ansatz (5.220) for the Hall conductivity, cycling adiabatically the magnetic flux in the solenoid from the value 0 to the value of q times the
unit of flux quantum transfers a charge equal to q ξ times the electron
charge from one edge to the other. Of this charge, only the integer
part p times the electron charge can be brought back to its original
edge through the conducting wires, for we assume that only charge in
units of the electron charge can be transported along these wires.
The
p
final state is thus not equal to the initial state since a charge q ξ − q
is left over on the “wrong” edge. The final state must then be an excited state of Ĥ(r, R), the energy of which is nothing but the left-hand
side of the inequality (5.223). However, we have constructed a state
of Ĥ(r, R) with an energy below the energy gap ∆(r, R) that we had
assumed between the ground state and all excited states. This is a contradiction with assumption L2. The irrational Ansatz (5.221) is thus
not permissible.
The conclusion of this thought experiment is that the Hall conductivity for a two-dimensional Hamiltonian satisfying conditions L1,
L2, and L3 must take rational values in units of e2 /h. Which rational value is selected goes beyond Laughlin flux insertion argument,
i.e., the ground state and low-lying excited states of Ĥ(r, R) must be
computed. The constraint (5.220) is nevertheless a powerful one that
severely limits the admissible effective low-energy field theories for the
Hamiltonian lim r→0 Ĥ(r, R) supporting a non-vanishing Hall conducR→∞
tance. The discovery of the fractional quantum Hall effect (FQHE)
5.5. PROBLEMS
277
showed that strong many-body effects can stabilize a phase of matter
with a non-integer but rational value of h σH /e2 . [62]
Exercise 2.6: Convince yourself that the same conclusion would
hold if we replace condition L2 by condition L20 .
L20 : There exists a mobility gap above the many-body ground
state of the bulk many-body eigenstates.
The notion of a mobility gap covers the case when translation symmetry is broken by disorder in such a way that the spectral gap of
condition L2 (that would apply in the ideal limit when the total crystal momentum is a good quantum number) has been filled by impurity
states, but all these impurity states are localized, i.e., insensitive to any
change in the boundary conditions. The role of disorder is essential to
explain the observation of plateaus of the Hall conductivity at rational
values in units of e2 /h. [59, 61]
Hint: Consider the Hilbert space of smooth functions with support
on a circle of radius one obeying periodic boundary conditions. Use the
polar angle −π ≤ φ < π. Verify that the wave function ψpw (φ) := eiφ
and the smooth wave function ψloc (φ) := 1 if |φ| < (∆φ − )/2 π
and ψloc (0) := 0 if (∆φ + )/2 < |φ| < π obey periodic boundary
conditions. Here, the support ∆φ + ≈ ∆φ of ψloc is a non-vanishing
interval of the circle while is the small positive number over which
range ψloc changes in magnitude from 1 to zero. Compute the current
density carried by ψpw and ψloc , respectively, assuming that these wave
functions represent a point particle of mass m moving freely around
the unit circle. Connect this exercise to the value taken by the polar
components of the conductivity tensor in the Corbino geometry if all
states in the bulk are localized.
CHAPTER 6
Jellium model for electrons in a solid
Outline
The jellium model for a three-dimensional Coulomb gas is defined.
The path-integral representation of its grand-canonical partition function is presented. Collective degrees of freedom are introduced through
a Hubbard-Stratonovich transformation. The low-energy and longwavelength limit of the effective theory for the collective degrees of freedom that results from integration over the fermions is derived within
the random-phase approximation (RPA). A diagrammatic interpretation to the RPA approximation is given. The ground-state energy in
the RPA approximation is calculated. The dependence on momenta
and frequencies of the RPA polarization function is studied. A qualitative argument is given for the existence of a particle-hole continuum
and for a branch of sharp excitations called plasmons. The quasistatic and dynamic limits of the polarization function are studied. The
quasi-static limit is characterized by screening, Kohn effect, and Friedel
oscillations. The dynamic limit is characterized by plasmons and Landau damping. The physical content of the RPA approximation for a
repulsive short-range interaction is derived. The physics of zero-sound
is discussed. The feedback effect of phonons on the RPA effective interaction between electrons is sketched.
6.1. Introduction
The so-called jellium model is a very naive model for interacting
electrons hosted in a three-dimensional solid. Electronic interactions
are taken to be of the Coulomb type. The ions making up the solid are
treated in the simplest possible way, namely as an inert positive background of charges that insures overall charge neutrality. In spite of its
simplicity the jellium model is very instructive as it displays interesting
many-body effects. We shall see that screening of the Coulomb interaction takes place, there are Friedel oscillations, and there are collective
excitations called plasmons.
The method that we employ to derive these phenomena is very
general and powerful although it is not “elementary”. The idea is
to introduce a collective degree of freedom for which a low-energy and
long-wavelength-effective theory is derived. The effective theory cannot
279
280
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
be obtained exactly. Instead, the effective theory is derived within the
random-phase approximation (RPA).
6.2. Definition of the Coulomb gas in the Schrödinger
picture
6.2.1. The classical three-dimensional Coulomb gas. The
definition of the classical and three-dimensional jellium model is given
by the Hamiltonian
H :=
N
X
X
p2i
+
e2 Vcb (r i − r j )
2m
i=1
1≤i<j≤N
(6.1a)
that describes N identical electrons of charge e and mass m. The
electronic interaction is mediated by the electrostatic potential Vcb (r)
that satisfies the Poisson equation [~ = 1 and ∆ is Laplace operator in
(d = 3)-dimensional Euclidean space],
1
− ∆Vcb (r) = 4π δ(r) −
.
(6.1b)
V
All N electrons are confined to a box of linear size L and volume
V = L3 . The condition of total (electronic and ionic) charge neutrality,
i.e.,
Z
N
3
0 = d r ρ(r) −
,
(6.2)
V
V
where ρ(r) is the local electronic density and N/V is the local ionic
density, is the only remnant of the underlying ions in the solid. If
periodic boundary conditions are imposed on any solution to Eq. (6.1b),
one verifies that
1 X 4π iq·r
1 X 4π iq·r
e
≡
e
(6.3)
V L
q2
V q q2
3
q,
2π
q∈Z
is a solution to the Poisson equation
− ∆ϕ(r) = 4πδ(r)
(6.4)
and that
1 X 4π iq·r
e
V q6=0 q 2
1 X 4π
=
1 − δq,0 eiq·r
2
V q q
"
#
3
Z
1
d3 q 4π
2π
→ 3
1−
δ(q) eiq·r ,
L
(2π/L)3 q 2
L
Vcb (r) =
as L → ∞,
(6.5)
6.2. DEFINITION OF THE COULOMB GAS IN THE SCHRÖDINGER PICTURE
281
is a solution to Poisson equation (6.1b). Because of translation invariance, it is more advantageous to use the representation
X
1 X 4πe2 iq·(ri −rj )
e
2
V
q
1≤i<j≤N
q6=0
" N
! N
!
#
X
X
1 X 4πe2 1
=
e−iq·rj
e+iq·ri − N
V q6=0 q 2 2
j=1
i=1
X
e2 Vcb (r i − r j ) =
1≤i<j≤N
1 X 2πe2
=
ρ
ρ
−
N
,
+q
−q
V q6=0 q 2
(6.6)
whereby we have introduced the Fourier transform in reciprocal space
of the local electronic density
ρq :=
N
X
e−iq·rj
j=1
Z
3
d re
=
−iq·r
N
X
δ(r − r j )
(6.7)
j=1
V
Z
=:
d3 r e−iq·r ρ(r).
V
Hence,
N
X
1 X 2πe2
p2i
+
ρ+q ρ−q − N .
H=
2
2m V q6=0 q
i=1
(6.8)
6.2.2. The quantum three-dimensional Coulomb gas. The
quantum jellium model in three-dimensional position space is defined
by the Schrödinger equation
i~ ∂t Ψ = ĤN Ψ,
N
X
p̂2j
1 X 2πe2
ĤN =
+
ρ̂+q ρ̂−q − N ,
2
2m V q6=0 q
j=1
ρ̂q :=
N
X
exp −iq · r̂ j ,
j=1
a b
r̂i , p̂j = i~ δij δ ab ,
q=
2π
l,
L
i, j = 1, · · · , N,
(6.9)
l ∈ Z3 ,
a, b = 1, 2, 3.
The N -electrons wave functions Ψ spanning the Hilbert space H(N ) are
antisymmetric under exchange of any two electrons and obey periodic
boundary conditions in the box of volume V = L3 that defines the
solid. Equilibrium properties at inverse temperature β are obtained
282
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
from the canonical partition function
h
i
Zβ,N := TrH(N ) exp −β ĤN .
(6.10)
What makes the task of solving for the eigenvalues of the Hamiltonian (6.9), that defines the jellium model, difficult is the competition
between the kinetic energy and the potential energy. The kinetic energy
favors extended or delocalized states in position space. The potential
energy favors localized states in position space. Accordingly, the kinetic energy is local in momentum space whereas the potential energy
is non-local in momentum space.
The choice to work in the momentum space representation in Eq. (6.9)
is motivated by the strategy to solve the non-interacting problem first
and then to treat the Coulomb interaction by a perturbative expansion. If so, what could be the small expansion parameter that would,
a priori, justify treating the Coulomb interaction perturbatively? We
can try the following estimates. Identify the microscopic length scale
−1/3
N
a∼
,
the classical interparticle separation. (6.11)
V
With this characteristic microscopic length scale in hand, the following
estimates of the characteristic kinetic and Coulomb energies
e2
~2
and
E
∼
,
(6.12)
cb
m a2
a
respectively, can be made. The ratio of the characteristic Coulomb and
kinetic energies defines a dimensionless number rs ,
Ekin ∼
Ecb
m e2
∼ 2 a
Ekin
~
a
=
aB :=
aB
=: rs .
~2
m e2
being the Bohr radius
(6.13)
When rs 1, the kinetic energy is the largest energy scale. When
rs 1, the potential energy is the largest energy scale. We will only
treat the jellium model in the limit rs 1. The two limits rs 1
and rs 1 are not smoothly connected. In the limit rs 1, the
electronic ground state is known to break spontaneously translation
invariance and is called a Wigner crystal. Although it is believed that
Wigner crystals could be realized in some regimes of the quantum Hall
effect, the limit rs 1 seems to be the relevant one for a majority of
materials. Discussion of the transition to and from the Wigner crystal
is beyond the scope of this book. The physics of screening, Friedel
oscillations, and plasmons that emerge from the so-called RPA in the
regime rs 1 will be seen to depend in an non-analytic way on rs .
By this measure, the RPA is a highly sophisticated approximation. In
6.2. DEFINITION OF THE COULOMB GAS IN THE SCHRÖDINGER PICTURE
283
fact, the first calculation of the ground-state energy within the RPA
relied on perturbation theory and was a tour de force in diagrammatics.
Fortunately, it is now days possible to reproduce the RPA in a more
economical, although perhaps less “elementary”, way. The “modern”
method that we will follow here present the advantage of being of more
general use than the pioneering methods of the 50’s (see chapters 5
and 6 of Refs. [11] and [9], respectively, for a historical perspective.)
The price to be paid is that some machinery to reformulate quantum
mechanics as a path integral has to be introduced.
In order to apply the rules for second quantization to the jellium
model, we introduce the single-particle momenta and energies (V = L3 )
k=
2π
n,
L
n ∈ Z,
εk =
k2
.
2m
(6.14a)
We then postulate the equal-time fermionic algebra
{ĉk,σ , ĉ†k0 ,σ0 } = δk,k0 δσ,σ0 ,
{ĉk,σ , ĉk0 ,σ0 } = {ĉ†k,σ , ĉ†k0 ,σ0 } = 0. (6.14b)
If we define the Fourier transforms
1 X † −ik·r+iεk t
ψ̂σ† (r, t) = √
ĉk,σ e
,
V k
1 X
ψ̂σ (r, t) = √
ĉk,σ e+ik·r−iεk t ,
V k
(6.14c)
there follows the equal-time fermionic algebra
n
o
ψ̂σ (r, t), ψ̂σ† 0 (r 0 , t) = δσ,σ0 δ(r − r 0 ),
n
o n
o
†
0
†
0
ψ̂σ (r, t), ψ̂σ0 (r , t) = ψ̂σ (r, t), ψ̂σ0 (r , t) = 0.
The Fock space F on which these operators act is
(
Y † mj F := span
ĉkj ,σ
|0i mj = 0, 1, ĉk,σ |0i = 0,
j
j
(6.14d)
L
k ∈ Z3 ,
2π
(6.14e)
Define the (total number) operator
Q̂ :=
XX
σ=↑,↓
ĉ†k,σ ĉk,σ .
(6.15)
k
The Hilbert space H(N ) defined in Eq. (6.9) is the subspace of F for
which
Q̂ = N
(6.16)
)
σ =↑, ↓ .
284
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
holds. Application of the rules for second quantization to the jellium
model yields the identifications (~ = 1)
X
ρ̂(r, t) →
ψ̂σ† (r, t) ψ̂σ (r, t),
(6.17a)
σ=↑,↓
1 X +iq·r
e
ρ̂q ≡ ρ̂(r),
V q
Z
XX †
ρ̂q → d3 r e−iq·r ρ̂(r) =
ĉk,σ ĉk+q,σ ,
ρ̂(r, 0) →
σ=↑,↓
V
(6.17b)
(6.17c)
k
and
ĤN
X k2 †
1 X 2πe2
in Eq. (6.9) −→
ĉkσ ĉkσ +
ρ̂
ρ̂
−
N
+q
−q
2m
V q6=0 q 2
k,σ
X k2 †
1 X 2πe2 X X †
ĉk+q,σ ĉ†k0 −q,σ0 ĉk0 ,σ0 ĉk,σ .
ĉkσ ĉkσ +
=
2
2m
V
q
0 σ,σ 0
q6=0
k,σ
k,k
(6.17d)
Observe that the total particle number operator Q̂ can only be replaced
by the C-number N in the subspace H(N ) of F.
6.3. Path-integral representation of the Coulomb gas
The exact number of electrons is not accessible experimentally in a
macroscopic piece of metal. Hence, we can choose the grand-canonical
ensemble instead of the canonical ensemble to describe the jellium
model. In the grand-canonical ensemble, the uniform density of electrons is allowed to fluctuate around its average,
ρ0 ≡ N/V,
(6.18)
held fixed by the choice of the chemical potential µ as the temperature
β is varied or as the thermodynamic limit N, V → ∞ is taken. In
mathematical terms, the transition from the canonical to the grandcanonical ensemble is encoded by the identification of the pair of triplets
(Hamiltonian, Hilbert space, and partition function)
ĤN , H(N ) , Zβ,N −→ Ĥµ , F, Zβ,µ ,
(6.19a)
where the Hamiltonian entering the grand-canonical partition function
is
X k2
1 X 2πe2 X X †
ĉk+q,σ ĉ†k0 −q,σ0 ĉk0 ,σ0 ĉk,σ ,
Ĥµ :=
− µ ĉ†k,σ ĉk,σ +
2
2m
V
q
0 σ,σ 0
k,σ
q6=0
k,k
(6.19b)
the Hilbert space over which the trace entering the grand-canonical
partition function is to be performed is the Fock space (6.14e), and the
6.3. PATH-INTEGRAL REPRESENTATION OF THE COULOMB GAS
grand-canonical partition function is
h
i
Zβ,µ := TrF exp −β Ĥµ .
Of course, one must demand that
*
+
1 XX †
1 ∂ ln Zβ,µ
≡ ρ0
ĉk,σ ĉk,σ
:=
V
β
V
∂µ
k σ=↑,↓
285
(6.19c)
(6.19d)
Zβ,µ
is held fixed in order to specify the dependence of µ on the inverse
temperature β.
Observe that the Coulomb interaction is normal ordered in Eq. (6.19a),
i.e., with all creation to the left of annihilation operators. With the definition of ĤN on the first line of Eq. (6.17d) and the definition (6.19b)
of Ĥµ , the expectation value in the empty state |0i of their difference
differ by
D E
N X 2πe2
0 ĤN − Ĥµ 0 = −
.
(6.20)
V q6=0 q 2
This C-number does not enter observables obtained from taking logarithmic derivatives of the partition function, but it has to be accounted
for when evaluating the logarithm of the partition function, say, as is
the case for the ground-state energy.
Instead of calculating the grand-canonical partition function or its
logarithmic derivatives by performing the trace over the fermionic Fock
space F, we choose to represent the grand-canonical partition function
as a path integral over Grassmann coherent states. As is shown in
appendices E.1 and E.2, the creation ĉ†k,σ (t = −iτ ) and annihilation
ĉk,σ (t = −iτ ) operators in the Heisenberg picture are replaced, at nonvanishing temperature, by the imaginary-time dependent Grassmann
∗
fields ψk,σ
(τ ) and ψk,σ (τ ), respectively. Whereas ĉ†k,σ (t = −iτ ) is the
∗
adjoint of ĉk,σ (t = −iτ ), the two Grassmann fields ψk,σ
(τ ) and ψk,σ (τ ),
∗
are independent of each other. The symbol should here not be construed as implying some dependence, as occurs for complex conjugation
say. With the help of the Grassmann integration rules defined in appendices E.1 and E.2, the partition function in the grand-canonical
ensemble is given by
Z
Zβ,µ = D[ψ ∗ ] D[ψ] exp −Sβ,µ .
(6.21a)
Here, the Boltzmann weight is the exponential of the Euclidean action
"
#
Zβ
2
X
X 2πe2
∂
k
1
∗
Sβ,µ = dτ
ψk,σ
(τ )
+
− µ ψk,σ (τ ) +
ρ+q (τ ) ρ−q (τ ) ,
2
∂τ
2m
V
q
k,σ
q6=0
0
(6.21b)
286
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
where the notation
ρ+q (τ ) :=
X
∗
ψk,σ
(τ ) ψk+q,σ (τ )
(6.21c)
k,σ
is used for the counterpart to the Fourier component (6.17c) of the
electronic density operator, and the Grassmann integration variables
obey antiperiodic boundary conditions in imaginary time,
∗
∗
(τ ),
(τ + β) = −ψk,σ
ψk,σ
ψk,σ (τ + β) = −ψk,σ (τ ).
(6.21d)
Of course, one must demand that
1 1 ∂ ln Zβ,µ
ρq=0 (τ ) Z :=
≡ ρ0
β,µ
V
βV
∂µ
(6.21e)
is held fixed in order to specify the dependence of µ on the inverse
temperature β. Finally, the Grassmann measure D[ψ ∗ ] D[ψ] for the
two independent Grassmann fields
1 X ∗
∗
ψk,σ
(τ ) = √
ψ
e+i ωn τ
(6.21f)
β n∈Z k,ωn ,σ
and
1 X
ψk,σ (τ ) = √
ψ
e−i ωn τ ,
β n∈Z k,ωn ,σ
(6.21g)
where
π
(2n + 1),
n ∈ Z,
(6.21h)
β
are the discrete (fermionic)
frequencies, is to be understood
Q Matsubara
∗
as the product measure k,n,σ dψk,ωn ,σ dψk,ωn ,σ .
Observe that the Fourier series
1 X 1 X ∗
√
ψσ∗ (r, τ ) = √
ψk,ωn ,σ e−i(k·r−ωn τ )
(6.22a)
β n∈Z V k
ωn =
and
1 X 1 X
√
ψσ (r, τ ) = √
ψ
e+i(k·r−ωn τ ) ,
β n∈Z V k k,ωn ,σ
(6.22b)
can be inverted to yield the position-space and imaginary-time representation of the Euclidean action Sβ,µ ,
"
#
Zβ
Z
X
∆
Sβ,µ = dτ d3 r
ψσ∗ (r, τ ) ∂τ −
− µ ψσ (r, τ )
2m
σ
0
1
+
2
V
Zβ
Z
dτ
0
3
Z
dr
V
d3 r 0 [ρ(r, τ ) − ρ0 ]
e2
[ρ(r 0 , τ ) − ρ0 ] .
|r − r 0 |
V
(6.22c)
6.4. THE RANDOM-PHASE APPROXIMATION
287
6.4. The random-phase approximation
6.4.1. Hubbard-Stratonovich transformation. We begin with
the extension to infinite-product measures of the Gaussian identity
r
Z+∞
1
2π + 1 z2
2
e 2a =
dx e− 2 ax +x z ,
∀z ∈ C,
(6.23)
a
−∞
valid for any a > 0. Second, we introduce the real-valued scalar field
1 XX
ϕq,$l ei(q·r−$l τ ) ,
(6.24a)
ϕ(r, τ ) = √
β V l∈Z q
where
L
β
q ∈ Z3 ,
ϕq=0,$l = 0,
$ ∈ Z.
(6.24b)
2π
2π l
By construction, this scalar field obeys periodic boundary condition
in the imaginary-time direction and in position space. In addition,
it obeys the charge-neutrality condition. This scalar field is the field
conjugate to the local electronic density
ρ(r, τ ) =
X
(ψσ∗ ψσ )(r, τ )
(6.25)
σ
through the Hubbard-Stratonovich identity
−
e
Rβ
0
dτ V1
P
q6=0
2πe2
ρ+q (τ ) ρ−q (τ )
q2
Z
∝
D[ϕ] e
1
− 8π
−
Rβ
dτ
ie
√
2 V
−
Rβ
dτ
ie
√
2 V
Rβ
dτ
P
q6=0
0
q 2 ϕ+q (τ ) ϕ−q (τ )
[ϕ+q (τ ) ρ−q (τ )+ρ+q (τ ) ϕ−q (τ )]
q6=0
×e 0
Rβ
Z
P 2
1
dτ
q ϕ+q (τ ) ϕ−q (τ )
− 8π
q
0
by charge neutrality ϕ+q=0 (τ ) = 0 =
D[ϕ] e
×e
0
P
[ϕ+q (τ ) ρ−q (τ )+ρ+q (τ ) ϕ−q (τ )]
P
q
.
(6.26)
The physical interpretation of ϕ is that of the scalar potential associated to charge fluctuations. Charge neutrality is here implemented by
the condition
Z
1
0 = ϕ+q=0 (τ ) = √
d3 r ϕ(r, τ ) = 0,
∀τ.
(6.27)
V
In this way, the grand-canonical partition function becomes
+1/2 Z
Z
∆
0
Zβ,µ = Det −
× D[ϕ] D[ψ ∗ ] D[ψ] e−Sβ,µ ,
4π
(6.28a)
288
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
where
0
Sβ,µ
=
Zβ
Z
dτ
0
V
"
#
X 1
∆
d3 r
(−ϕ ∆ϕ) +
ψσ∗ ∂τ −
− µ + ie ϕ ψσ (r, τ )
8π
2m
σ
(
=
X
q2
∗
ϕ−q,−$l +
ψk+q,ω
n +$l ,σ
8π
q,l
k,n,σ
)
k2
ie
× −iωn +
− µ δq,0 δ$l ,0 + √
ϕ
ψ
.
2m
β V +q,+$l k,ωn ,σ
(6.28b)
X
ϕ+q,+$l
By analogy to the Gaussian integration over Grassmann
numbers, the
R
product measure D[ϕ] is normalized so that D[ϕ] exp − 21 ϕAϕ =
√ 1
for any symmetric bilinear form (symmetric kernel) A.
Det A
6.4.2. Integration of the electrons. With the extension of the
Gaussian identity for Grassmann integration,
Z
∗
dψ ∗ dψ e−ψ a ψ = a,
a ∈ C,
(6.29)
to infinite-product Grassmann measures, integration over the fermions
in the background of the real-valued scalar field ϕ yields
+1/2 Z
∆
00
Zβ,µ = Det −
× D[ϕ] e−Sβ,µ ,
4π
β
Z
Z
∆
00
3 1
− µ + ie ϕ .
Sβ,µ = dτ d r (−ϕ ∆ϕ) (r, τ ) − 2 Tr ln ∂τ −
8π
2m
0
V
(6.30a)
The functional trace Tr · appears when exponentiating the fermionic
determinant,
2
∆
∆
Det ∂τ −
− µ + ie ϕ
= exp 2 ln Det ∂τ −
− µ + ie ϕ
2m
2m
∆
= exp 2 Tr ln ∂τ −
− µ + ie ϕ
.
2m
(6.30b)
The prefactor of 2 multiplying the trace is due to the spin degeneracy.
6.4.3. Gaussian expansion of the fermionic determinant.
We need to approximate
Tr ln M := Tr ln(M0 + M1 )
= Tr ln M0 1 + M0−1 M1
= Tr ln M0 + Tr ln 1 + M0−1 M1 ,
(6.31a)
6.4. THE RANDOM-PHASE APPROXIMATION
289
where
∆
− µ + ie ϕ,
2m
∆
M0 := ∂τ −
− µ,
2m
M1 := ie ϕ.
M := ∂τ −
(6.31b)
(6.31c)
(6.31d)
Define the unperturbed Green function
G0 := −M0−1 = −
1
∂τ −
∆
2m
−µ
.
(6.32)
The sign is convention. Perform the expansion
Tr ln M = Tr ln(−G−1
0 + M1 )
= Tr ln −G−1
+ Tr ln (1 − G0 M1 )
0
∞
X
1
−1
= Tr ln(−G0 ) −
Tr (G0 M1 )n
n
n=1
(6.33)
to the desired order. The RPA truncates this expansion to second
order.
Use the short-hand notations
k := (k, ωn )
(6.34a)
for the fermionic momenta and fermionic Matsubara frequencies and
k2
(M0 )kk0 :=
−iωn +
− µ δk,k0 ,
(6.34b)
2m
1
1
(G0 )kk0 :=
δk,k0 ≡
δ 0 ≡ G0k δk,k(6.34c)
0,
k2
iωn − ξk k,k
iωn −
+µ
2m
(M1 )kk0
ie
ϕ 0.
:= √
β V k−k
(6.34d)
To first order,
− Tr (G0 M1 ) = −
X
(G0 )kk0 (M1 )k0 k
k,k0
= −
X
G0k δk,k0 (M1 )k0 k
k,k0
= −
X
G0k (M1 )kk .
(6.35)
k
To second order,
1
1X
− Tr (G0 M1 )2 = −
G (M1 )kk0 G0k0 (M1 )k0 k
2
2 k,k0 0k
1X
G (M1 )k(k+q) G0(k+q) (M1 )(k+q)k
(6.36)
.
= −
2 k,q 0k
290
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
We see that charge neutrality insures that the first-order contribution
vanishes. The first non-vanishing contribution
to the expansion of the
√
fermionic determinant in powers of e/ β V is Gaussian and given by
!
X e2 X
1
1
2
G0k G0(k+q) ϕ+q ϕ−q .
− Tr (G0 M1 ) = (−1)2
2
2 q βV
k
(6.37)
In summary, expansion of the fermionic logarithm in
in powers
of ie ϕ/[∂τ − (∆/2m) − µ] yields the first non-vanishing contribution to
second order only, owing to the charge-neutrality condition ϕq=0 (τ ) =
0. Truncation of this expansion to this order yields the RPA to the
jellium model. The RPA partition function in the grand-canonical
ensemble is
RPA
,
(6.38a)
Zβ,µ ≈ Zβ,µ
where
+1/2 2 Z
∆
∆
RPA
RPA
Zβ,µ = Det −
× Det ∂τ −
× D[ϕ] exp −Sβ,µ
−µ
.
4π
2m
(6.38b)
The RPA action in imaginary time is
2
X 1
q
L
β
RPA
2 RPA
ϕ+q
− e Πq
ϕ−q ,
q ∈ Z3 ,
$ = l ∈ Z,
Sβ,µ =
2
4π
2π
2π l
00
Sβ,µ
q=(q,$l )
(6.38c)
where we have introduced the RPA kernel
2 X
ΠRPA
:=
+
G G
,
q
β V k 0k 0(k+q)
and the single-particle Green function
1
1
L
G0k :=
≡
,
k ∈ Z3 ,
k2
iωn − ξk
2π
iωn −
+µ
2m
(6.38d)
ωn =
π
(2n+1), n ∈ Z.
β
(6.38e)
Comments:
• The kernel ΠRPA
that results from the integration over the
q
fermions within the RPA approximation is called the polarization function. It endows the field ϕ with a non-trivial dynamics, i.e., ϕ is no longer instantaneous. This kernel is a property
of the occupied states of a Fermi gas at temperature T = β −1 ,
i.e., of all states within a small window of energy T /εF 1
around the Fermi energy εF = kF2 /2m, kF = (3π 2 N/V )1/3 [recall Eq. (5.38)] at sufficiently low temperatures relative to the
Fermi energy.
00
• It is important to stress that the exact effective action Sβ,µ
in Eq. (6.30) for the order parameter ϕ has been expanded
up to quadratic order in ϕ within the RPA approximation.
6.4. THE RANDOM-PHASE APPROXIMATION
291
However, since this expansion takes place in the argument of
an exponential, this is clearly not second-order perturbation
theory in the electric charge e.
• We are using the terminology of an order parameter for the
field ϕ to stress that our treatment of the Coulomb interaction
is nothing but a mean-field theory with the inclusion of Gaussian fluctuations around the mean-field value of the order pa00
rameter. Indeed, assume that Sβ,µ
can be “Taylor expanded”
around some field configuration ϕ0 ,
∞
00 X
1 δ m Sβ,µ
00
Sβ,µ [ϕ0 + δϕ] =
(δϕ)m ,
(6.39)
m
m! δϕ
ϕ=ϕ
m=0
0
and impose the “selfconsistency” condition that ϕ0 be a lo00
cal extrema of Sβ,µ
. Then, the mean-field value of the order
parameter is the one solving
*
+
+ieδ(r − r 0 )δ(τ − τ 0 ) 1
0=
(−1)(∆ϕ0 )(r, τ ) − 2 r 0 τ 0 r0 τ 0 .
∂ 0 − ∆r0 − µ + ie ϕ (r 0 , τ 0 ) 8π
0
τ
2m
(6.40)
The Ansatz that we made to solve this condition is ϕ0 (r, τ ) = 0
for all r and τ , i.e., we extended the charge-neutrality condition ϕq=0 = 0 to all q’s. Since mean-field theory is here nothing
but a saddle-point approximation, 1 it should be exact in the
β → ∞ limit, provided the saddle-point is an absolute minimum.
00
• One a posteriori justification for the truncation of Sβ,µ
in
Eq. (6.30) up to second order in ϕ consists in verifying that
. If so, the mean-field configuration ϕ = 0
q 2 /(4π) > e2 ΠRPA
q
is, at the very least, a local minimum of the exact effective
00
action Sβ,µ
in Eq. (6.30). This is not to say that ϕ = 0 is
00
a global minimum of Sβ,µ
. Unfortunately, short of an exact
solution of the problem or the identification of an instability,
for example another mean-field Ansatz with lower energy than
the energy of the Fermi gas (the ground state when ϕ = 0), it
is impossible to prove that ϕ = 0 is an absolute minimum of
00
Sβ,µ
.
• Mean-field theory with eventual inclusion of fluctuations within
the RPA approximation or to more than Gaussian order should
not be thought of as a systematic method to solve an interacting many body problem. Rather, it is a practical method
based on physical intuition or prejudice that can only be justified a posteriori by comparison with experiments.
1
The terminology of steepest descent or stationary phase approximation can
also be found in the mathematics literature.
292
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
q
(a) D0q =
(b)
Dq =
(c)
G0q =
(d) ie =
q
q
k+q
q
k
(e)
q
=
q
q
+
⌃q
q
k+q
(f )
(g)
e2 ⇧RPA
=
q
q
RPA
=
⌘
k
q
+
RPA
q
q
RPA
RPA
Figure 1. (a − d) Rules to construct (Feynman) diagrams. (e) Dyson’s equation. (f ) RPA electron-hole
bubble. (e) RPA propagator follows from (e) with the
substitution of the self-energy Σq by the electron-hole
.
bubble −e2 ΠRPA
q
6.5. Diagrammatic interpretation of the random-phase
approximation
We derived in section 6.4 the random-phase approximation (RPA)
to the jellium model. The RPA amounts to expanding the logarithm of
the fermionic determinant up to quadratic order in the electron charge
in the effective action for the collective field ϕ . The collective field
ϕ was introduced through a Hubbard-Stratonovich transformation. It
couples to local electronic charge fluctuations as the scalar potential
does in electrodymanics. Hence, ϕ can be thought of as an effective
scalar potential. The RPA thus trades the fermionic partition function
for the jellium model in the grand-canonical ensemble in favor of the
effective bosonic partition function (6.38).
The polarization function ΠRPA
encodes the effects of the Coulomb
q
interaction within RPA. The limit e → 0 tells us that if we insert in
the non-interacting Fermi gas two static (infinitely heavy) unit point
charges at r and r 0 , respectively, then they interact through the (instantaneous) bare Coulomb potential δ(τ − τ 0 )/|r − r 0 |. The bare Coulomb
potential is renormalized by the response of the Fermi sea to switching
on e.
6.5. DIAGRAMMATIC INTERPRETATION OF THE RANDOM-PHASE APPROXIMATION
293
The RPA has a straightforward interpretation in terms of diagrams.
The Euclidean propagator for the scalar potential in the RPA is, by
q2
definition and up to a sign, the inverse of the kernel 4π
in
− e2 ΠRPA
q
Eq (6.38c),
1
DqRPA := − q2
,
q ≡ (q, $l ).
(6.41)
2 ΠRPA
−
e
q
4π
The sign is convention. In the absence of Coulomb interaction, e = 0,
the Euclidean propagator D0 q is instantaneous as it is independent of
the Matsubara frequency $l ,
4π
D0 q := − 2 .
(6.42)
q
This is nothing but the bare Coulomb potential in Fourier space. We
thus have
1
DqRPA =
−1
(D0 q ) + e2 ΠRPA
q
1
= D0 q
1 + e2 D0 q ΠRPA
q
∞
X
n
= D0 q
(−1)n e2 D0 q ΠRPA
q
= D0 q
n=0
∞
X
2 RPA
n
(ie) Πq D0 q .
(6.43)
n=0
Equation (6.43) is the approximate solution to Dyson’s equation,
1
Dq = D0 q + D0 q Σq Dq ⇐⇒ Dq =
D ,
(6.44a)
1 − D0 q Σq 0 q
where Dq is the exact propagator, i.e., (the sign is convention)
R
00
D[ϕ] ϕ+q ϕ−q exp −Sβ,µ
(6.44b)
Dq := − R
00
D[ϕ]
exp −Sβ,µ
with
00
Sβ,µ
Zβ
=
Z
dτ
0
1
∆
d r (−ϕ∆ϕ)(r, τ ) − 2Tr ln ∂τ −
− µ + ie ϕ ,
8π
2m
3
V
(6.44c)
and the right-hand side defines the so-called self-energy Σq of the collective field ϕ. The RPA replaces the self-energy Σq of ϕ by the RPA
polarization function,
Σq → (ie)2 ΠRPA
.
q
(6.45)
The diagrammatic or perturbative definition of the self-energy of ϕ
goes as follows.
294
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
• Draw a dotted thin line for the unperturbed bosonic propagator D0 q [Fig. 1(a)].
• Draw a dotted thick line for the exact bosonic propagator Dq
[Fig. 1(b)].
• Draw a thin line for the unperturbed fermionic propagator G0 q
[Fig. 1(c)].
• The rule for connecting dotted and non-dotted lines is that
a dotted (thin or thick) line can only be connected to two
distinct non-dotted lines. The connection point is a vertex.
It is to be associated with the factor ie [Fig. 1(d)]. Momenta
and energies on the three lines meeting at a vertex must obey
momentum and energy conservation. The electronic spin index
is a by-stander.
• Each perturbative contribution to the exact propagator Dq is
related to a connected diagram that has been built from dotted
and non-dotted lines according to the preceding rules with no
more and no less than two open ended lines. These two lines
are dotted thin lines that are called external legs. External
legs carry the momentum and energy q = (q, $l ). All other
lines are called internal lines. Momenta and energies on the
internal lines that differ from q = (q, $l ) are said to be virtual
and are to be summed over. One must also sum over the spin
degrees of freedom of the fermionic (non-dotted) internal lines.
This gives an extra degeneracy factor of 2 for all diagrams.
• An irreducible diagram contributing to the exact propagator
is a connected diagram that cannot be divided into two subdiagrams joined solely by a single dotted line.
• An irreducible self-energy diagram is an irreducible diagram
with the two external legs removed (amputated).
• The irreducible self-energy Σq is the sum of all irreducible selfenergy diagrams.
The diagrammatic counterparts to Eqs. (6.44a) and (6.43) are given in
Figs. 1(e) and 1(g), respectively.
In terms of the original electrons, Dq is closely related to the densitydensity correlation function
R
ρ+q ρ−q
Sβ,µ
:= R
D[ψ ∗ ] D[ψ] ρ+q ρ−q exp −Sβ,µ
D[ψ ∗ ] D[ψ]
exp −Sβ,µ
(6.46a)
with
Sβ,µ =
X
k,σ
∗
ψk,σ
k2
1 X 2πe2
−iωn +
− µ ψk,σ +
ρ ρ ,
2m
β V q6=0,$ q 2 +q −q
l
(6.46b)
6.5. DIAGRAMMATIC INTERPRETATION OF THE RANDOM-PHASE APPROXIMATION
295
and
ρ+q :=
X
∗
ψk,σ
ψk+q,σ ,
k = (k, ωn ),
k,σ
L
k ∈ Z3 ,
2π
ωn =
π
(2n+1),
β
(6.46c)
for n ∈ Z. To see this, note that the saddle-point equation
0
δSβ,µ
0=
δϕq
(6.47)
applied to the exact partition function
+1/2 Z
Z
∆
0
Zβ,µ = Det −
× D[ϕ] D[ψ ∗ ] D[ψ] exp −Sβ,µ
,
4π
(6.48a)
where
"
#
Zβ
Z
X ∆
1
0
ψσ∗ ∂τ −
Sβ,µ
(−ϕ ∆ϕ) +
− µ + ie ϕ ψσ (r, τ )
= dτ d3 r
8π
2m
σ
0
V
(
X
X
q2
∗
ψk+q,ω
ϕ−q,−$l +
ϕ+q,+$l
=
n +$l ,σ
8π
k,n,σ
q,l
)
k2
ie
× −iωn +
− µ δq,0 δ$l ,0 + √
ϕ
ψ
,
2m
β V +q,+$l k,ωn ,σ
(6.48b)
yields
δ ln Zβ,µ
δϕq
Z
Z
0
δSβ,µ
1
0
=
D[ϕ] D[ψ ∗ ] D[ψ]
e−Sβ,µ
Zβ,µ
δϕq
2
q
ie
ϕ−q + √
ρ
.
=
4π
β V −q S 0
0= −
(6.49)
β,µ
Equation (6.49) suggests the identification
2
4π
(ie)2 Dq −→
ρ
ρ
.
+q −q S
β,µ
q2
βV
(6.50)
More generally, any m-point correlation function for the scalar potential ϕ corresponds to a (n = 2m)-point correlation function for
electrons. The converse is not true. Not all n-point and fermionic
correlation functions can be written as m-point correlation functions
for the scalar field ϕ. For example, the two-point fermionic correlation
296
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
function (electron propagator)
R
D[ψ ∗ ] D[ψ] ψk ψk∗ exp −Sβ,µ
Gk = R
D[ψ ∗ ] D[ψ]
exp −Sβ,µ
(6.51)
has no simple expression in terms of correlation functions for the scalar
field ϕ. The electronic density-density correlation function can be measured by inelastic X-ray scattering. The electronic two-point function can be measured by angular resolved photoemission scattering
(ARPES). At zero temperature and as a function of the Matsubara
frequency analytically continued to the imaginary axis, poles of the
propagator Dq are interpreted as collective excitations of the underlying jellium model. Similarly, poles of the two-point fermionic Green
function Gk are called quasiparticle excitations.
6.6. Ground-state energy in the random-phase
approximation
The ground-state energy follows from the partition function by taking the zero temperature limit β → ∞
1
(6.52)
lim − ln Zβ,µ =: EGS .
β→∞
β
Remember that we chose to define Ĥµ to be normal ordered from the
outset in section 6.3, i.e., that
D E
N X 2πe2
0 ĤN − Ĥµ 0 = −
.
(6.53)
V q6=0 q 2
We need to account for the shift in the energy due this choice. The
RPA provides an upper bound to the exact ground-state energy. After
evaluating the partition function (6.38), a Gaussian integral in the RPA
approximation, we infer that
RPA EGS

−1/2
q2
e2 RPA
X 2πe2
Y 8π
−
Π
N
q,$
2
RPA l 
+ lim (−1)β −1 ln 
− EGS
= −
e6=0
e=0
β→∞
V
q2
q 2 /(8π)
q6=0
q6=0,$l
 2

q
e2 RPA
1 X X  8π − 2 Πq,$l 
N X 2πe2
= −
+ lim
ln
β→∞ 2β
V
q2
q 2 /(8π)
$
q6=0
q6=0 l


+∞

Z
X  N 2πe2
d$
4πe2 RPA
=
−
+
ln 1 − 2 Πq ($)
.
 V q2

4π
q
q6=0
−∞
(6.54)
In the limit
as :=
3 V
4π N
1/3
aB :=
~2
,
m e2
(6.55)
6.6. GROUND-STATE ENERGY IN THE RANDOM-PHASE APPROXIMATION
297
Im z
k
i !n
Re z
@U⇠k
@U⇠k+q
i$l
Figure 2. Poles of Euclidean polarization function on
and off imaginary z-axis in the representation of Eq.
(6.60) arising from an arbitrarily chosen k contribution.
Hole- (particle-) like poles are off the imaginary axis and
denoted by an empty (filled) circle. Poles on the imaginary axis at the Matsubara frequencies ωn are denoted
by smaller filled circles. Closed integration paths Γk ,
∂Uξk+q −i$l and ∂Uξk are also drawn.
it can be shown that (see Ref. [63])
RPA
EGS
=
N
2.21
0.916
+ 0.062 ln(as /aB ) − 0.096 + O (as /aB ) ln(as /aB ) Ry,
−
(as /aB )2 (as /aB )
(6.56)
where
Ry :=
~2
e2
=
.
2ma2B
2aB
(6.57)
The first term is called the Hartree term. The first two terms are the
Hartree-Fock terms (see chapter 17 of Ref. [64]). In conventional metals
as /aB range from 2 to 6 which indicates that electronic interactions need
to be accounted for to calculate the energy of a metal with any hope
of precision. The RPA gives the next two leading corrections in an
expansion in powers of as /aB . [63] Evidently, it is doubtful that such
an expansion is of relevance to metals in a computational sense. The
RPA is, however, instructive conceptually and was, historically, the first
attempt to calculate systematically the effects of electron interactions
in a metal. Our next task is to identify the excitation spectrum above
the RPA ground state.
298
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
6.7. Lindhard response function
It is time to evaluate the polarization function
1 X1X
1
,
:=
2
ΠRPA
q,$l
V k β ω (iωn − ξk ) iωn + i$l − ξk+q
q 6= 0.
n
(6.58)
The first step consists in performing the summation over fermionic
Matsubara frequencies ωn = π(2n + 1)/β, n ∈ Z for any given k =
2πl/L, l ∈ Z3 . As an intermediary step, observe that the Fermi-Dirac
distribution function
1
f˜FD (z) := βz
,
z ∈ C,
(6.59a)
e +1
has equidistant first-order poles at
zn = iωn ,
n ∈ Z,
(6.59b)
with residues
1
=− .
(6.59c)
β
iωn
For any given k, let Γk be the path running antiparallel to the imaginary axis infinitesimally close to its left and parallel to the imaginary
axis infinitesimally close to its right, i.e., it goes around the imaginary
axis in a counterclockwise fashion. By the residue theorem,
Z
1 X
dz
f˜FD (z)
RPA
.
Πq,$l = −2
(6.60)
V k
2πi (z − ξk ) z + i$l − ξk+q
˜
Res fFD (z)
Γk
Since q 6= 0, the integrand with k fixed has, asides from first-order
poles along the imaginary axis, two isolated and first-order poles at
zk := ξk ,
zk+q,$l := ξk+q − i$l ,
(6.61)
with residues
1
f˜FD (ξk )
2πi ξk − ξk+q + i$l
(6.62)
f˜FD (ξk+q )
1 f˜FD (ξk+q − i$l )
1
=−
,
2πi ξk+q − ξk − i$l
2πi ξk − ξk+q + i$l
(6.63)
and
respectively. These two first-order poles merge into a second-order
pole when $l = 0 and q → 0. By Cauchy theorem, the contour of
integration can be deformed into two small circles ∂Uzk and ∂Uzk+q,$
l
encircling zk and zk+q,$l , respectively, in a clockwise fashion (see Fig.
2),
Γk → ∂Uzk ∪ ∂Uzk+q,$ .
(6.64)
l
6.7. LINDHARD RESPONSE FUNCTION
299
A second application of the residue theorem gives [be aware of the extra
(−1)]
2
ΠRPA
q,$l = (−1) 2
1 X f˜FD (ξk ) − f˜FD (ξk+q )
V k
ξk − ξk+q + i$l
˜
˜
1 X fFD (ξk− q2 ) − fFD (ξk+ q2 )
= +2
.
V k
ξk− q − ξk+ q + i$l
2
(6.65)
2
(Strictly speaking, the change of variable needed to reach the second
line is only legal in the thermodynamic limit V → ∞.) It is customary
to define the (Euclidean) dielectric constant εq to be the proportionality
constant between the bare, Eq. (6.42), and renormalized propagators
in Dyson’s equation (6.44a),
D0 q = 1 − D0 q Σq Dq =: εq Dq .
(6.66)
[Compare with Eq. (E.150).] The RPA for the (Euclidean) dielectric
constant is obtained with the substitution (6.45),
εRPA
q,$l = 1 −
4πe2 RPA
Πq,$l
q2
˜
˜
4πe2 1 X fFD (ξk− q2 ) − fFD (ξk+ q2 )
= 1−2 2
.
q V k
ξk− q − ξk+ q + i$l
2
(6.67)
2
Equation (6.67) is known as the Lindhard dielectric constant. Equation
(6.67) was first derived in the static limit $l = 0.[65] The static limit
$l → 0 of the (Euclidean) polarization function is called the Lindhard
function.
A useful property of the Euclidean dielectric function at zero temperature is that an excitation in the jellium model with momentum q
and real-time frequency $
e q shows up as a zero of the analytic continuation of the Euclidean dielectric function to the negative imaginary
axis 2
lim + εq,$ = 0.
(6.69)
$→−i$
e q +0
Indeed, a pole of Dq at some q 6= 0 implies a zero of εq at some q 6= 0
in Eq. (6.66), as the left-hand side of Eq. (6.66) is a non-vanishing and
finite number for any q 6= 0. Alternatively, the physical interpretation
of Eqs. (6.69) and (6.66) is that a harmonic perturbation with arbitrarily small amplitude induces a non-vanishing response of the Fermi sea
2 Remember that real time t is related to imaginary time τ by τ = it. A
Matsubara frequency $l that enters as $l τ in the imaginary-time Fourier expansion
of fields is related to the real-time frequency $
fl by
$l τ = (+i$l )(−iτ ) ≡ $
fl t,
$
fl := +i$l ,
t := −iτ.
(6.68)
300
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
2
4⇡e2 1 X f˜FD (⇠k ) f˜FD (⇠k+q )
⌘
⇣
q2 V
e
⇠k+q ⇠k i0+
k $
q fixed with |q| > 2kF
=$
e q max
=$
e q min
k
k
q
1
q
$
ePq
0
$
e
/ 1/L
Figure 3. Qualitative plot of (4πe2 /q 2 )ΠRPA
+ as a
q,−i$+0
e
function of the real-time frequency $
e at fixed momentum
q, |q| > 2kF . The polarization function decays like 1/$
e
when |$|
e |$
e qmin |, |$
e qmax |. The number of intercepts
2
2
RPA
between (4πe /q )Πq,−i$+0
+ and the constant line at 1
e
for $
e qmin ≤ $
e ≤$
e qmax scales like the inverse of the level
spacing 1/L, i.e., like L = V 1/3 . There can be one more
intercept between (4πe2 /q 2 )ΠRPA
+ and the constant
q,−i$+0
e
line at 1 for $
e qmax < $.
e This intercept takes place at the
plasma frequency $
e Pq .
in the form of a non-vanishing renormalization of the Coulomb potential. 3 The jellium model supports free modes of oscillations with the
dispersion $
e q since these oscillations need not be forced by an external
probe to the electronic system. The excitation spectrum within the
RPA is obtained from solving
0=
lim
$→−i$
e q +0+
εRPA
q,$
4πe2 1 X f˜FD (ξk ) − f˜FD (ξk+q )
.
=1 − 2 2
q V k $
e q − ξk+q − ξk − i0+
3
(6.70)
The harmonic perturbation can be imposed, for example, by forcing a charge
fluctuation in the electron gas that varies periodically in space and time with wave
vector q and real-time frequency $
e q , respectively. The infinitesimal frequency 0+
in Eq. (6.69) ensures that the perturbation on the jellium model is switched on
adiabatically slowly.
6.7. LINDHARD RESPONSE FUNCTION
$
e
$
e q max
301
$
e q min
$
eP
2kF
|q|
Figure 4. Qualitative excitation spectrum for the jellium model within the RPA approximation. The dashed
region represents the particle-hole continuum. The line
emanating from (|q| = 0, $P ) is the plasmon branch of
collective excitations.
Figure 3 displays a graphical solution to Eq. (6.70). One distinguishes
two types of excitations. There is a continuum of particle-hole excitations when, for $ ≥ 0 and zero temperature say,
$
e qmin ≤ $
e ≤$
e qmax ,
|q|2 kF |q|
−
,
|q| > 2kF
2m m
|q|2
kF |q|
= inf
+ cos θ
,
0≤θ<2π
2m
m
=
inf
ξk+q − ξk ,
$
e qmin :=
|k|<kF ,|k+q|>kF
|q| > 2kF
|q| > 2kF ,
(6.71)
2
k |q|
|q|
+ F
2m m
|q|2
kF |q|
= sup
+ cos θ
2m
m
0≤θ<2π
=
sup
ξk+q − ξk .
$
e qmax :=
|k|<kF ,|k+q|>kF
There is another branch of excitations called plasmons that merges
into the continuum for sufficiently large momentum transfer. Figure 4
sketches the excitation spectrum for the jellium model within the RPA.
References have been made to the Fermi sea and Fermi wave vector
kF , see section 5.3. We recall that at zero temperature, the Fermi-Dirac
distribution becomes the Heaviside step function,
(
0, if ξ > 0,
lim f˜FD (ξ) = Θ(−ξ) =
(6.72)
β→∞
1, otherwise,
302
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
and
h
i
lim f˜FD (ξk− q ) − f˜FD (ξk+ q ) 6= 0 ⇐⇒ ξk− q × ξk+ q < 0.
β→∞
2
2
2
2
(6.73)
We also recall that the interpretation of Eq. (6.72) is that all singleparticle states with an energy smaller than the chemical potential
(Fermi energy) are occupied in the Fermi sea at zero temperature.
The interpretation of Eq. (6.73) is that for the difference of the FermiDirac distributions at two single-particle energies to be non-vanishing
at zero temperature, one of the two single-particle states must be above
the Fermi energy while the other single-particle state must be below
the Fermi energy. The difference of the Fermi-Dirac distributions in
Eq. (6.73) selects an electron-hole pair that is represented by the bubble diagram in Fig. 1(e).
At low temperatures, the polarization function (6.65) is thus controlled by the geometrical properties of the Fermi sea, the unperturbed
ground state of the Fermi gas. We need some characteristic scales of
the Fermi sea. The Fermi wave vector kF is defined by filling up all
available single-particle energy levels,
XX
N
= V −1
Θ(−ξk )
V
σ=↑,↓ k
X k2
k2
F
−1
=: 2V
Θ
−
2m
2m
k
1 4π
(k )3
(2π)3 3 F
(kF )3
=
.
3π 2
The Fermi velocity and Fermi energy are
1/3
2/3
kF
N
N
(kF )2
vF :=
∝
,
εF :=
∝
,
m
V
2m
V
= 2×
(6.74)
(6.75)
respectively. The Fermi velocity appears naturally when expanding the
numerator in powers of |q|/kF ,
∂ f˜FD (ξk )
= −δ(ξk )
β→∞
∂ξk
2
k
kF2
= −δ
−
2m 2m
1
= − δ(|k| − kF ).
vF
lim
(6.76)
At temperatures much smaller than the Fermi energy, only those singleparticle electronic states within a distance β −1 of the Fermi surface
contribute to the polarization function. At zero temperature and in
6.7. LINDHARD RESPONSE FUNCTION
303
the infinite-volume limit, the Lindhard function can be calculated explicitly, (we reinstate ~)
mkF
1 1 − x2 1 + x |q|
RPA
lim lim Πq,$l = −
+
ln ,
x=
.
2
2
β→∞ $l →0
~π
2
4x
1−x
2kF
(6.77)
Note the presence of logarithmic singularities when the magnitude of
the momentum transfer |q| is twice the Fermi wave vector. These
logarithmic singularities are responsible for the so-called Friedel or
Ruderman-Kittel-Kasuya-Yosida oscillations. Note also that (we reinstate ~)
mkF
RPA
lim lim lim Πq,$l = −
× 1 ≡ −νF ,
(6.78)
β→∞ q→0 $l →0
~2 π 2
where νF is the single-particle density of states per unit energy and
per unit volume at the Fermi energy [see Eq. (5.33)]. In fact the full
dependence of the polarization function on transfer momentum q and
transfer energy $l can be expressed in terms of elementary functions
(see section 12 of chapter 4 in Ref. [12]). We will restrict ourselves to
the derivation of some limiting cases below.
From now on, both the zero-temperature and infinite-volume limits
are understood,
Z
Z
Z
1 X
1 X1X
d3 k
d3 q
d$
−→
,
−→
. (6.79)
3
3
V k
(2π)
V q β $
(2π)
2π
R3
R3
R
Furthermore, the long-wavelength limit
|q| kF
(6.80)
will also be assumed for some given transfer momentum q. Choose a
spherical coordinate system in k-space with the angle between q and
k the polar angle θ:
k · q = |k||q| cos θ ≡ |k||q|ν,
Z
Z+∞
Z2π Zπ
Z+∞
Z2π Z+1
3
2
2
d k=
d|k||k|
dφ dθ sin θ ≡
d|k||k|
dφ dν.
0
R3
0
0
0
0
−1
(6.81)
Insertion of
ξk+ q − ξk− q =
2
2
k·q
,
m
∂ f˜ (ξ ) k · q
f˜FD (ξk+ q ) − f˜FD (ξk− q ) = FD k
+ O |q|3
2
2
∂ξk
m
1
k·q
+ O |q|3 ,
= − δ(|k| − kF )
vF
m
(6.82)
304
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
into the polarization function (6.65) yields
ΠRPA
q,$
˜
˜
1 X fFD (ξk− q2 ) − fFD (ξk+ q2 )
= +2
V k
ξk− q − ξk+ q + i$
2
Z∞
= +2
d|k|
|k|2
(2π)3
0
m kF
= +2
(2π)2
2
Z2π
Z+1
dφ
0
dν
1
kF
δ(|k| − kF ) |k| |q| ν + O (|q|3 )
i$ −
−1
|k| |q|
m
ν
Z+1
v |q| ν + O (|q|3 /(vF )3 )
dν F
i$ − vF |q| ν
−1
4m kF
$
= −
1−
arctan
2
(2π)
vF |q|
vF |q|
$
"
+O
|q|
vF
2 #
,(6.83)
with the help of Eqs. (1.622) and (2.112) from Ref. [57] to reach the
last equality. Next, two limits of Eq. (6.83) are considered:
• The quasi-static limit,
|$| vF |q|,
|q| m vF .
(6.84)
In this limit, the RPA encodes the physics of screening.
• The dynamic limit,
vF |q| |$|,
|q| m vF .
(6.85)
In this limit, the RPA encodes the physics of plasma oscillations.
6.7.1. Long-wavelength and quasi-static limit at zero temperature. The regime
|$| vF |q|,
|q| m vF ,
(6.86)
suggests using the expansion
π 1
1
v |q|
− + 3 + ··· ,
|z| > 1,
z→ F ,
2 z 3z
$
in Eq. (6.83). To leading order in this expansion,
" 2 #
2
4mk
π
$
|q|
$
F
ΠRPA
1−
+O
,
.
q,$ = −
(2π)2
2 vF |q|
vF
vF |q|
arctan z =
(6.87)
(6.88)
In turn, the RPA propagator in Eq. (6.41) is approximated by
1
RPA
Dq,$
= − q2
− e2 ΠRPA
q,$
4π
" 2 #
2
4π
|q|
$
+O
= −
,
(6.89).
2mk
vF
vF |q|
|q|2 + 8πe2 (2π)F2 1 − π2 v $|q|
F
6.7. LINDHARD RESPONSE FUNCTION
305
6.7.1.1. The physics of screening. Analytical continuation of Eq.
(6.89) onto the negative imaginary axis yields
lim
$→−i$+0
e
RPA
Dq,$
=−
+
4π
2mk
8πe2 (2π)F2
|q|2
i π2 v $e|q|
F
+
1+
" 2 #
2
|q|
$
+O
,
.
vF
vF |q|
(6.90)
The imaginary part of the denominator indicates that the “lifetime” of
the field ϕ is non-vanishing in the quasi-static limit. The bare Coulomb
interaction is thus profoundly modified by the Fermi sea. The Fermi sea
is characterized by a continuum of particle-hole excitations causing a
non-vanishing lifetime of ϕ at non-vanishing frequencies and screening
in the static limit. As we shall see, screening is non-perturbative in
powers of e.
6.7.1.2. Thomas-Fermi approximation. In the static limit, $
e = 0,
the field ϕ acquires an infinite lifetime,
" 2 #
2
4π
$
|q|
RPA
lim Dq,$
=− 2
+O
,
,
(6.91)
$→0
vF
vF |q|
|q| + (λTF )−2
where we have introduced the Thomas-Fermi screening length
−1/2
2 2 m kF
λTF := 8π e
(2π)2
(6.92)
−1/2
2 m kF
= 4e
.
π
The dependence on e2 of the Thomas-Fermi screening length is nonanalytic in the vicinity of e2 = 0. A position-space Fourier transformation of the right-hand side yields the Yukawa potential
|r|
e
−λ
TF
|r|
.
(6.93)
However, this Fourier transform extends the range of validity of Eq.
(6.91) beyond the long-wavelength limit. Fourier transform to position
space of the Lindhard function amounts to the replacement
|r|
−λ
e
TF
|r|
−→ |r|−3 cos(2 kF |r|).
(6.94)
This oscillatory behavior is known as a Friedel oscillation. We will
rederive this result by “elementary” means below.
306
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
I
q
II
|q| < 2kF
I
q
II
|q| = 2kF
I
q
II
|q| > 2kF
Figure 5. Two Fermi spheres are drawn to represent pictorially the Kohn effect at zero temperature. The center of the
Fermi spheres are shifted in reciprocal space by the transfer momentum −q. Contributions to the dielectric constant are only
possible when f˜FD (ξk ) = 1 and f˜FD (ξk−q ) = 0 or f˜FD (ξk ) = 0
and f˜FD (ξk−q ) = 1. The condition f˜FD (ξk ) = 1 defines the interior of the Fermi sphere centered at the origin, the condition
f˜FD (ξk−q ) = 0 defines the outside of the Fermi sphere centered at
+q, combining those two conditions yields region I. The condition
f˜FD (ξk ) = 0 defines the outside of the Fermi sphere centered at the
origin, the condition f˜FD (ξk−q ) = 0 defines the inside of the Fermi
sphere centered at +q, combining those two conditions yields region II. The union of regions I and II is the Fermi surface ξk = 0
to a very good approximation for very small momentum transfer
q. As the momentum transfer increases in magnitude so does the
volume of the union of region I and II. The volume of the union of
region I and II saturates to twice the Fermi volume at and beyond
the value |q| = 2kF .
6.7.1.3. Kohn effect. We would like to revisit the Thomas-Fermi
approximation to the RPA propagator DqRPA and the condition under
which it breaks down. We have seen that in the static limit,
ξk+q − ξk = + (q · ∇k ξk ) + O(q 2 ),
!
˜
∂
f
FD
f˜FD (ξk+q ) − f˜FD (ξk ) =
(q · ∇k ξk ) + O(q 2 ),
∂ξ
"
!
#
2 Z
3
˜
4πe
d
k
∂
f
εRPA
− FD + O(q 2 )
q,$=0 = 1 + 2
q2
(2π)3
∂ξ
(6.95a)
(6.95b)
(6.95c)
(kTF )2
=1 +
+ O(q 0 ),
2
q
where the Thomas-Fermi wave vector kTF is proportional to the density
of states at the Fermi energy νF defined in Eq. (6.78). What happens
for larger |q|’s? We can use the Lindhard function (6.77), (we reinstate
~)
4πe2 mkF
1 4(kF )2 − q 2 2kF + |q| RPA
εq,$=0 = 1 + 2
+
ln , (6.96)
q
~2 π 2
2
8kF |q|
2kF − |q| to infer that the effective screening length increases with the momentum transfer |q|. It is becoming more and more difficult to make electrons screen out potentials on shorter wavelengths. Moreover, when
6.7. LINDHARD RESPONSE FUNCTION
307
|q| = 2kF , the dielectric constant becomes singular. This singularity comes about from the fact that the summand in the polarization
function is proportional to
f˜ (ξ ) − f˜ (ξ ).
(6.97)
FD
k+q
FD
k
Only those single-particle states with momenta k and k + q contribute
to the sum in the polarization function provided either of one is occupied but not both simultaneously. For small values of |q| the pairs of
single-particle states k and k + q contributing to
f˜ (ξ ) − f˜ (ξ )
(6.98)
FD
k+q
FD
k
belong to two regions I and II that are essentially equal to the surface
of the Fermi sea (see Fig. 5). As |q| is increased, regions I and II
increase in size and converge smoothly
to the Fermi sea.
i There thus
Ph˜
˜
fFD (ξk+q ) − fFD (ξk ) upon a small
exists a functional change of
k
variation δq of q,
i
δ Xh˜
˜
|q| < 2kF =⇒
fFD (ξk+q ) − fFD (ξk ) 6= 0.
δq k
(6.99)
However, as soon as q equals in magnitude twice hthe Fermi wave vectori
P ˜
fFD (ξk+q ) − f˜FD (ξk )
and beyond, there is no functional change of
k
anymore upon a small variation δq of q,
i
δ Xh˜
|q| ≥ 2kF =⇒
fFD (ξk+q ) − f˜FD (ξk ) = 0.
δq k
(6.100)
Hence, transfer momenta obeying |q| = 2kF must be singular points.
This argument does not depend on the shape of the Fermi surface.
It also has consequences for the ability of electrons to screen out the
electrostatic potential set up by collective modes propagating through
the solids. For example, a phonon of wave vector K sets up an external
potential due to the coherent motion of ions. The electrons respond by
screening the electric field induced by the phonon. Evidently, screening of the ions by the much more mobile electrons changes the effective
interaction between the ions in a nearly instantaneous way. This last
change should thus be encoded by the electronic dielectric constant
in the static limit. Moreover, any singularity in the electronic dielectric constant should show up in the phonon spectrum thereby opening
the possibility to measure the Fermi wave vector by inspection of the
phonon spectrum. This phenomenon is called the Kohn effect.
6.7.1.4. Friedel or Ruderman-Kittel-Kasuya-Yosida oscillations. The
dependence on position r of the RPA propagator in the static limit is
given by
2
−1
Z
q
RPA
3
+iq·r
2
lim D$ (r) = − d q e
+ e χq
,
(6.101)
$→0
4π
308
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
where χq is, up to a sign, the static limit of the polarization function,
i.e., (~ = 1)
mkF
|q|
1 1 − x2 1 + x χq = 2 ×
,
x=
+
ln . (6.102)
2
2π
2
4x
1−x
2kF
We have explicitly factorized a factor of 2 arising from the two-fold
spin degeneracy. As already noted in Eq. (6.77), the logarithmic singularity of χq when |q| = 2kF shows up as an oscillatory behavior
at long distances. This oscillatory behavior is known as a Friedel oscillation in the context of the jellium model. It is also known as the
Ruderman-Kittel-Kasuya-Yosida (RKKY) oscillation of the static spin
susceptibility induced by a magnetic impurity in a free electron gas.
We are going to sketch an alternative derivation of the Friedel oscillations. In this derivation, the emphasis is on the response of the electron gas to a s-wave static charge impurity. Consider the Schrödinger
equation
2
p
i∂t Ψ =
+ V (|r|) Ψ
(6.103)
2m
for a (spinless) particle subjected to a spherically symmetric potential V (|r|) that decays faster than 1/|r| for large |r|. The boundary
conditions
Ψ ∼ outgoing plane wave for large |r|,
(6.104)
are imposed. Stationary states have an energy spectrum {ε|k|,l } that
depends on the angular momentum quantum number l and on the
magnitude of the momentum k of the outgoing wave. For large |r|, we
do the expansion in terms of the spherical waves
∞
X
A|k|,l (t) ψ|k|,l (r),
(6.105a)
Ψk (r, t) =
Ψ(r = 0, t) finite,
l=0
where the stationary states behave at large |r| as
1
π
ψ|k|,l (r) ∼
sin |k||r| − l + ηl (|k|) Pl (cos θ),
|r|
2
l = 0, 1, · · · .
(6.105b)
Here, θ is the angle between the momentum k and r and Pl is a Legendre polynomial. The phase shifts ηl (|k|), which are functions of |k|,
encode all informations on the impurity potential V (|r|). For V = 0,
ηl (|k|) = 0 and stationary states behave at large |r| as
π 1
(0)
ψ|k|,l (r) ∼
sin |k||r| − l Pl (cos θ),
l = 0, 1, · · · . (6.106)
|r|
2
The energy spectrum is discrete if we impose the hard wall boundary condition
lim ψ|k|,l (r) = 0
(6.107)
|r|→R
6.7. LINDHARD RESPONSE FUNCTION
l given, ⌘l (|k|) = 0
⇡/R
(0)
|kn,l |
|k1 |
309
(0)
|kn+1,l |
|k2 |
|k|
l given, ⌘l (|k|) 6= 0
|kn,l |
|k1 |
|k2 |
|k|
Figure 6. By switching on the spherical impurity potential V , eigenvalues are shifted along the momentum
quantization axis that characterizes the large |r| asymptotic behavior of energy eigenstates. This shift of the
spectrum can cause a net change in the number of eigenvalues in the fixed interval |k1 | ≤ |k| ≤ |k2 |. The shift in(0)
duced by the spherical impurity potential between |kn,l |
(0)
and |kn,l | is ηl (|kn,l |)/R.
where R is the radius of a large sphere centered about the origin, i.e.,
about the impurity, since
1
π
|k| =
nπ + l − ηl (|k|) ,
n ∈ Z,
l = 0, 1, · · · , (6.108)
R
2
must then hold. Notice that in the absence of the impurity, i.e., when
ηl (|k|) = 0 with l = 0, 1, · · · , the quantization condition
π 1
nπ + l ,
n ∈ Z,
l = 0, 1, · · · ,
(6.109)
|k| =
R
2
yields the same number of energy eigenstates below the Fermi energy
εF as if we had chosen to impose periodic boundary conditions in a box
of volume 4πR3 /3 instead. We will denote solutions to Eq. (6.108) by
(0)
|kn,l | and solutions to Eq. (6.109) by |kn,l |.
Consider the momentum range |k1 | ≤ |k| ≤ |k2 | as is depicted in
Fig. 6. In the absence of the impurity at the origin we can enumerate
all eigenfunctions that decay like
1
π (0)
sin |kn,l ||r| − l Pl (cos θ)
(6.110)
|r|
2
by the integers l and n allowed by the hard wall boundary condition
on the very large sphere of radius R and for which
l π
(0)
|k1 | ≤ |kn,l | = n +
≤ |k2 |
(6.111)
2 R
must hold. If we fix l, the spacing in momentum space between neighboring eigenvalues is π/R. Under the terminology of adiabatic switching of the s-wave impurity potential one understands the hypothesis
310
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
that there exists a one-to-one correspondence between the eigenfunctions (6.110) with the quantization condition (6.111) and all eigenfunctions that decay like
1
π
sin |kn,l ||r| − l + ηl (|kn,l |) Pl (cos θ)
(6.112)
|r|
2
with the quantization condition
ηl (|kn,l |) π
l
|k1 | ≤ |kn,l | = n + −
≤ |k2 |,
2
π
R
(6.113)
up to few states with wave vectors in the vicinity of |k1 | and |k2 |. In
the spirit of adiabaticity, the phase shift ηl (|kn,l |) should be thought of
(0)
as a function of |kn,l |. Moreover, it is worthwhile to keep in mind that
the shift
ηl (|kn,l |)
(0)
(6.114)
|kn,l | − |kn,l | = −
R
vanishes in the thermodynamic limit R → ∞.
In the thermodynamic limit R → ∞, the change in the number of
energy eigenvalues with fixed l in the range |k1 | ≤ |k| ≤ |k2 | before
and after adiabatically switching the s-wave impurity potential V (|r|)
is
1
[η (|k |) − ηl (|k1 |)] .
(6.115a)
π l 2
If |k1 | and |k2 | are chosen to be infinitesimally far apart, i.e., |k1 | → |k|
and |k2 | → |k| + d|k|, then the number (6.115a) takes the differential
form
1
dηl
×
× d|k|.
(6.115b)
π d|k|
Let us further assume that:
(1) First,
lim ηl (|k|) = 0.
(6.116)
|k|→0
(2) Second, the Fermi momentum kF or, more generally, the volume of the Fermi sea, is left unchanged by switching on V (|r|).
We can then integrate Eq. (6.115b) to obtain the total number
∞
1X
2×
(2l + 1) ηl (kF )
π l=0
(6.117)
of new electrons required to fill up all single-particle energy levels up
to the Fermi energy after switching on the s−wave impurity potential
V (|r|). (The factor of 2 accounts for the two-fold spin degeneracy. The
factor of (2l + 1) accounts for the spherical geometry of the impurity
potential.) If we further require that the electric charge of a s-wave
6.7. LINDHARD RESPONSE FUNCTION
311
impurity must be neutralized by an excess of electrons within a nonvanishing distance R, then the difference Z of the valency between the
impurity and the metallic host is given by
∞
1X
Z =2×
(2l + 1) ηl (kF ).
π l=0
(6.118)
Equation (6.118) is known as Friedel sum rule.
Associated to the phase shifts ηl there are changes in the local
electronic density. In the thermodynamic limit and at large distances
from the s-wave impurity, the excess charge is given by
δρ(|r|) ∝ lim 2 × e
R→∞
∞
X
∞
X
ZkF
(2l + 1)
l=0
0
kF
1
∝e
(2l + 1) 2
r
l=0
∝e
i
dk h
2
2
|ψk,l;ηl 6=0 (|r|)| − |ψk,l;ηl =0 (|r|)|
π/R
Z
h
π
π i
dk sin2 k r − l + ηl (|k|) − sin2 k r − l
2
2
0
h i
l
∞
(2l
+
1)(−1)
sin
η
(k
)
cos
η
(k
)
−
cos
2k
r
+
η
(k
)
X
F
l F
l F
l F
r3
l=0
(6.119)
in the static limit. 4 The Yukawa decay predicted by the Thomas-Fermi
approximation is replaced by the slower algebraic decay with superimposed periodic oscillations (quantum interferences) with periodicity of
twice the Fermi wave vector.
6.7.2. Long-wavelength and dynamic limit at T = 0. The
regime
vF |q| |$|,
|q| m vF ,
(6.120)
suggests using the expansion
arctan z = z −
z3 z5
+
− ··· ,
3
5
|z| < 1,
z→
vF |q|
,
$
in Eq. (6.83). To leading order in this expansion,
" 2
4 #
2
4mk
v
|q|
|q|
v
|q|
F
F
ΠRPA
+O
, F
.
q,$ = −
3(2π)2
$
vF
$
4
(6.121)
(6.122)
The normalization of a wave function decaying like r−1 sin(k r) in a sphere of
radius R is proportional to R−1/2 .
312
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
In turn, the RPA propagator in Eq. (6.41) is approximated by
RPA
Dq,$
= − q2
4π
1
− e2 ΠRPA
q,$
"
4π
= −
4mk
|q|2 + 4πe2 3(2π)F2
|q|2
h
1+
vF |q|
$
2 + O
"
4π
= −
$P 2
$
i +O
|q|
vF
|q|
vF
2 4 #
vF |q|
,
$
2 4 #
vF |q|
,
,
$
(6.123)
whereby the so-called plasma frequency is
2
4mkF
4 2 N e2
N e2
4
2
3e
(k
)
=
3π
=
4π
.
(v
)
=
3(2π)2 F
3π F m
3π
V m
V m
(6.124)
Observe that the factorization of q 2 in the denominator of Eq. (6.123)
is special to the Coulomb interaction.
6.7.2.1. The physics of plasmons. Analytical continuation of Eq.
(6.123) onto the negative imaginary axis yields
" 4 #
2
4π
|q|
v
|q|
RPA
h
lim + Dq,$
=
, F
. (6.125)
2 i +O
$
$→−i$+0
e
v
$
e
P
2
F
q 1 − $e
($P )2 := 4πe2
After this analytical continuation, we find poles whenever
$
e q = $P ,
∀q.
(6.126)
Of course the independence on the momentum transfer q is an artifact of truncating the gradient expansion to leading order. Including
higher-order contributions in the gradient expansion gives, up to some
numerical constant #, the so-called plasmon branch of excitations
"
#
2
vF
$
e q = $P 1 + #
q + ··· ,
(6.127)
$P
provided the momentum transfer is not too large.
6.7.2.2. Landau damping. Once the dispersion curve of plasmons
enters in the particle-hole continuum, plasmons become unstable to
decay into an electron-hole pair. This phenomenon is signaled by
RPA −1
lim + Dq,$
(6.128)
$→−i$+0
e
acquiring an imaginary part and thus a non-vanishing lifetime), [use
(x − i0+ )−1 = P1/x + iπδ(x)]
Z
i
d3 k
k·q h
RPA −1
˜ (ξ ) .
Im
lim + Dq,$
∝
δ
$
e
−
q
·
∇
f
k FD k
q
$→−i$+0
e
(2π)3
m
(6.129)
6.8. RANDOM-PHASE APPROXIMATION FOR A SHORT-RANGE INTERACTION
313
The factor
k·q
δ $
eq −
m
(6.130)
select electrons whose velocities |k|/m are close to the phase velocity
$
e q /|q| of the plasmon density wave in that k·q/m = $
e q . There is thus
a small range of electron velocities for which the electrons are able to
surf the plasmon wave. Electrons moving initially slightly more slowly
than the plasmon wave will pump energy from the plasmon wave as
they are accelerated up to the wave speed by the wave leading edge.
Conversely, electrons moving initially faster than the plasmon wave will
give up energy to the plasmon wave as they are decelerated up to the
wave speed by the wave trailing edge. Because the velocity distribution
of electrons
h
i
˜
q · ∇k fFD (ξk )
(6.131)
is skewed in favor of low energy electrons, the net effect is to damp the
wave. This damping is called Landau damping.
6.8. Random-phase approximation for a short-range
interaction
So far, we have been dealing exclusively with the two-body repulsive
potential
e2
Vcb (r 1 − r 2 ) = +
.
(6.132)
|r 1 − r 2 |
(The coupling constant e2 has units of energy × length.) What if we
work instead with a short-range repulsive potential, say
Vλ (r 1 − r 2 ) = +λδ(r 1 − r 2 )?
(6.133)
(The coupling constant λ has units of energy × volume.) This type of
modeling of a two-body interaction is made, for example, to describe
the interaction between 3 He atoms in liquid 3 He. Since
1 X +iq·r
δ(r) =
e
(6.134)
V q
in a box of volume V with the imposition of periodic boundary conditions, we have the Fourier transforms
Vcb q =
4πe2
,
q2
(6.135a)
and
Vλ;q = +λ,
(6.135b)
for the Coulomb and contact repulsive interactions, respectively. RPA
fluctuations of the order parameter ϕ around the mean field ϕmf = 0
314
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
is now encoded by the effective action
X 1
1
RPA
RPA
Sλ =
ϕ
− Πq
ϕ−q ,
2 +q λ
(6.136a)
q=(q,$l )
instead of [compare with Eq. (6.38c) and note that the convention for
the (engineering) dimension of ϕ has been changed]
2
X 1
q
RPA
RPA
Scb =
ϕ
ϕ−q .
(6.136b)
− Πq
2 +q 4πe2
q=(q,$l )
The locations of the poles of
−1
DqRPA := − (Vq )−1 − ΠRPA
q
(6.137)
depend dramatically on the short distance behavior of Vq in the dynamic limit |$l | vF |q|, |q| kF . Indeed, the plasma dispersion
(6.127) becomes gapless for our naive modeling of 3 He as
" 4 #
2
−1
v
|q|
|q|
RPA
lim + Dλ;q,$
=
, F
, (6.138)
2 + O
$→−i$+0
e
v
$
e
c|q|
F
1 − $e
where
λ
× ($P )2
4πe2
N λ
.
=
V m
c2 : =
Eq. (6.124)
(6.139)
These excitations are just above the particle-hole continuum and are
called zero-sound. Collective modes whose energies go to zero at large
wavelength are the general rule. A non-vanishing energy mode at large
wavelengths such as the plasmon is the exception as it is associated with
long-range forces. Long-range forces are very special. In the context
of phase transitions, they cause the breakdown of Goldstone theorem,
i.e., of the existence of excitations with arbitrary small energies when
a continuous symmetry is spontaneously broken.
Coming back to zero-sound, one can show that zero-sound is a coherent superposition of particle-hole excitations near the Fermi surface tantamount to some q-resolved periodic oscillation of the local (in
space) Fermi surface (see section 5.4 in Ref. [16]). Zero-sound is thus
completely different from thermodynamic sound in a Fermi gas. Thermodynamic sound is a classical phenomenon that can only be observed
on time scales much larger than the typical time scale τ (smallest between microscopic time scale and inverse temperature) for particles
to interact. Indeed the adjective “thermodynamic” requires thermodynamic equilibrium. In turn, thermodynamic equilibrium can only
6.9. FEEDBACK EFFECT ON AND BY PHONONS
315
be achieved due to interactions, i.e., interactions are needed to relax any initial (non-interacting, say) state into thermodynamic equilibrium. Conventional (i.e., thermodynamic) sound results from a timedependent perturbation whose characteristic time 1/ω is much larger
than τ ,
ω τ 1.
(6.140)
On such time scales, quasiparticle and collective modes have already
decayed at non-vanishing temperatures and thus are unrelated to thermodynamic sound. From a geometrical point of view, thermodynamic
sound can be viewed as an isotropic pulsating local (in position space)
Fermi sphere (see section 5.4 in Ref. [16]). Zero-sound is the opposite extreme to thermodynamic sound. Zero-sound is built out of
quasiparticles. At zero temperature, the coherent superposition of
quasiparticles responsible for zero-sound acquires an infinite lifetime.
Hence, zero-sound can propagate at non-vanishing frequencies. At nonvanishing temperatures, a necessary condition for the observation of
zero-sound is that the frequency ω of the laboratory probe be large
enough for the characteristic observation time 1/ω to be smaller than
the lifetime of quasiparticles,
ω τ 1.
(6.141)
6.9. Feedback effect on and by phonons
We now consider a jellium model for ions. Ions are point charges
of mass M immersed in an (initially) uniform electron gas of density
ρ0 = N/V . The electric charge per ion is denoted Z e. The averaged
number of ions per unit volume is denoted
ρ̄ion ≡ Nion /V.
(6.142a)
The number of ions per unit volume
ρion = ρ̄ion + δρion
(6.142b)
is allowed to weakly fluctuate in space and time through δρion . Charge
neutrality reads
Z Nion = N,
Z ρ̄ion = ρ0 .
(6.143)
In the absence of any electronic motion but allowing the ionic density
to fluctuate in space and time according to
M v̇ ion = (Z e) E,
∇ · E = 4π e (Z ρion − ρ0 ) ,
0 = ρ̇ion + ∇ · J ion ,
J ion := ρion v ion ,
(6.144)
we find, up to linear order in δρion , the plasma oscillation
0 = δ ρ̈ion + Ω2P δρion
(6.145a)
316
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
with the characteristic plasma frequency [compare with Eq. (6.124)]
Nion (Z e)2
.
(6.145b)
V
M
The ionic plasma frequency ΩP is much lower then the electronic one
since
2
2
N
4π Vion (ZMe)
ΩP
Zm
=
1.
(6.146)
=
2
(−e)
N
$P
M
4π
.
Ω2P := 4π
V
m
Ions move much more slowly that electrons. Electrons can thus
adapt to the motion of ions. In particular, any (infinitesimal) local
excess of ionic charge δρion induced by a collective motion of the ions
that solves Eq. (6.144) is screened by the electrons. To account for
this physics, we may set up the following model for the coupled system
of ions and electrons in the presence of an external density ρext . The
dynamical response of the ions with electrons providing screening to
an external charge density e ρext is governed by the classical model
M δr̈ ion = −(Z e) ∇φ,
− ∆φ + k02 φ = 4π (Z e) δρion + 4π e ρext ,
(6.147)
Nion
(Z e) δρion = −
(Z e) ∇ · δr ion .
V
Here, δr ion is the deviation of the position of an ion with regard to
its equilibrium position, the divergence ∇ · δr ion is proportional to
the small deviation δρion in the number of ions per unit volume relative
to the uniform density ρ̄ion of ions at equilibrium, and k0 =
p
4e2 m kF /π = λ−1
TF is the inverse Thomas-Fermi screening length.
We have assumed that the characteristic frequency ΩP that enters the
electronic polarization function is, for all intent and purposes, so small
that we can use the Thomas-Fermi approximation to account for the
screening by the electrons. Fourier transformation of
Nion (Z e)2
∆φ + 4π e ρ̈ext
V
M
with respect to position space and time gives
− ∆φ̈ + k02 φ̈ = 4π
(6.148)
q 2 (−$2 ) φq,$ + k02 (−$2 ) φq,$ = Ω2p (−q 2 ) φq,$ + 4π e (−$2 ) ρext q,$ ,
(6.149)
i.e.,
1 4π e
1
φq,$ =
φ
,
ρext q,$ ≡
2
εq,$ q
εq,$ ext q,$
1
εq,$
=
$q2 =
1
$2
,
1 + (k02 /q 2 ) $2 − $q2
Ω2P
.
1 + (k02 /q 2 )
(6.150)
6.10. PROBLEMS
317
We conclude that
• The response to an external test charge diverges when $2 =
($q )2 . A longitudinal density fluctuation can thus propagate
at this frequency. For small |q|,
$ ≈ c |q|,
(6.151a)
where the sound velocity is given by
c2 = (ΩP /k0 )2
(6.151b)
1 m 2
= Z
vF .
3 M
This approximate relation between the speed of sound c and
the Fermi velocity vF is called the Bohm-Staver relation.
• The effective Coulomb propagator mediating the interaction
between electrons is modified by the slow motion of the ions.
It becomes [recall Eqs. (6.42), (6.44a), and (E.150)]
1
Dq,$ =
D
,
εq,$ 0 q,$
$2
1
(6.152)
,
εq,$
1 + (k02 /q 2 ) $2 − $q2
4π
D0 q,$ = − 2 .
q
This propagator is frequency dependent, i.e., the force between
two electrons is not instantaneous anymore. More importantly,
whenever
$2 < $q2 ,
(6.153)
the force between electrons has effectively changed signed; it
has become attractive. This arises because the passage of an
electron nearby an ion draws the ion to the electron. However,
in view of the difference in the characteristic energy scales
ΩP /$P 1, the ion relaxes to its equilibrium position on
time scales much larger than the time needed for the electron
to be far away. In the mean time, another electron can take
advantage of the gain in potential energy caused by moving
in the wake of the positive charge induced by the displaced
ion. As both ΩP , or, more generally, the Debye energy, are
small compared to the Fermi energy εF , only electrons near
the Fermi surface can take advantage of the gain in potential
energy induced by the ionic motion.
1
≡
6.10. Problems
6.10.1. Static Lindhard function in one-dimensional position space. The retarded density-density correlation function for an
318
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
electron gas is defined by [recall Eq. (E.102c)]
D
E
i
χ(q, t − t0 ) := − Θ(t − t0 ) e2 [ρ̂I (+q, t), ρ̂I (−q, t0 )] .
(6.154)
~
Here, −e < 0 is the charge of the electron and ρ̂I (q, t) is the Fourier
transform of the local electronic density operator evaluated at the (real)
time t and wave vector q in the interaction picture. The infinite volume
limit V → ∞ has been taken and dimensionality of position space is
d = 1, i.e., q ∈ R. The angular brackets refer to averaging in the grand
canonical statistical ensemble and the function Θ is the Heaviside step
function.
The function (6.154) gives a linear response of the electron gas.
It is related to the dielectric response function by Eq. (E.150b). The
static limit ω → 0 of the time Fourier transform of the function (6.154)
reduces to the momentum integral
Z
dk fFD (εk−q/2 ) − fFD (εk+q/2 )
χ(q, ω = 0) ∝ +2
(6.155a)
2π εk−q/2 − εk+q/2 + i0+
in the Random Phase Approximation (RPA). This function is known
in the literature as the Lindhard function. Here, fFD denotes the Fermi
function
1
1
fFD (ε) := β(ε−µ)
,
β :=
.
(6.155b)
e
+1
kB T
Exercise 1.1:
(a) Show that at zero temperature and for a linearized dispersion,
the Fermi function is given by
fFD (εk ) = Θ(k + kF ) − Θ(k − kF ),
(6.156)
where kF is the magnitude of the Fermi wave number.
(b) Show that at T = 0 and for q in the vicinity of ±2kF ,
2 kF + q 1
sgn(q),
χ(q) ≈ −
(6.157)
ln
π ~ vF 2 kF − q where vF = ~ kF /m denotes the Fermi velocity. Conclude that
the static Lindhard function χ(q) diverges at q = ±2 kF .
(c) To evaluate the static Lindhard function at a finite temperature we approximate the Fermi-Dirac function by

1,
for ε < εF − 2 kB T ,






(6.158a)
fFD (ε) = gFD (ε), for |ε − εF | ≤ 2 kB T ,






0,
for ε > εF + 2 kB T ,
6.10. PROBLEMS
319
where
0
gFD (ε) := fFD
(εF )(ε − εF ) + fFD (εF )
(6.158b)
0
with fFD
the derivative of fFD . Use approximation (6.158)
to show that the static Lindhard function at a non-vanishing
but low (compared to the Fermi energy) temperature and at
q = 2 kF is given by
2
2εF − kB T
kB T
χ(2 kF ) ≈ −
ln
+O
(6.159)
π ~ vF
kB T
εF
to leading order in kB T /εF .
(d) How would approximation (6.159) change had one chosen a
different slope for gFD (ε)?
(e) How are the singularities q = ±2 kF affected by a finite temperature?
6.10.2. Luttinger theorem revisited: Adiabatic flux insertion.
Introduction. For a Fermi liquid as defined by the effective Hamiltonian Eq. (F.15), Luttinger theorem holds in the form of Eq. (F.14). [66]
The proof of Luttinger theorem for spinless fermions involves the following ingredients. Let Λ be a finite lattice made of NΛ = L3 /a3 sites,
where a3 is the volume of the elementary unit cell of the lattice. Let Ĥ
be the many-body Hamiltonian acting on the Fock space F of dimension 2NΛ for identical spinless fermions. We impose periodic boundary
conditions and assume that translation invariance holds so that the
total momentum P̂ is conserved. The eigenvalues of P̂ are countable
because of the periodic boundary conditions. The total number operator N̂ is also conserved by assumption. The partition function
Z (NΛ ) (β, µ, λ) := TrF e−β (Ĥ−µN̂ )
(6.160)
in the grand-canonical ensemble is the sum of 2NΛ analytic functions
of the inverse temperature β (the Boltzmann constant kB = 1), the
chemical potential µ, and all intrinsic coupling constants of Ĥ that we
denote collectively by the symbol λ. For any given NΛ , the expectation
value of the total number operator divided by the volume, i.e.,
n(NΛ ) (β, µ, λ) := (L3 β)−1 ∂µ ln Z (NΛ ) (β, µ, λ),
(6.161)
is for the same reason an analytic function of β, µ, and λ. At zero
temperature, assuming that translation symmetry is not spontaneously
broken and that the ground state is non-degenerate, the expectation
value of N̂ is an integer, i.e., an analytic function of µ and λ taking
discrete values. Consequently, for any given NΛ , n(NΛ ) (β = ∞, ·, ·) is
constant as a function of µ and λ, while holding β = ∞. Assume that
n̄ = n(NΛ ) (β = ∞, µ, λ),
(6.162a)
320
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
for some given density
n̄ :=
Nf 1
,
NΛ a3
Nf = 1, 2, · · · , NΛ ,
(6.162b)
of spinless fermions, defines implicitly the real value µ(λ). A mathematically rigorous proof of Luttinger theorem is achieved once the
existence of the limiting function
(6.163)
n̄ = lim n(NΛ ) β = ∞, µ(·), ·
Nf →∞
NΛ →∞
in an open neighborhood of λ = 0 is proved.
The proof of Eq. (6.163) is a formidable task because interactions
distort the shape of the Fermi surface. Consequently, uniform convergence of perturbation theory for the two-point Green function in powers of the two-point Green function for non-interacting fermions breaks
down in the thermodynamic limit very much in the same way as uniform convergence of perturbation theory for the two-point function of
the O(3) NLσM was shown to break down in section 3.5. As noted by
Luttinger, [66] the cure to this problem demands a non-perturbative
definition of the Fermi surface, which he defines by the location in
momentum space at which the dependence on the single-particle momentum p of the occupation number
nβ=∞,µ (·, λ) : R3 → [0, 1]
p 7→ nβ=∞,µ (p, λ)
(6.164)
[i.e., the ground state expectation value of the number operator ĉ† (p) ĉ(p)
for the spinless fermions] is discontinuous. The strategy is here similar to the one used in section 3.6 when defining non-perturbatively
the renormalization point of the O(3) NLσM. If the Fermi surface for
interacting fermions exists, this definition allows to estimate its distortion relative to the non-interacting Fermi surface to any given order
in perturbation theory. The thermodynamic limit can then be taken
order by order in perturbation theory in powers of two-point functions
with poles on the (perturbatively) renormalized Fermi surface, so as to
prove Luttinger theorem as can be found in Ref. [67].
Although the mathematically rigorous proof of Luttinger theorem
is challenging, there is a tautological flavor to it once the existence of
the Fermi surface for interacting fermions is proved (or assumed). This
suggests that assuming adiabatic continuity between the Fermi surface
for non-interacting fermions and the one for interacting fermions should
allow to rationalize Luttinger theorem by elementary means. Indeed,
it is possible to relate Eq. (F.14) to spectral flows under an adiabatic
insertion of magnetic fluxes without invoking perturbation theory if the
Fermi surface exists in the thermodynamic limit, following an argument
developed by Oshikawa in Ref. [68]
6.10. PROBLEMS
(a)
321
(b)
E(t)
0
VFS
(t)
(t)
E1 (t)
t
k=
2⇡
e 0
=~
c L1
L1
Figure 7. (a) A two-torus T 2 is a surface generated
by revolving a circle in three-dimensional space about
an axis coplanar with this circle. Topologically, a twotorus T 2 is homeomorphic to the Cartesian product of
two circles, T 2 ∼ S 1 × S 1 . Assign the radius r to the
revolving circle. Assign the radius R to the circle traced
by the center of the revolving circle. These are the radii
of the circles in the homeomorphism T 2 ∼ S 1 × S 1 . By
taking the radius R to infinity holding the radius r fixed,
one obtains locally a cylinder whose symmetry axis is
the limit of the circle with radius R → ∞. If the circle
with radius R is identified with an infinitesimal solenoid
in which a magnetic field varies in time, there follows a
time-dependent magnetic flux Φ(t) that generates a timedependent electric field E(t) tangent to the surface of
revolution. (b) Spectral flow of a circular Fermi surface
induced by twisting boundary conditions along the e1
direction in Cartesian coordinates.
We shall make the following assumptions:
• Identical spinless fermions are confined to a hypercube of volume
V = L1 × · · · × Ld ⊂ R d .
(6.165a)
The Cartesian basis of Rd will be denoted eµ with µ = 1, · · · , d.
• Their many-body quantum dynamics is governed by the conserved (i.e., Hermitean) Hamiltonian Ĥ, an operator-valued
function of the position operator r̂ i and momentum operator
p̂i obeying the canonical algebra
[r̂iµ , p̂νj ] = δij δ µν i~
(6.165b)
for any pair of fermions labeled by i and j and for any µ, ν =
1, · · · , d.
• The total number N̂ of spinless fermions is conserved,
[Ĥ, N̂ ] = 0.
(6.165c)
322
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
• Periodic boundary conditions are assumed, i.e., for any manybody state |Ψi from the Hilbert space of Ĥ with a given number of spinless fermions,
T̂µ |Ψi = |Ψi,
(6.165d)
where
T̂µ := ei Lµ P̂ ·eµ /~ ,
µ = 1, · · · , d,
(6.165e)
is the translation operator across the volume V along the direction eµ generated by the total momentum operator
X
P̂ :=
p̂i .
(6.165f)
i
The geometry of position space is thus, in the thermodynamic
limit, that of a d-dimensional torus T d as is illustrated in
Fig. 7(a) for the two-torus T 2 embedded in R3 .
• The total momentum operator P̂ commutes with Ĥ,
[Ĥ, P̂ ] = 0.
(6.165g)
• The low-energy theory of Hamiltonian Ĥ is that of a Fermi
liquid. This means that the notion of quasiparticle is well defined for excited states close to the ground state of Ĥ in the
sense that their number operators {n̂p } approximately commute with Ĥ,
[Ĥ, n̂p ] ∼ [ĤFL , n̂p ] = 0,
where ĤFL is defined by [compare with Eq. (F.15)]
X
1 XX
f 0 n̂ n̂ 0 .
εp n̂p + 3
ĤFL :=
2L p p0 p,p p p
p
(6.165h)
(6.165i)
The interpretation of the operator n̂p is that it measures the occupancy of the quasiparticle state labeled by the single-particle
momentum p in a Fermi liquid. Hence, the eigenvalue np of
n̂p is either 0 or 1. This operator should not be confused with
the number operator n̂p for the original spinless fermions. The
latter operator does not commute with ĤFL , whereas Landau
postulates that n̂p does. The many-body eigenstates of ĤFL
take the form | · · · , np , · · · i. Their eigenenergies depend on
the phenomenological single-particle dispersion εp and on the
residual fermion-fermion interaction encoded by the Landau
function fp,p0 . The ground state of ĤFL is the Fermi sea defined to be the state of the form | · · · , np , · · · i with the lowest
energy eigenvalue. It is characterized by a Fermi surface, a
(d − 1)-dimensional surface in momentum space that bounds
6.10. PROBLEMS
323
a volume, the Fermi sea. In the Fermi sea of ĤFL , np takes the
value 1 if p belongs to the Fermi sea and zero otherwise.
The assumption that the fermions are spinless is not necessary. It
allows to simplify notations.
The notion of Fermi liquid is rooted in the notion of an adiabatic
response to switching on a many-body interaction, as explained in appendix F. A Fermi liquid demands the existence of a Fermi surface and
of quasiparticles whose lifetimes diverge upon approaching the Fermi
surface. If we assume that a Fermi surface exists and that quasiparticles are well defined in its neighborhood, the Fermi surface and the
quasiparticles must respond adiabatically to a perturbation. The idea
of Oshikawa is to choose a periodic adiabatic perturbation and track
the spectral flow undergone by the Fermi surface and the quasiparticles
during the period that takes the Fermi liquid back to itself. This periodic and adiabatic perturbation is a one-body perturbation by which
a unit of magnetic flux is threaded through the torus. It can be implemented by twisting the boundary conditions as we now explain.
Magnetic flux as twisted boundary conditions. We momentarily work
in three-dimensional position space, d = 3. We recall that the homogeneous Maxwell equations are
1
∇ · B = 0,
∇ ∧ E + ∂t B = 0,
(6.166)
c
while the inhomogeneous Maxwell equations are
4π
1
j,
(6.167)
∇ · E = 4πρ,
∇ ∧ B − ∂t E =
c
c
Gaussian units are used.
We consider a cylinder Ω ⊂ R3 in position space with the radius
L1 /(2π), height L2 , and symmetry axis parallel to e3 ,
L1
L1
3
Ω := r =
cos ϕ e1 +
sin ϕ e2 + r3 e3 ∈ R 0 ≤ ϕ < 2π, 0 ≤ r3 ≤ L2 .
2π
2π
(6.168)
Here, we are using the cylindrical coordinates
r = ρ cos ϕ e1 + ρ sin ϕ e2 + r3 e3 ,
(6.169)
where
r2
,
(6.170)
r1
and the orthonormal cylindrical basis vectors at r are given by
q
ρ := + r12 + r22 ,
eρ = + cos ϕ e1 + sin ϕ e2 ,
ϕ := arctan
eϕ = − sin ϕ e1 + cos ϕ e2 ,
e3 .
(6.171)
We are going to construct time-dependent electric and magnetic fields
such that the electric field is pointing along the direction eϕ and is constant in magnitude on the surface of the cylinder, whereas the magnetic
324
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
field vanishes everywhere in space except along the symmetry axis of
the cylinder, where it is singular.
Exercise 1.1:
(a) Show that the homogeneous Maxwell equations are satisfied if
the magnetic and electric fields are expressed according to
1
B = ∇ ∧ A,
E = −∇ · A0 − ∂t A,
(6.172)
c
in terms of the vector A and scalar A0 potentials.
(b) Show that
hc
,
h = 2π ~,
(6.173)
Φ0 :=
e
where −e < 0 is the electron charge has the units of a magnetic
flux.
(c) What are the units of the vector potential expressed in terms
of units of length and magnetic flux?
(d) Assume the time-dependent magnetic and electric fields
Φ
Φ
B(r, t) = − 0 φ(t) δ(ρ) e3
E(r, t) = + 0 (∂t φ)(t) eϕ
2π ρ
2π c ρ
(6.174)
where φ : R → R, t 7→ φ(t) is some dimensionless real-valued
function of time. Show that these magnetic and electric fields
follow from choosing
Φ
A0 = 0,
A(r, t) = − 0 φ(t) eϕ .
(6.175)
2π ρ
We now return to the case of d-dimensional position space with the
assumptions (6.165). We modify the many-body Hamiltonian Ĥ as
follows. Motivated by Eq. (6.175), we couple the vector gauge field
Φ0
(~ c/e)
φ e1 = −2π
φ e1 ,
(6.176)
L1
L1
where φ is a dimensionless number, by the minimal coupling, i.e.,
through the substitution
(−e)
p̂i → p̂i −
A(φ),
(6.177)
c
to all electrons (with the label i) carrying the electric charge −e < 0.
The resulting many-body Hamiltonian is denoted Ĥ(φ). It acts on a
Hilbert space with given number of spinless fermions.
Exercise 1.2:
(a) Verify that, as was the case with Eq. (6.165d), any many-body
state |Ψi from the Hilbert space with given number of spinless
fermions of Ĥ(φ) obeys the periodic boundary conditions
A(φ) := −
T̂µ |Ψi = |Ψi
(6.178)
6.10. PROBLEMS
325
with T̂µ defined by Eq. (6.165e) for µ = 1, · · · , d.
(b) Verify that, as was the case with Eq. (6.165g),
[Ĥ(φ), P̂ ] = 0.
(6.179)
(c) How does Ĥ(φ) change under the local gauge transformation
|Ψi =: e
=e
−i ~ec A(φ)·
φ
+i 2π
L
1
P
i
P
i
r̂ i
r̂ i ·e1
|Θ(φ)i
(6.180)
|Θ(φ)i?
(d) What twisted boundary conditions obeys the transformed state
|Θ(φ)i?
The local gauge transformation (6.180) is called a large gauge transformation, for it changes the boundary conditions. We have shown
that we can trade the one-body coupling between the fermions and the
vector potential (6.176) for twisted boundary conditions.
Spectral flows. Define the vector from Rd
φ = φµ eµ
(6.181)
and the vector potential
A(φ) := −
Φ0
(~ c/e)
φµ eµ = −2π
φµ e µ
Lµ
Lµ
(6.182)
with the summation convention implied on the repeated indices µ =
1, · · · , d. Define Ĥ(φ) through the minimal coupling
(−e)
p̂i → p̂i −
A(φ).
(6.183)
c
Any many-body wave function from the Hilbert space with given number of spinless fermions of Ĥ(φ) obeys periodic boundary conditions.
Define the local gauge transformation
Û (φ) := e
−i ~ec A(φ)·
P
i
r̂ i
+i
=e
2π φµ
Lµ
P
i
r̂ i ·eµ
(6.184a)
with the summation convention implied on the repeated indices µ =
1, · · · , d. We can then rewrite Eq. (6.180) as
|Ψi = Û (φ) |Θ(φ)i.
(6.184b)
Exercise 2.1:
(a) Show that the transformation law of the operator Ô in the |Ψi
basis under the unitary transformation (6.184) is
Ô → Û † (φ) Ô Û (φ).
(6.185)
(b) With the help of Eq. (6.165e) and the Baker-Campbell-Hausdorff
formula, show that
Û † (φ) T̂µ Û (φ) = e+i 2π φµ N̂ T̂µ ,
µ = 1, · · · , d,
(6.186)
where N̂ is the total number operator for the spinless fermions.
326
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
(c) Show that
Tµ |Θ(φ)i = e−i 2π φµ N̂ |Θ(φ)i,
µ = 1, · · · , d.
(6.187)
Compare this result to the one obtained in exercise 1.2(d).
(d) Show that
Û † (φ0 ) Ĥ(φ + φ0 ) Û (φ0 ) = Ĥ(φ)
(6.188)
0
for any φ and φ of the form (6.181).
Exercise 2.2: Let |Θ0 (φ)i be the ground state of Ĥ obeying the
twisted boundary condition implied by Eq. (6.186). By assumption,
this ground state is non-degenerate. Hence, it can be chosen to be
an eigenstate of the total momentum operator P̂ with the eigenvalue
P 0 (φ). Assume adiabatic continuity and show the spectral flow
P 0 (φ + φ0 ) − P 0 (φ) = −2π~ Nf
φ0µ
e
Lµ µ
(6.189)
with the summation convention implied on the repeated indices µ =
1, · · · , d and where Nf , the number of spinless fermions in the Hilbert
space of Ĥ, was defined in Eq. (6.162b). Hint: Relate |Θ0 (φ + φ0 )i to
|Θ0 (φ)i with the help of Eq. (6.188). For any position vector r, apply
the translation operator T̂ (r) := e+iP̂ ·r/~ to this relation and make use
of the proper generalization of Eq. (6.186).
Spectral flow of the Fermi sea. The relation (6.189) makes no reference to a Fermi sea. It applies to the non-degenerate ground state of
any Hamiltonian that commutes with the total number operator and
the total momentum operator, and for which adiabatic continuity with
respect to twisting boundary conditions holds. The number of spinless
electrons Nf is fixed by the choice of the filling fraction of the lattice Λ
made of NΛ unit cells in position space.
To derive Luttinger theorem as stated by Eq. (F.14), we are going
to compute the change in the total momentum of the Fermi sea of a
Fermi liquid under the assumption of adiabatic continuity under the
parametric change of the single-particle momenta given by
φµ
p̂i → p̂i − 2π~
e ,
(6.190)
Lµ µ
as follows from Eqs. (6.183) and (6.182). [The summation convention is
implied on the repeated indices µ = 1, · · · , d.] Let P FS (φ) denote the
total momentum of the Fermi sea, with the latter defined to be an eigenstate of all the quasiparticle number operators n̂p , see Eqs. (6.165h)
and (6.165i), with the eigenvalues 1 for NFS single-particle momenta
and 0 for all NΛ − NFS remaining single-particle momenta. Which of
the single-particle momenta are occupied may change as the interaction
is changed or as the adiabatic parameter φ is changed, but not NFS .
Exercise 3.1:
6.10. PROBLEMS
327
(a) Find the transformation on the number operators in Eq. (6.165i)
that brings ĤFL to the form given in Eq. (F.15).
(b) Show that
P FS (φ + φ0 ) − P FS (φ) = −2π~ NFS
φ0µ
e .
Lµ µ
(6.191)
The summation convention is implied on the repeated indices
µ = 1, · · · , d.
(c) After comparing Eq. (6.189) to Eq. (6.191), deduce that
Nf
N
= FS + nµ ,
µ = 1, · · · , d,
(6.192)
Lµ
Lµ
where n1 , · · · , nd are integers.
(d) What is the origin of the integers nµ with µ = 1, · · · , d?
Thermodynamic limit. We assume that the thermodynamic limit is
well defined, i.e., that the limit NΛ , Nf , NFS → ∞, holding the ratios
Nf /NΛ and NFS /NΛ fixed, exists and is unique.
Exercise 4.1:
(a) To simplify notation, we set the unit of length a and hence
the volume of the unit cell ad to unity. If so, the macroscopic
side lengths L1 , · · · , Ld are integers. Take advantage of the
existence of the thermodynamic limit by choosing L1 , · · · , Ld
to be pairwise mutually prime positive integers (two integers
are mutually prime if the only positive integer that evenly
divides both of them is 1). Show that
Nf = NFS + n
d
Y
µ=1
Lµ ⇐⇒ n̄ =
VFS
+n
(2π)d
(6.193a)
with n some integer, where the density n̄ was defined in Eq. (6.162b)
and
VFS
NFS
:=
,
(6.193b)
d
(2π)
L1 × · · · × Ld
where VFS is the volume of the Fermi sea in d-dimensional
momentum space. We have recovered Luttinger theorem, as
stated by Eq. (F.14), for a Fermi liquid.
(b) What is the interpretation of the integer n on the right-hand
side of Eq. (6.193a)?
6.10.3. Fermionic slave particles. In strongly correlated physics,
one is often confronted with quantum Hamiltonians expressed in terms
of local operators that obey an algebra that is different from that
obeyed by bosons or fermions. Wick theorem (or Leibniz theorem of
differentiation within the path integral formalism) does not apply for
such operators, i.e., perturbation theory is very complicated. The simplest physical example of local operators whose algebra differs from the
328
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
ones bosons or fermions obey is that of quantum spin-S operators. To
circumvent this difficulty, we used Holstein-Primakoff bosons in section
2.6.1 to represent the spin operators in quantum Heisenberg magnets.
However, there is no unique choice of auxiliary operators to represent
spin operators.
The choice made to represent the spin operators is of no consequences as long as no approximation is performed. However, different
choices can deliver qualitatively different predictions as soon as approximations are made. The mean-field approximations based on the
Holstein-Primakoff bosons representation of quantum spins become exact in the classical limit S → ∞ for the quantum spin number. It
has been used successfully to describe colinear magnetic long-range order. It is doubtful that an approximation based on Holstein-Primakoff
bosons would be useful to describe a putative magnetic ground state
without magnetic long-range order (a so-called spin liquid) in the extreme quantum limit by which the quantum spin number S = 1/2 and
the lattice has a low coordination number or is geometrically frustrated.
The “slave-particle” method was introduced in condensed matter
physics by Read [69] and Coleman [70] in order to study a local quantum spin-1/2 immersed in a sea of conduction electrons (Kondo problem). Later the method was used to treat the large-U Hubbard model
in Refs. [71] and [72], which is believed to describe the physics of hightemperature superconductors and has become the cornerstone of the socalled Resonating-Valence-Bond (RVB) approach to high-temperature
superconductors.
Both with the Kondo model or with the large-U Hubbard model, it
is possible to find a representation of the Hamiltonian purely in terms
of bosons or fermions, in which case the applicability of Wick theorem
is restored. A price must however be paid as the underlying Hilbert
space for the slave particles has been enlarged by the introduction of
unphysical degrees of freedom, namely gauge degrees of freedom. In
effect, the original problem is traded for a problem in lattice gauge
theory at infinite bare gauge coupling.
Here, we are going to present the fermionic “slave-particle” method
for the problem of two quantum spin-1/2 particles interacting through
the Heisenberg exchange interaction. The following exercises go step by
step through the exact calculation of the partition function using a representation of the spin-1/2 algebra in terms of slave-fermions. First, the
spin problem is mapped onto a fermion problem. Second, a fermionic
path integral representation of the partition function is derived. Third,
this path integral is explicitly computed using a diagrammatic method.
Slave-fermion Representation of Heisenberg Exchange Interaction.
Let Ŝ 1 and Ŝ 2 be two spin-1/2 operators satisfying the commutation
6.10. PROBLEMS
relations (~ = 1)
h
i
Ŝia , Ŝjb = i δij abc Ŝjc ,
a, b, c = x, y, z ≡ 1, 2, 3,
329
i, j = 1, 2.
(6.194a)
The quantum dynamics for these two spins is governed by the Heisenberg exchange Hamiltonian
J
(6.194b)
Ĥ = Ŝ 1 · Ŝ 2 .
2
330
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
Exercise 1.1:
(a) Show that the Hilbert space H on which Ĥ is defined is four
dimensional and decomposes (irreducibly) into the singlet and
triplet sectors.
(b) Show that the partition function
Z := TrH e−β Ĥ
(6.195a)
can be written as
Z = e+β
3J
8
J
+ 3 e−β 8 ,
(6.195b)
where β is the inverse temperature in units for which the Boltzmann constant is unity.
In order to map this spin problem into a fermion problem, we use
the following fermion representation of the spins (~ = 1) 5
1 †
ĉ σ ĉ ,
i = 1, 2.
(6.196)
2 iα αβ iβ
The summation convention over repeated labels is assumed throughout
this section from now on. Here, the ĉ’s are operators labeled by a site
(i) and spin (α) index. They satisfy the anticommuting relations
Si →
{ĉiα , ĉ†jβ } = δij δαβ ,
{ĉiα , ĉjβ } = {ĉ†iα , ĉ†jβ } = 0,
(6.197)
The three Pauli matrices are denoted by the vector σ. Equation (6.196)
must be handled with care, for the Hilbert space of the spins H and
the Fock space F spanned by the c-operators do not share the same
dimension.
Exercise 1.2:
(a) Show that the Hilbert space H and the Fock space F do not
have the same dimension.
(b) Construct explicitly an isomorphism between H and the restricted Fock space
(
)
X
Fphys := |ψi ∈ F ĉ†iα ĉiα |ψi = |ψi ,
i = 1, 2 . (6.198)
α=1,2
(c) Verify that the right-hand side of Eq. (6.196) reproduces the
angular momentum algebra for spin s = 1/2 in the restricted
Fock space Fphys .
As a side note, the restriction to Fphys implies that each site i = 1, 2
in the fermion representation is occupied by a single fermion with either
up or down spin.
Exercise 1.3:
5Repeated
indices are to be summed over.
6.10. PROBLEMS
331
(a) Prove the useful identity
σ ab · σ cd = 2δad δbc − δab δcd ,
(6.199)
and show that, in the fermion representation, the Heisenberg
exchange Hamiltonian (6.194b) becomes the quartic fermion
interaction
J
J
(6.200)
Ĥf = − ĉ†1α ĉ†2β ĉ1β ĉ2α − ĉ†1α ĉ1α ĉ†2β ĉ2β .
4
8
(b) Show that
J † †
ĉ ĉ ĉ ĉ
4 1α 2β 1β 2α
commutes with the local fermionic density
Ĥf0 := −
n̂i := ĉ†iα ĉiα .
(6.201)
(6.202)
Infer from this observation that the partition function (6.195)
for the two interacting spins is proportional to
0
Zfphys := TrFphys e−β Ĥf ,
(6.203)
the proportionality factor being exp +β J8 .
Grassmann-path-integral representation. In order to compute explicitly Zfphys with the use of a Grassmann path integral, we must replace the trace over the physical subspace by the trace over the entire
Fock space. This can be done by integration over two local Lagrange
multipliers ϕi , for each site i = 1, 2.
Exercise 2.1:
(a) Show that
Z
phys
phys
Zf
= Dµϕ TrF e−β Ĥf ,
(6.204a)
where
Ĥfphys
:=
Ĥf0
+i
2
X
ϕi n̂i ,
(6.204b)
i=1
and the integration measure Dµϕ is given by (Gutzwiller projection)
2
Y
β
dϕi e+iβ ϕi .
(6.204c)
Dµϕ :=
2π
i=1
(b) What is the range of integration for the Lagrange multipliers?
Why?
Exercise 2.2: Now that the trace is over the entire Fock space
and that the Hamiltonian is normal ordered, we can construct the
Grassmann path integral following sections E.1 and E.2.
332
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
(a) By use of the completeness relation for the coherent states |ηi
!
P ∗
Z Y
−
ηj ηj
∗
j∈I
dηi dηi e
|ηi hη| = 1,
(6.205)
i∈I
TrF
where the set I runs over the two sites i = 1, 2 and the spin
index α =↑, ↓≡ 1, 2, show that


(α)
∗
Z
M
−ηimα Aim ηjnα
phys

 Y
jn
e−β Ĥf = lim
Dµη∗ η 
e

M →∞
→0


×
Dµη∗ η
M
Y

J
e+ 4
(6.206a)
∗
∗
η1mα
η2mβ
η1(m−1)β η2(m−1)α 
,
m=1
α,β=1,2
where M = β is kept fixed, and the Grassmann valued fields
∗
and ηimα obey antiperiodic boundary conditions in the
ηimα
imaginary-time direction. Here, we have introduced the notation
M
M
M Y
Y
Y Y
Y Y
∗
∗
∗
≡
dηm dηm ≡
dηim dηim ≡
dηimα
dηimα ,
m=1
(α)
Aim
jn
m,n=1
i,j,α=1,2
i=1,2 m=1
= δij δmn − δ(m−1)n 1 − iϕj ,
(α)
A i1 = δij δ1n − (−1)δM n (1 − iϕj ) ,
i=1,2 m=1 α=↑,↓
1 < m ≤ M, 1 ≤ n ≤ M,
1 ≤ n ≤ M.
jn
(6.206b)
phys
The representation (6.206) of the partition function TrF e−β Ĥf
can
be visualized as follows (see Fig. 8). One can think of a ladder whose
rungs are labeled by the index m running from 1 to M . A vertex of the
ladder corresponds to a point (i, m) in position space and imaginary
time with the spatial coordinate taking two possible values i = 1, 2
and the imaginary-time coordinate taking the M values m = 1, · · · , M .
Actually, the ladder is toroidal in the imaginary-time direction due to
the antiperiodic boundary condition obeyed by the Grassmann fields,
as it should be for any partition function. The left (right) frame of the
ladder is labeled by i =D1 (i = 2). For example, with
E this convention,
phys
† the expectation value 0 ĉ1↑ exp(−β Ĥf ) ĉ1↑ 0 corresponds to a
(possibly) broken line that obeys (1, m + M ) ≡ (1, m) for any m =
1, · · · , M . The line or rather the world line for a fermion with spin up
and initially located on site i = 1 is constructed with the following rule.
If the fermion has reached the space-time point (i, m), 1 ≤ m ≤ M ,
then the next point in position space and imaginary time is either
(i, m + 1) or (i + 1, m + 1) provided 2 + 1 ≡ 1 is understood. It is clear
6.10. PROBLEMS
(a)
333
(b)
m=3
m=2
m=1
m=0
i=1 i=2
i=1 i=2
Figure 8. (a) A ladder with M = 3 rungs representing
position space and imaginary time. (b) A possible world
line of a fermion with spin up initially located on site
i = 1.
that only the Heisenberg interaction can cause a zig zag in the world
line of a fermion.
Exercise 2.3: In order to establish a connection with a lattice
gauge theory, we now introduce an additional auxiliary complex-valued
field, Q 2m , for each diagonal link between the sites (i, m) and (i +
1(m−1)
1, m + 1) of the ladder. This is achieved by a Hubbard-Stratonovich
transformation on the quartic contribution of Ĥfphys . Recall that the
identity
Z
dq ∗ dq −q∗ q+√Aw∗ q+√Azq∗
e
=
2πi
Z
dq ∗ dq −(q−√Aw)∗ (q−√Az)+w∗ Az
e
.
2πi
(6.207)
holds for any positive number A and for any pair of complex numbers z
and w with their complex conjugate denoted by z ∗ and w∗ , respectively.
Here, dq ∗ dq/(2πi) is nothing but the usual (Riemann) infinitesimal
integration area in√the complex q plane. If we choose z = w, we can do
the shift q → q + Az of the integration variable q and then perform
the integration over the complex plane. Because this integration is over
a normalized Gaussian, the multiplicative integration constant is unity
and one finds the desired identity
e
z∗ A z
Z
=
dq ∗ dq −q∗ q+√Az∗ q+√Azq∗
e
.
2πi
(6.208)
334
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
We now decree that the identity
Z
J ∗
1
∗
exp
η
η
η
η
=
dQ∗ 2m dQ 2m
4 1mα 2(m−1)α 2mβ 1(m−1)β
2πi
1(m−1)
1(m−1)
!
r
2 r
J
J
∗
∗
∗
,
× exp − Q 2m +
η
η
Q 2m +
η
η
Q 2m
4 1mα 2(m−1)α 1(m−1)
4 2mβ 1(m−1)β 1(m−1)
1(m−1)
(6.209)
holds. Here, Q
2m
1(m−1)
is treated as if it were a complex number (the aux-
iliary Hubbard-Stratonovich field). The justification of Eq. (6.209) will
be that we can reproduce the partition functions (6.195b) and (6.203)
using this identity.
(a) Show that there is an ambiguity in decoupling the quartic
interaction in that the decoupling is not unique. For the choice
we have taken in Eq. (6.209), the auxiliary field Q 2m is
1(m−1)
called the Affleck-Marston order parameter.
(b) Show that the final version of the Grassmann path integral for
the partition function is


2π/β
(α)
∗
Z
Z
Z∞ Z2π
M
−η
Ã
η
Y  Y
imα im jnα

jn
Dµϕ lim
DµQ∗ Q
Dµη∗ η
e
Zfphys =

,
M →∞
→0 0
0
α=↑,↓
0
i,j=1,2
m,n=1
(6.210a)
where
Dµϕ =
2
Y
i=1
dϕi +iβ ϕi
e
(2π/β)
(6.210b)
is the Riemann measure of the Lagrange multiplier,
DµQ∗ Q =
M
Y
m=1
dK
2
2m
1(m−1)
exp −K 2
2m
1(m−1)
1
dφ 2m
2π 1(m−1)
(6.210c)
is the Riemann measure of the Hubbard-Stratonovich field,
Q 2m ≡ K 2m exp −iφ 2m
1(m−1)
1(m−1)
1(m−1)
(6.210d)
≡ K 2m exp +iφ 1m
1(m−1)
∗
≡ Q 1m
2(m−1)
2(m−1)
is the polar decomposition of the Affleck-Marston auxiliary
field Q (there is an amplitude 0 ≤ K < ∞ and a phase 0 ≤
6.10. PROBLEMS
(a)
(b)
(c)
335
(d)
(e)
Figure 9. (a) Example of a diagram which is not saturated. (b) Saturated diagram in the sector of total occupation number 0. (c) Example of a saturated diagram
in the sector of total occupation number 1. (d) Example
of a saturated diagram in the sector of total occupation
number 3. (e) Example of a saturated diagram in the
sector of total occupation number 4.
φ < 2π),
Dµη∗ η =
2 Y
M Y
Y
∗
dηimα
dηimα
(6.210e)
i=1 m=1 α=↑,↓
is the Grassmann measure, and
r
(α)
J ∗
Q im δ(m−1)n ,
4 j(m−1)
r
J ∗
= δij δ1n − (−1) 1 − iϕj δM n − (−1)
Q i1 δM n .
4 jM
(6.210f)
Ãim = δij δmn − 1 − iϕj δ(m−1)n −
jn
(α)
à i1
jn
To shorten the notation, it is understood that Q∗
im
j(m−1)
vanishes
when i = j.
Diagrammatic interpretation of the Grassmann path integral. Exercise 3.1: The integrand of Eq. (6.210) is a polynomial of degree 8M
in the η’s and η ∗ ’s with monomials weighted
ϕ and comQ by the phase
∗
plex number Q. Only the coefficient of imα ηimα ηimα
, a monomial
of degree 8M in the η’s and η ∗ ’s, contributes to the integral over the
Grassmann variables. This coefficient does not depend on the phase of
the Affleck-Marstonorder parameter and is proportional to the factor
exp − iβ(ϕ1 + ϕ2 ) . In the following we use a diagrammatic method
to evaluate this coefficient. Verify that the integrand in Eq. (6.210)
336
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
can be rewritten as the product over the M factors
#
"
r
Y
J ∗
(α)
(α)
(α)
,
Pm =
1 − Iim + (1 − iϕi ) T im +
Q im
T im
4 (i−1)(m−1) (i−1)(m−1)
im
i(m−1)
α=↑,↓
i=1,2
(6.211a)
with the following Grassmann bilinears
∗
∗
Iim = ηimα
ηimα = e−ηm ηm hηm |ĉ†iα ĉiα |ηm i,
(α)
im
(α)
T im
i(m−1)
T
∗
∗
= ηimα
ηi(m−1)α = e−ηm ηm−1 hηm |ĉ†iα ĉiα |ηm−1 i,
∗
(α)
im
(i−1)(m−1)
∗
η(i−1)(m−1)α = e−ηm ηm−1 hηm |ĉ†iα ĉ(i−1)α |ηm−1 i.
= ηimα
(6.211b)
The factor Pm connects the four vertices on the plaquette {(1, m −
1); (2, m − 1); (2, m); (1, m)} of the ladder. The Grassmann bilinears
can be interpreted as follows:
(α)
(i) Iim counts the number of Grassmann degrees of freedom with
im
spin α on site (i, m).
(α)
(ii) T im transfers one Grassmann degree of freedom with spin
i(m−1)
α from site (i, m − 1) to site (i, m), i.e., in the imaginary-time
direction of the mth plaquette.
(α)
transfers one Grassmann degree of freedom with
(iii) T im
(i−1)(m−1)
spin α from site (i − 1, m − 1) to site (i, m), i.e., along the
diagonal of the mth plaquette and thereby rounding the corner
(i, m − 1).
We are now ready to describe diagrammatically the many contributions to the integrand. We keep track of the I (↑(↓)) ’s by assigning
the cross × to the appropriate site. To each T (↑(↓)) , we assign an arrow
↑, or %, or - linking appropriate sites. The spin index of the I’s
and T ’s is fixed by coloring appropriately the crosses and the arrows
(say green for spin
R up andQred for spin down). The only non-vanishing
contributions to Dµη∗ η m Pm are the fully saturated diagrams, i.e.,
those for which any given site of the ladder is such that either (i) two
× of different colors are present, or (ii) one × of a given color is present
together with one arriving and one departing arrow of the other color
or (iii) two arrows with different colors arrive and two arrows with different colors depart (see Fig. 9). The boundary condition forces the set
of arrows to be closed in the imaginary-time direction. We interpret
any closed sequence of arrows as the world line of a fermion of a given
spin.
Exercise 3.2:
6.10. PROBLEMS
337
(b)
+···
(c)
+···
(a)
Figure 10. (a) Example of a saturated diagram in the
sector of total occupation number 2 which is unphysical.
(b) Examples of physical diagrams for indistinguishable
world lines. (c) Examples of physical diagrams for distinguishable world lines.
(a) Convince yourself that fully saturated diagrams with the initial
condition that n (n = 0, 1, . . . , 4) arrows depart from the rung
m = 1 contribute to the sector of the Fock space with total
occupation number n (see Fig. 9).
(b) Show that integration over Dµϕ in Eq. (6.210) cancels the contribution of all diagrams to the physical partition function,
except of some diagrams that correspond to the Fock space
with total occupation number 2.
(c) Show that integration over the Affleck-Marston order parameter Q selects those fully saturated diagrams such that any zig
from site (1, m) to site (2, m + 1) takes place together with a
zag from site (2, m) to site (1, m + 1). Conclude that the physical diagrams (i.e. those diagrams that survive the integration
over ϕi and Q) are the fully saturated diagrams with the initial condition that one and only one arrow departs from each
vertices (1, 1) and (2, 1) and the dynamical constraint that at
a given imaginary time any zig takes place with a zag (see
Fig. 10).
If the world lines of a given physical diagram are of the same color,
then there can be exchange in the sense that the number d of plaquettes covered by two diagonal arrows is odd, in which case the ladder
is covered by a single world line. When the number d of plaquettes
covered by two diagonal arrows is even, the ladder is covered by two
338
6. JELLIUM MODEL FOR ELECTRONS IN A SOLID
world lines of the same color. The fact that any d = 0, · · · , M is allowed reflects the indistinguishability of two fermions of the same spin.
There are Md different ways to cover d plaquettes with two diagonal
arrows of the same color.
On the other hand, when the two word lines are of different colors,
then the number of plaquettes covered with diagonal arrows is always
even.
Exercise 3.3: Convince yourself that the physical diagrams can
be separated into two different classes. The first class consists of all
physical diagrams for which any ladder site is the arrival and departure
of one and only one arrow of a given color. The second class consists of
all physical diagrams with two world lines which can be distinguished
by their color (see Fig. 10).
Exercise 3.4: Before calculating the contribution to the partition
function of the two classes of physical diagrams we need some few
identities that greatly simplify performing the Grassmann integral.
(a) Show that
M
Y
∗
ηimα
ηimα = (−1)M
m=1
M
Y
∗
ηimα ηimα
,
i = 1, 2,
α =↑, ↓ .
m=1
(6.212)
(b) Show that
M
Y
∗
ηimα
ηi(m−1)α =
m=1
M
Y
∗
ηimα ηimα
,
i = 1, 2,
α =↑, ↓ . (6.213)
m=1
(c) Show that
T
(α)
1m
1(m−1)
T
(α)
2m
2(m−1)
(α)
1m
2(m−1)
= −T
T
(α)
2m ,
1(m−1)
α =↑, ↓ .
(6.214)
(d) What is the effect of the boundary conditions in imaginary
time in the physical sector?
Consequently, the only physical diagrams which contribute negatively to the path integral are those with an odd number of plaquettes
supporting two diagonal arrows of the same color. We will account for
this sign by assigning to each diagonal arrow the imaginary factor i.
The full diagrammatic prescription is thus:
• Construct all physical diagrams by covering exactly once all
the ladder sites by either one or two closed word lines of the
same color or two world lines of different colors and add to
each ladder site a cross colored differently from the world line
already there.
• Every crosses from a physical diagram corresponds to the factor −1.
• Every vertical arrow from a physical diagram which links (i, m−
1) to (i, m) corresponds to the factor [1 − iϕi ].
6.10. PROBLEMS
339
• Every diagonal arrow from an physical diagramq
which links site
(i−1, m−1) to (i, m) corresponds to the factor i
J
4
Q∗
.
im
(i−1)(m−1)
Exercise 3.5: Compute the Grassmann path integral (6.210) by
explicitly summing over all the physical diagrams. Compare the result
with Eq. (6.195b) or Eq. (6.203).
CHAPTER 7
Superconductivity in the mean-field and
random-phase approximations
Outline
The notion of a pairing order is introduced. A repulsive interaction
is decoupled through the pairing-order parameter. It is shown that the
coupling of a repulsive interaction decreases upon momentum shell integration, whereas the coupling of an attractive interaction increases.
An effective action for a uniform and static pairing-order parameter is
calculated and analyzed perturbatively as well as non-perturbatively.
The BCS mean-field theory for a uniform and static pairing-order parameter is described. Effective theories for the superconducting order
parameter are derived in the vicinity of T = 0 and T = Tc , respectively.
First, an effective action is derived that describes in the vicinity of
T = 0 the long-wavelength and low-frequency fluctuations of the phase
of the superconducting order parameter. Second, the Gross-Pitaevskii
non-linear Schrödinger equation, from which follows the Meissner effect, is derived at T = 0. Third, the polarization tensor in the pairing
state is calculated to quadratic order in the expansion of the fermionic
determinant and shown to encode the Anderson-Higgs mechanism by
which the photon acquires an effective mass in a superconductor. Finally, the space-dependent Ginzburg-Landau functional is calculated
in the vicinity of Tc .
7.1. Pairing-order parameter
In the context of the jellium model in the canonical ensemble with
the uniform electronic density N/V , we opted to decouple the fourfermion interaction in
Z
Z
1
N
3
3
d r 1 d r 2 Vcb (r 1 − r 2 ) ρ̂(r 1 ) ρ̂(r 2 ) − δ(r 1 − r 2 )
,
2
V
V
V
(7.1)
X
†
ρ̂(r) :=
ĉσ (r)ĉσ (r),
σ=↑,↓
by trading the local electronic density operator ρ̂(r) for the fluctuations
of the order parameter ϕ(r) (interpreted as an effective scalar potential)
341
7.
342SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
about the vanishing mean-field value (charge neutrality).
0=V
−1/2
ϕq=0
1
=
V
Z
d3 r ϕ(r)
(7.2)
V
However, this choice for decoupling the four-fermion interaction is not
unique.
For example, if the interaction potential is spin dependent,
1
2
Z
3
d r1
V
ρ̂σ (r) :=
Z
3
d r2
X
σ1 ,σ2
V
ĉ†σ (r)ĉσ (r),
N
Vintσ1 ,σ2 (r 1 − r 2 ) ρ̂σ1 (r 1 )ρ̂σ2 (r 2 ) − δσ1 ,σ2 δ(r 1 − r 2 )
2V
σ =↑, ↓,
(7.3)
it might be better to introduce an order parameter that is sensitive
to ordering of the spins at low temperatures. Even for an interaction
isotropic in spin space, there are many other possible ways to decouple
the four-fermion interaction.
One could imagine an order parameter that corresponds to a ground
state in which the Fourier transform ρ̂q of the local electronic density
ρ̂(r) acquires a non-vanishing expectation value for some special value
q = Q, i.e., translation invariance could be spontaneously broken if the
interaction favors a charge-density wave with momentum Q over the
Fermi-liquid ground state of section 6.4.
Another possible instability of the Fermi-liquid ground state of section 6.4 is to a superconducting ground state. The superconducting
order parameter is associated with the development of a ground-state
expectation value for the pair of adjoint operators
Φ̂†σ1 σ2 (r 1 , r 2 ) := ĉ†σ1 (r 1 ) ĉ†σ2 (r 2 ),
Φ̂σ1 σ2 (r 1 , r 2 ) := ĉσ2 (r 2 ) ĉσ1 (r 1 ).
(7.4)
The operator Φ̂σ1 σ2 (r 1 , r 2 ) is quadratic in the electronic annihilation
operator. Its adjoint Φ̂†σ1 σ2 (r 1 , r 2 ) creates out of the empty state |0i a
pair of electrons at r 1 and r 2 with the spin quantum numbers σ1 and σ2 ,
respectively. The operator Φ̂σ1 σ2 (r 1 , r 2 ) is thus called a pairing operator
and its (complex-valued) expectation value is called a pairing-order
parameter. Choosing the pairing order parameter is made plausible by
,
7.1. PAIRING-ORDER PARAMETER
343
normal ordering the interaction,
Z
Z
X
1
N
3
3
d r1 d r2
=
Veffσ1 ,σ2 (r 1 − r 2 ) ρ̂σ1 (r 1 )ρ̂σ2 (r 2 ) − δσ1 ,σ2 δ(r 1 − r 2 )
2
2V
σ1 ,σ2
V
V
Z
Z
X
1
3
d r 1 d3 r 2
Veffσ1 ,σ2 (r 1 − r 2 ) Φ̂†σ1 σ2 (r 1 , r 2 )Φ̂σ1 σ2 (r 1 , r 2 ),
2
σ1 ,σ2 =↑,↓
V
V
Z
N
σ =↑, ↓,
d3 r ρ̂σ (r) = ,
ρ̂σ (r) ≡ ĉ†σ (r)ĉσ (r),
2
V
(7.5)
when the effective potential
Veffσ1 ,σ2 (r 1 − r 2 )
(7.6)
is attractive in some channel. Development of an expectation value
for (condensation of) Φ̂†σ1 σ2 (r 1 , r 2 ) Φ̂σ1 σ2 (r 1 , r 2 ) in the attractive channel would then lower the interaction energy. Thus, one decouples the
effective interaction by introducing the order parameters
∆σ1 σ2 (r 1 , r 2 ),
[∆∗σ1 σ2 (r 1 , r 2 )]
(7.7)
Φ̂†σ1 σ2 (r 1 , r 2 )
[Φ̂σ1 σ2 (r 1 , r 2 )]
(7.8)
for
respectively.
The strategy of this chapter consists in constructing an effective
theory for the order parameter ∆σ1 σ2 (r 1 , r 2 ) conjugate to Φ̂†σ1 σ2 (r 1 , r 2 )
by integrating out electrons, once the four-fermion decoupling has been
performed, and to verify that the order parameter indeed develops longrange order below some transition temperature. This strategy is identical to the one applied to the jellium model. As before, the effective
theory for the order parameter is an approximate one. The difference
with the RPA on the jellium model will be the physical content of this
approximation, namely the phenomenon of superconductivity. As with
section 6.4, the approximation to be performed is uncontrolled. This
approximation is only to be justified by comparison with experiments.
We begin this chapter with an interpretation of any non-vanishing expectation value for the operators in Eq. (7.4) as a signature of phase
ordering or phase stiffness.
7.1.1. Phase operator. Consider the Fock space
(↠)n F := span |ni := √ |0i n = 0, 1, 2, · · · ,
â|0i = 0
n!
that is generated by the bosonic algebra
[â, ↠] = 1,
[â, â] = [↠, ↠] = 0.
(7.9a)
(7.9b)
7.
344SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
We are already familiar with the number operator
N̂ := ↠â.
(7.10)
Define the operator θ̂ by taking the “square root” of N̂ , i.e., define
on the Fock space F \ {λ|0i, λ ∈ C}
e−iθ̂ := (N̂ )−1/2 ↠,
e+iθ̂ := â (N̂ )−1/2 ,
(7.11a)
â = e+iθ̂ (N̂ )+1/2 .
(7.11b)
or, upon inversion,
↠= (N̂ )+1/2 e−iθ̂ ,
Equation (7.11b) resembles the polar representation of complex numbers. Computation of the matrix elements
hm|e+iθ̂ N̂ e−iθ̂ |ni = hm|â (N̂ )−1/2 ↠â (N̂ )−1/2 ↠|ni
a† |ni =
√
n + 1|n + 1i
= hm + 1|↠â |n + 1i
= (n + 1)δm,n ,
m, n = 1, 2, · · · , (7.12a)
implies that
e+iθ̂ N̂ e−iθ̂ = N̂ + 1
(7.12b)
holds on F \ {λ|0i, λ ∈ C}. Extend the definition of θ̂ to all of F by
demanding that
e−iθ̂ |0i = |1i,
e+iθ̂ |0i = 0.
(7.13)
Hence, the “phase” operator exp(−iθ̂) turns an eigenstate of the
number operator N̂ with eigenvalue n into an eigenstate of the number
operator N̂ with eigenvalue n + 1. Conversely, the “phase” operator
exp(+iθ̂) turns an eigenstate of the number operator N̂ with eigenvalue
n into an eigenstate of the number operator N̂ with eigenvalue n − 1.
The phase operator exp(−iθ̂) is not quite unitary because of the vacuum
state |0i that makes it not norm preserving for all states in the Fock
space. The commutator between N̂ and θ̂ is obtained from
1 = [â, ↠]
h
ih
i h
ih
i
= e+iθ̂ (N̂ )+1/2 (N̂ )+1/2 e−iθ̂ − (N̂ )+1/2 e−iθ̂ e+iθ̂ (N̂ )+1/2
= e+iθ̂ N̂ e−iθ̂ − N̂
Z1
d +iαθ̂ −iαθ̂ =
dα
e
N̂ e
dα
0
Z1
= −i
dα e+iαθ̂ [N̂ , θ̂] e−iαθ̂ ,
(7.14a)
0
i.e.,
[N̂ , θ̂] = +i,
[N̂ , (α θ̂)] = +iα,
∀α ∈ C.
(7.14b)
7.1. PAIRING-ORDER PARAMETER
345
This is the same algebra as the one obeyed by the position, x̂, and momentum, p̂, operators except for the important caveat that the eigenvalues of θ̂ are defined on the circle as opposed to the real line for p̂.
Correspondingly, the eigenvalues of N̂ are discrete instead of continuous for x̂. The operator θ̂ is called the phase operator. The operator θ̂
is canonically conjugate to the number operator N̂ .
What is the counterpart of θ̂ for the fermionic algebra
{ĉσ , ĉ†σ0 } = δσσ0 ,
{ĉσ , ĉσ0 } = {ĉ†σ , ĉ†σ0 } = 0?
(7.15)
The fermionic counterparts are [recall Eq. (7.4)]
exp(−2iθ̂) −→ ĉ†↑ ĉ†↓ ,
↠â −→ ĉ†↑ ĉ↑ + ĉ†↓ ĉ↓ ,
exp(+2iθ̂) −→ ĉ↓ ĉ↑ .
(7.16)
Indeed, note that
N̂ ≡ ĉ†↑ ĉ↑ + ĉ†↓ ĉ↓ ,
N̂ 2 = N̂ + 2ĉ†↑ ĉ†↓ ĉ↓ ĉ↑ ,
(7.17a)
so that
ĉ↓ ĉ↑
ĉ†↑ ĉ↑
+
ĉ†↓ ĉ↓
ĉ†↑ ĉ†↓ = ĉ↓ ĉ↑ ĉ†↑ ĉ↑ ĉ†↑ ĉ†↓ + (−1)2 ĉ↑ ĉ↓ ĉ†↓ ĉ↓ ĉ†↓ ĉ†↑
= ĉ↓ ĉ↑ ĉ†↑ ĉ†↓ + (↑↔↓)
= ĉ↓ ĉ†↓ − ĉ↓ ĉ†↑ ĉ↑ ĉ†↓ + (↑↔↓)
= 1 − ĉ†↓ ĉ↓ − (−1)2 ĉ†↑ ĉ↓ ĉ†↓ ĉ↑ + (↑↔↓)
= 1 − ĉ†↓ ĉ↓ − ĉ†↑ ĉ↑ + ĉ†↑ ĉ†↓ ĉ↓ ĉ↑ + (↑↔↓)
By Eq. (7.17a)
= 2 − 2N̂ + N̂ (N̂ − 1)
= (N̂ − 1)(N̂ − 2).
(7.17b)
Thus, as it should be, the only non-vanishing matrix element of Eq.
(7.17b) in the fermionic Fock space
o
n
(7.18)
F := span |0i, ĉ†↑ |0i, ĉ†↓ |0i, ĉ†↑ ĉ†↓ |0i ĉ↑ |0i = ĉ↓ |0i = 0
is the expectation value in the vacuum |0i.
At the level of quantum field theory, we deduce from the identifications (Schrödinger picture)
Φ̂†↑↓ (r, r) =: e−2iθ̂↑↓ (r,r) ,
Φ̂↑↓ (r, r) =: e+2iθ̂↑↓ (r,r) ,
(7.19a)
the equal-time (Heisenberg picture)
[ρ̂(r, t), Φ̂†↑↓ (r 0 , r 0 , t)] = +2Φ̂†↑↓ (r 0 , r 0 , t) δ(r − r 0 ),
[ρ̂(r, t), Φ̂↑↓ (r 0 , r 0 , t)] = −2Φ̂↑↓ (r 0 , r 0 , t) δ(r − r 0 ),
(7.19b)
commutators. By expanding the pairing operators in Eq. (7.19a) to
linear order in θ̂↑↓ (r, r) in the commutators (7.19b), the commutator
h
i
ρ̂(r, t), θ̂↑↓ (r 0 , r 0 , t) = iδ(r − r 0 )
(7.19c)
7.
346SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
follows. Hence, a sharp ground-state expectation value of the density
operator ρ̂(r, t) implies a broad uncertainty in the expectation value
of the pairing operator Φ̂†↑↓ (r, r, t). 1 This scenario is the one realized
in section 6.4. In this and chapter 8, we realize the superconducting
scenario in which a sharp ground-state expectation value of the Fourier
transform of the pairing operator Φ̂†↑↓ (r 1 , r 2 , t) implies maximum uncertainty in the expectation value of the density operator ρ̂(r, t).
7.1.2. Center-of-mass and relative coordinates. In this subsection, we want to argue that the characteristic separation r 1 − r 2
is of the order of the inverse Fermi momentum in the pairing operator Φ̂†↑↓ (r 1 , r 2 ), if the pairing operator signals the instability of the
non-interacting Fermi sea induced by some interaction to a state that
preserves translation invariance. To see this trade r 1 and r 2 for the
relative coordinate r and the center of mass coordinate R,
r := r 1 − r 2 ,
R :=
r1 + r2
,
2
r
r1 = R + ,
2
r
r2 = R − ,
2
(7.20)
respectively. Fourier transformation gives
Φ̂†σ1 σ2 (r 1 , r 2 )
2 X
1
√
=
e−ik1 ·r1 e−ik2 ·r2 ĉ†σ1 k1 ĉ†σ2 k2
V
k1 ,k2
X
r
r
1
=
e−ik1 ·(R+ 2 ) e−ik2 ·(R− 2 ) ĉ†σ1 k1 ĉ†σ2 k2
V k ,k
1
2
1 X −i(k1 +k2 )·R − i (k1 −k2 )·r †
=
e
e 2
ĉσ1 k1 ĉ†σ2 k2
V k ,k
1
2
1 X −iQ·R −iq·r †
e
e
ĉσ Q +q ĉ†σ Q −q , (7.21a)
=
) 2( 2 )
V q,Q
1( 2
whereby
q :=
k1 − k2
,
2
Q := k1 + k2 ,
1
Q
+ q,
2
Q
k2 =
− q.
2
k1 =
(7.21b)
By sharp or broad expectation values, we mean small or large mean square
root deviations about expectation values, respectively.
7.1. PAIRING-ORDER PARAMETER
347
Define Φ̂†σ1 σ2 Q (r) by
r
r
1 X −iQ·R †
Φ̂†σ1 σ2 R + , R −
=√
e
Φ̂σ1 σ2 Q (r),
2
2
V Q
1 X −iq·r †
Φ̂†σ1 σ2 Q (r) := √
e
ĉσ Q +q ĉ†σ Q −q .
) 2( 2 )
1( 2
V q
(7.22)
We are only interested in a temperature range well below the Fermi
energy εF ,
T εF .
(kB = 1)
(7.23)
All relevant energy, time, and length scales are controlled by the Fermi
momentum kF of the Fermi sea in this range of temperature and in
the non-interacting limit. For example, relevant momenta k1 and k2
entering the Fourier expansion of Φ̂†σ1 σ2 (r 1 , r 2 ) must obey
|k1 | ≥ kF ,
|k2 | ≥ kF ,
(7.24)
if Φ̂†σ1 σ2 (r 1 , r 2 ) applied to the Fermi sea does not annihilate it at zero
temperature. If we presume that the Fermi sea is made unstable by
interactions to a many-body state in which some Fourier components
of Φ̂†σ1 σ2 (r 1 , r 2 ) acquire a non-vanishing expectation value, it is reasonable to assume that these Fourier components have a vanishing
center-of-mass momentum Q = 0, for translation symmetry would be
spontaneously broken otherwise. If so, the relative momenta in the
Fourier expansion (7.21a) must obey
|q| ≥ kF .
(7.25)
If the single-particle momenta k1 and k2 entering the Fourier expansion (7.21a) are close to the Fermi surface, it then follows that the
characteristic size of the relative coordinate r in Φ̂†σ1 σ2 Q (r) is
|r| ∼
1
.
kF
(7.26)
For a good metal,
1
∼ a,
kF
(7.27)
a the lattice spacing. Hence, the characteristic size of the relative coordinate r in Φ̂†σ1 σ2 (R + r2 , R − r2 ) is the lattice spacing for a good
metal made unstable by some
interaction to a many-body state in
which Φ̂†σ1 σ2 R + r2 , R − r2 acquires an expectation value that does
not break translation invariance.
7.
348SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
7.2. Scaling of electronic interactions
7.2.1. Case of a repulsive interaction. Consider the electronic
Coulomb interaction
Ĥint :=
Z
1
2
d3 r 1
V
=
Z
1
2
d3 r
Z
ZV
Vcb (r 1 − r 2 ) Φ̂†σ1 σ2 (r 1 , r 2 ) Φ̂σ1 σ2 (r 1 , r 2 )
σ1 ,σ2 =↑,↓
X
d3 R
σ1 ,σ2
V
2V
X
d3 r 2
r
r
r
r
Vcb (r) Φ̂†σ1 σ2 R + , R −
Φ̂σ1 σ2 R + , R −
.
2
2
2
2
(7.28)
We have seen in section 6.4 that the response of the Fermi sea to the
Coulomb interaction is to screen the algebraic tails of the Coulomb
interaction in its position-space representation. Moreover, as we have
argued in section 7.1.2, we expect that the response of the Fermi sea will
be dominated by pairs of electrons (holes) whose relative coordinates
r are of the order of the inverse Fermi momentum, which, for a good
metal, is the lattice spacing, i.e., r = 0 on macroscopic length scales.
On the lattice scale, the diverging short-distance Coulomb interaction
might as well be approximated by a repulsive delta function,
Vcb (r) −→ U δ(r),
U > 0.
(7.29)
We thus arrive at the approximate interacting Hamiltonian
Ĥint
U
≈
2
Z
3
Z
dr
2V
Z
=U
3
dR
V
X
δ(r)
Φ̂†σ1 σ2
σ1 ,σ2 =↑,↓
r
r
r
r
R + ,R −
Φ̂σ1 σ2 R + , R −
2
2
2
2
d3 R Φ̂†↑↓ (R, R) Φ̂↑↓ (R, R) .
V
(7.30)
What is the fate of the repulsive residual interaction U δ(r) upon partial
integration over electrons, whereby only high-energy electrons in a thin
shell around the Fermi surface are successively integrated out? We shall
see that integration of high-energy electrons induces a renormalization
of the interaction strength U such that U decreases with lower energy
cutoff!
From now on, we will rely on the Grassmann-path-integral representation of the grand-canonical partition function. The only rules that
7.2. SCALING OF ELECTRONIC INTERACTIONS
349
need to be kept in mind are the substitutions
ĉ†σ (r) −→ ψσ∗ (r, τ ) = −ψσ∗ (r, τ + β),
ĉσ (r) −→ ψσ (r, τ ) = −ψσ (r, τ + β),
Φ̂†σ1 σ2 (R, R) −→ Φ∗σ1 σ2 (R, τ ) := ψσ∗1 (R, τ )ψσ∗2 (R, τ ),
(7.31a)
Φ̂σ1 σ2 (R, R) −→ Φσ1 σ2 (R, τ ) := ψσ2 (R, τ )ψσ1 (R, τ ),
Z
−β Ĥβ,µ
Zβ,µ := TrF e
−→ D[ψ ∗ ]D[ψ] e−Sβ,µ ,
where we decompose additively the Euclidean action
Sβ,µ = S0 + SU
(7.31b)
into the non-interacting action
Zβ
S0 :=
Z
dτ
0
3
dr
X
σ=↑,↓
V
ψσ∗ (r, τ )
∇2
− µ ψσ (r, τ )
∂τ −
2m
(7.31c)
and the interacting action
Zβ
SU := U
Z
dτ
0
d3 R Φ∗↑↓ (R, τ ) Φ↑↓ (R, τ ).
(7.31d)
V
Here, ψσ∗ (R, τ ) and ψσ (R, τ ) are two independent Grassmann numbers, i.e., they are anticommuting numbers. The Grassmann integral
is defined so that it is invariant under a unitary transformation of the
ψ’s whereas it changes by the inverse of the Jacobian of a non-unitary
transformation of the ψ’s instead of the usual Jacobian of Riemann
integrals. 2
We can make use of the simplification brought upon by dealing with
Grassmann numbers as opposed to operators in the integrand of the
grand-canonical partition function. For example, we can freely write
e−S0 −SU = e−S0 × e−SU
(7.32)
in the path-integral representation of Zβ,µ whereas this step is illegal
in the trace over the Fock-space-representation of Zβ,µ . In turn, we
2
This is so because the Grassmann
is constructed
√ ∗ √such that
R integral
∗
dψ
√
√
dψ ∗ dψ exp(−ψ ∗ A ψ) = A, i.e., A dψ
exp
−
(
Aψ )( Aψ)
=
A
A
R ∗
∗
A dζ dζ exp(−ζ ζ) = A.
R
7.
350SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
introduce through the Hubbard-Stratonovich transformation
Zβ

exp(−SU ) = exp −U

Z
dτ
0
d3 R Φ∗↑↓ (R, τ ) Φ↑↓ (R, τ )
V

Z
∝
D[∆∗ , ∆] exp −
Zβ

Z
dτ
0
Zβ

× exp −
Z
dτ
0
3
d3 R
1 ∗
∆ (R, τ ) ∆(R, τ )
U
V

d3 R i∆∗ (R, τ ) Φ↑↓ (R, τ ) + i∆(R, τ ) Φ∗↑↓ (R, τ )  ,
V
(7.33)
the complex-valued order parameters
∆∗ (R, τ ) = |∆(R, τ )| e−iφ(R,τ ) .
(7.34a)
∆(R, τ ) = |∆(R, τ )| e+iφ(R,τ )
(7.34b)
and
These order parameters will shortly be interpreted as being closely
related to the expectation values
hΦ∗↑↓ (R, τ )iZβ,µ
R
D[ψ ∗ ]D[ψ] Φ∗↑↓ (R, τ ) e−S0 −SU
:= R
D[ψ ∗ ]D[ψ]
e−S0 −SU
(7.35a)
hΦ↑↓ (R, τ )iZβ,µ
R
D[ψ ∗ ]D[ψ] Φ↑↓ (R, τ ) e−S0 −SU
:= R
,
D[ψ ∗ ]D[ψ]
e−S0 −SU
(7.35b)
and
respectively. Observe the presence of the imaginary number i in the
second exponential on the right-hand side of Eq. (7.33). This imaginary
number originates in the interaction being repulsive. No imaginary
number would be needed for an attractive interaction. An upper cutoff
3
The measure for the auxiliary fields ∆∗ and ∆ is defined so as to insure
convergence. This implies that ∆∗ is the complex conjugate to ∆.
7.2. SCALING OF ELECTRONIC INTERACTIONS
351
in momentum space is imposed in the Fourier expansions
ψσ∗ (R, τ )
|K|<Λ
1 X X −i(K·R−ωn τ ) ∗
=: √
e
ψσKωn ,
βV ω
K
n
ψσ (R, τ ) =: √
|K|<Λ
1 X X +i(K·R−ωn τ )
e
ψσKωn ,
βV ω
K
n
∆∗ (R, τ ) =: √
1
βV
X |Q|<2Λ
X
$l
(7.36)
e−i(Q·R−$l τ ) ∆∗Q$l ,
Q
|Q|<2Λ
1 X X +i(Q·R−$l τ )
∆(R, τ ) =: √
e
∆Q$l .
βV $
Q
l
Remember that the Matsubara frequencies are
π
n ∈ Z,
(7.37)
ωn = (2n + 1),
β
and
2π
$l =
l,
l ∈ Z,
(7.38)
β
respectively.
In summary, the full partition function has the Grassmann-pathintegral representation
Z
Z
0
∗
Zβ,µ ∝ D[ψ ]D[ψ] D[∆∗ , ∆] e−Sβ,µ ,
(7.39a)
with the additive decomposition of the action
0
Sβ,µ
= Scond + S0 + SU0
(7.39b)
into a quadratic action for the order parameter,
XX 1
∆∗Q$l ∆Q$l
Scond =
U
$
Q
l
(7.39c)
Zβ
=
Z
dτ
0
1
d R ∆∗ (R, τ )∆(R, τ ),
U
3
V
a quadratic action for the Grassmann variables (the fermions),
S0 =
X |K|<Λ
X X
ωn
K
Zβ
=
Z
dτ
0
V
∗
ψ
(−iωn + ξK ) ψσKω
n σKωn
σ=↑,↓
d3 R
X
σ=↑,↓
(7.39d)
2
∇
ψσ∗ (R, τ ) ∂τ −
− µ ψσ (R, τ ),
2m
7.
352SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
and a coupling between the order parameter and bilinears in the Grassmann variables (the fermions),
SU0
|Q−K|<Λ
|K|<Λ
i X X X X ∗
=√
∆Q$l ψ↓(Q−K)($l −ωn ) ψ↑Kωn
βV $
ω
Q
K
n
l
i X
+√
βV $
l
Zβ
=i
Z
dτ
0
|Q−K|<Λ
X
Q
X |K|<Λ
X
ωn
∗
∆Q$l ψ↑Kω
ψ∗
n ↓(Q−K)($l −ωn )
K
d3 R ∆∗ (R, τ ) ψ↓ ψ↑ (R, τ ) + ∆(R, τ ) ψ↑∗ ψ↓∗ (R, τ ) .
V
(7.39e)
As always, the average number of electrons is N = β −1 ∂µ ln Zβ,µ . Finally,
ξK =
K2
− µ,
2m
L
K ∈ Z3 ,
2π
L
Q ∈ Z3 .
2π
(7.39f)
The classical equations of motion for the auxiliary fields ∆∗ (R, τ )
and ∆(R, τ ) are
0
∂Sβ,µ
1
= ∆(R, τ ) + i ψ↓ ψ↑ (R, τ ),
∗
∂∆ (R, τ )
U
0
∂Sβ,µ
1
= ∆∗ (R, τ ) + i ψ↑∗ ψ↓∗ (R, τ ).
0=
∂∆(R, τ )
U
0=
(7.40)
Hence, the following physical interpretation of the auxiliary fields ∆∗ (R, τ )
and ∆(R, τ ) follows. If we compute
the expectation value of Eq. (7.40)
R
0
with the partition function D[ψ ∗ ]D[ψ] e−S0 −SU , we find that the auxiliary fields ∆∗ (R, τ ) and ∆(R, τ ) are a mean-field approximation to
−iU times the expectation values in Eqs. (7.35a) and (7.35b), respectively.
The poor man’s scaling procedure that we follow consists in integrating out electrons with momenta K belonging to the momentum
shell
Λ − dΛ < |K| < Λ.
(7.41)
In doing so, the bare action
0
Sβ,µ
= Scond + S0 + SU0
(7.42)
will be modified. The renormalization-group (RG) method applies
when it is possible to absorb all these modifications through a renormalization of length scales and coupling constants in a way that preserves
the form of the action.
7.2. SCALING OF ELECTRONIC INTERACTIONS
353
Here, we limit ourselves to deriving the changes induced by a momentumshell integration for the contribution
0
S00;Λ
:=
1 ∗
∆ ∆
U 00 00
X |K|<Λ
X X
∗
+
(−iωn + ξK ) ψσKω
ψ
n σKωn
ωn
K
σ=↑,↓
|K|<Λ
i∆∗00 X X
ψ↓(−K)(−ωn ) ψ↑Kωn
+√
βV ω
K
(7.43)
n
|K|<Λ
i∆00 X X ∗
∗
+√
ψ↑Kωn ψ↓(−K)(−ω
,
n)
βV ω
K
n
to the action coming from momentum and energy transfer
Q = 0,
$l = 0,
(7.44)
respectively. This is the reduced action for space- and time-independent
configurations of the order parameter.
0
Observe that S00;Λ
can be rewritten as
0
S00;Λ
X |K|<Λ
X
1 ∗
∗
ψ↑Kω
ψ
= ∆00 ∆00 +
↓(−K)(−ω
)
n
n
U
ωn
K
!
i∆
√ 00
−iωn + ξK
ψ↑Kωn
βV
×
.
∗
i∆∗
ψ↓(−K)(−ω
√ 00
−iωn − ξK
n)
(7.45)
βV
Here, the property of inversion symmetry
ξK = ξ−K
(7.46)
of the single-particle dispersion was used. The 2 × 2 grading that has
been introduced is called the particle-hole grading. It plays a very
important role in the mean-field theory of superconductivity and for
fluctuations about it.
Integration over the fermions within the momentum shell defines
0
the new action S00;Λ−dΛ
,
Z
0
0
exp −S00;Λ−dΛ :=
D[ψ ∗ ]D[ψ] exp −S00;Λ
.
(7.47)
Λ−dΛ<|K|<Λ
But integration over the fermions is the functional determinant
!
2
i∆
√ 00
+∂τ − ∇
−
µ
2m
βV
Det
(7.48)
i∆∗
∇2
√ 00
+∂
+
+
µ
τ
2m
βV
7.
354SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
owing to the rules of Grassmann integration. Moreover, since a determinant can always be re-exponentiated, we can write
Λ−dΛ<|K|<Λ
(Scond )00;Λ−dΛ
X
1
:= ∆∗00 ∆00 −
U
ω
X
log det
i∆∗
√ 00
βV
K
n
i∆
√ 00
βV
−iωn + ξK
−iωn − ξK
Λ−dΛ<|K|<Λ
X
1
= ∆∗00 ∆00 −
U
ω
∗
2 ∆00 ∆00
2
2
log −ωn − ξK − i
.
βV
X
K
n
(7.49)
With the help of the Taylor expansion ln(1 − x) = −
+∞
P
j=1
1 j
x,
j
the right-
hand side becomes
(Scond )00;Λ−dΛ = −
X Λ−dΛ<|K|<Λ
X
ωn
2
log −ωn2 − ξK
K


Λ−dΛ<|K|<Λ
X
X
1
1
1
 ∆∗00 ∆00
+  + (−i2 )
2
2
U
βV ω
ω
+
ξ
n
K
K
n
+
Λ−dΛ<|K|<Λ +∞
X
X
1X
j=2
j
ωn
K
∆∗00 ∆00
1
(−i )
2
2
βV ωn + ξK
2
j
.
(7.50)
If we reinterpret (Scond )00;Λ as the infinite series
(Scond )00;Λ :=
+∞
X
a00;Λ,j (∆00 ∆∗00 )j ,
(7.51)
1
j = 2, 3, · · · ,
a00;Λ,0 = 0,
a00;Λ,1 = ,
a00;Λ,j = 0,
U
we see that the momentum-shell integration is encoded by a renormalization of the coefficients a00;Λ,j , j = 0, 1, 2, · · · ,
j=0
(Scond )00;Λ−dΛ =
+∞
X
a00;Λ−dΛ,j (∆00 ∆∗00 )j ,
j=0
a00;Λ−dΛ,0 = −
X Λ−dΛ<|K|<Λ
X
ωn
a00;Λ−dΛ,j
δj,1 1
=
+
U
j
2
,
log −ωn2 − ξK
(7.52)
K
1
βV
j X Λ−dΛ<|K|<Λ
X
ωn
K
1
2
2
ωn + ξK
j
,
where j = 1, 2, · · · .
The coefficient a00;Λ−dΛ,0 is a C number. It does not enter in any
correlation function. We need not worry about it anymore. A very
!!
7.2. SCALING OF ELECTRONIC INTERACTIONS
355
good estimate of a00;Λ−dΛ,j+1 can be done at low temperatures. When
β → ∞ the summation over frequencies can be replaced by an integral,
a00;Λ−dΛ,j+1 =
=
δj+1,1
1
+
β
U
j+1
δj+1,1
1
+
U
j+1
1
βV
1
βV
+∞
j+1 Λ−dΛ<|K|<Λ
Z
X
K
j
Ij+1
1
V
dω
2π
−∞
Λ−dΛ<|K|<Λ X
K
1
2
ω 2 + ξK
1
|ξK |
j+1
2j+1
(7.53a)
,
where
Z+∞
Ij+1 :=
dx
2π
1
2
x +1
j+1
−∞
page 254, section 4.8 from Ref. [73]
=
=
(2j)! −2j
1
×
2 π
2π
(j!)2
(2j)! −2j−1
2
,
j = 0, 1, 2, (7.53b)
··· .
(j!)2
With the help of the density of states per unit volume
ν̃(ξ) :=
1 XX
1 XX
δ(ξ − ξK ) =
δ(ξ − εK + µ), (7.54)
V σ=↑,↓ K
V σ=↑,↓ K
the momentum summation can be rewritten as an energy integral,
1
V
Λ−dΛ<|K|<Λ
X
|ξK |−(2j+1) =
K
2 /2m
ΛZ
dε
ν̃(ε − µ)
|ε − µ|−(2j+1)
2
(Λ−dΛ)2 /2m
εΛ > εF ≡ µ
ν̃ (εΛ − εF )
(εΛ − εF )−(2j+1) dεΛ + O (dΛ)2
2
νF
εF
−2j
=
(εΛ ) d ln εΛ + O
+ O (dΛ)2 .
2
εΛ
(7.55)
=
Here, the assumption that the density of states per unit volume varies
very slowly from the Fermi energy εF to the upper energy cutoff εΛ has
been made,
dν ν̃(εΛ − εF ) = ν̃(0) +
(ε − εF ) + · · ·
dε ε Λ
F
(7.56)
≡ νF + νF0 (εΛ − εF ) + · · ·
≈ νF .
7.
356SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
Specializing henceforth to the scaling of U , the coefficient a00;Λ−dΛ,1 ,
we have found that
1
1 νF
1
1
2 νF
+
d ln εΛ ≈
−i
d ln Λ .
(7.57)
≈
UΛ−dΛ
UΛ 2 2
UΛ
2
If dΛ is chosen positive, Λ − dΛ decreases and so does UΛ−dΛ according
to Eq. (7.57). For dΛ infinitesimal, the poor man’s scaling differential
equation
d UΛ−1
ν ν = − F = +i2 F
(7.58)
d ln Λ
2
2
equates the logarithmic derivative of the inverse, bare repulsive interaction strength with the negative of the density of states at the Fermi
energy. Integration of high-energy electrons decreases the strength of
the repulsive contact interaction. This suggests that at a low enough
electronic energy scale, an attractive force between electrons mediated
by phonons might overcome
the repulsive Coulomb interaction. The
coupling constant UΛ−1 increases with decreasing cutoff Λ as integration of Eq. (7.58) between 0 < Λ1 < Λ2 yields
UΛ2
νF
Λ2
−1
−1
. (7.59)
UΛ2 − UΛ1 = − ln
⇐⇒ UΛ1 =
ν
Λ
2
Λ1
1 + UΛ2 2F ln Λ2
1
In summary, neither does coupling electrons to phonons or coupling
electrons to the superconducting order parameter favor repulsion at
low energies. In view of this, what can we say if the effective electronic
interaction is attractive?
7.2.2. Case of an attractive interaction. We reverse the sign
of the contact interaction in Eq. (7.29) to make it attractive, i.e., the
interacting Hamiltonian in the canonical ensemble is given by
Ĥint
U
:= −
2
Z
3
Z
d r
2V
Z
= −U
V
d3 R
X
σ1 ,σ2 =↑,↓
r
r
r
r
Φ̂σ1 σ2 R + , R −
δ(r) Φ̂†σ1 σ2 R + , R −
2
2
2
2
d3 R Φ̂†↑↓ (R, R)Φ̂↑↓ (R, R).
U ≥ 0.
V
(7.60)
The only modification to the repulsive case is the necessity to remove
the imaginary number i in the Hubbard-Stratonovich transformation
(7.33). Carrying through this change leads to changing the sign on the
right-hand side of the poor man’s scaling differential equation (7.58),
d UΛ−1
ν
= + F.
(7.61)
d ln Λ
2
7.3. TIME- AND SPACE-INDEPENDENT LANDAU-GINZBURG ACTION 357
Equation (7.59) turns into
UΛ−1
1
−
ν
= + F ln
2
UΛ−1
2
Λ1
Λ2
UΛ1
⇐⇒ UΛ2 =
1 − UΛ1
νF
2
ln
Λ1
Λ2
.
(7.62)
Taken at face value, Eq. (7.62) predicts a divergence of the attractive
interaction when
!
2
Λ2 = Λ1 exp −
.
(7.63)
UΛ1 νF
However, this conclusion is incorrect since self-consistency of the initial
assumption that U is small is violated. Inconsistencies of this type are
often the signal that the non-interacting ground state, here the Fermi
sea, is not the true ground state of the interacting system.
7.3. Time- and space-independent Landau-Ginzburg action
From now on we will assume an attractive contact interaction. The
partition function in the grand-canonical ensemble is given by
Z
Z
0
∗
Zβ,µ ∝ D[∆ , ∆] D[ψ ∗ ]D[ψ] e−Sβ,µ ,
(7.64a)
with the additive decomposition of the action
0
Sβ,µ
= Scond + S0 + SU0
(7.64b)
into a quadratic action for the order parameter,
Scond =
XX 1
∆∗Q$l ∆Q$l
U
$
Q
l
(7.64c)
Zβ
=
Z
dτ
0
1
d R ∆∗ (R, τ )∆(R, τ ),
U
3
V
a quadratic action for the Grassmann variables (the fermions),
S0 =
XXX
ωn
K
Zβ
=
Z
dτ
0
∗
ψ
(−iωn + ξK ) ψσKω
n σKωn
σ
3
dR
V
X
σ
ψσ∗ (R, τ )
∇2
∂τ −
− µ ψσ (R, τ ),
2m
(7.64d)
7.
358SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
and a coupling between the order parameter and bilinears in the Grassmann variables (the fermions),
1 XXXX ∗
SU0 = √
∆ ψ
ψ
βV $ Q ω K Q$l ↓(Q−K)($l −ωn ) ↑Kωn
n
l
1 XXXX
∗
+√
∆Q$l ψ↑Kω
ψ∗
n ↓(Q−K)($l −ωn )
βV $ Q ω K
l
Zβ
=
Z
dτ
0
n
d3 R ∆∗ (R, τ ) ψ↓ ψ↑ (R, τ ) + ∆(R, τ ) ψ↑∗ ψ↓∗ (R, τ ) .
V
(7.64e)
As always, the average number of electrons is N = β −1 ∂µ ln Zβ,µ . Finally,
K2
L
L
− µ,
K ∈ Z3 ,
Q ∈ Z3 .
(7.64f)
2m
2π
2π
The only important difference with the repulsive case, see Eq. (7.39),
is the absence of the imaginary-valued multiplicative factors in SU0 ,
Eq. (7.64e). The order of path-integral integrations has been exchanged
relative to Eq. (7.39a) and the restriction on the summation over fermionic
momenta removed in Eqs. (7.64d) and (7.64e).
The classical equations of motion for the auxiliary fields ∆∗ (R, τ )
and ∆(R, τ ) are
0
∂Sβ,µ
1
0=
=
∆(R,
τ
)
+
ψ
ψ
(R, τ ),
↓
↑
∂∆∗ (R, τ )
U
(7.65)
0
∂Sβ,µ
1 ∗
∗ ∗
0=
= ∆ (R, τ ) + ψ↑ ψ↓ (R, τ ).
∂∆(R, τ )
U
ξK =
Hence, the following physical interpretation of the auxiliary fields ∆∗ (R, τ )
and ∆(R, τ ) follows. If we compute
the expectation value of Eq. (7.65)
R
0
∗
with the partition function D[ψ ]D[ψ] e−S0 −SU , we find that the auxiliary fields ∆∗ (R, τ ) and ∆(R, τ ) are a mean-field approximation to −U
times the expectation values in Eqs. (7.35a) and (7.35b), respectively.
Below, we focus exclusively on the intensive grand canonical potential Fβ,µ (∆∗ , ∆) obtained after integrating over all fermions in the
background of a space- and time-independent order parameter
1
1
∆(R, τ ) = √ ∆0,0 ≡ ∆,
∀R, τ.
∆∗ (R, τ ) = √ ∆∗0,0 ≡ ∆∗ ,
βV
βV
(7.66)
More precisely,
∗
∗
Z
∆ (R,τ )=∆
∗
∗
0
exp − βV Fβ,µ (∆ , ∆) :=
D[ψ ]D[ψ] exp −Sβ,µ .
∆ (R,τ )=∆
(7.67)
7.3. TIME- AND SPACE-INDEPENDENT LANDAU-GINZBURG ACTION 359
To compute Eq. (7.67), we can borrow Eq. (7.49), keeping in mind
that i2 together with the restriction on the momentum summation must
be removed. In a first application of Eq. (7.49), we calculate
Veff (∆∗ , ∆) := lim Fβ,µ (∆∗ , ∆)
β,V →∞
(7.68)
in closed form and derive the so-called BCS gap equation at zero
temperature. In a second application of Eq. (7.49), we expand the
fermionic determinant to study the intensive grand canonical potential
in the vicinity of the transition temperature below which ∆∗ and ∆
acquire expectation values. In the vicinity of the transition temperature, Fβ,µ (∆∗ , ∆) is called the time-independent and space-independent
Landau-Ginzburg free energy.
7.3.1. Effective potential at T = 0. The second line of Eq.
(7.49) yields, in the limit of infinite volume and vanishing temperature,
Z
Z 3
1 ∗
dK
dω
∗
2
Veff (∆ , ∆) = ∆ ∆ −
log −ω 2 − ξK
− ∆∆∗
3
U
2π
(2π)
R
R3
Z
Z 3
1 ∗
dK dω
2
2
∗
= ∆ ∆−
ln
ω
+
ξ
+
∆∆
+
iπ
.
K
U
2π
(2π)3
R3
R
(7.69)
Assuming that Veff (∆∗ , ∆) is differentiable,
Z
Z
ν̃(ξ)
dω
∆
∂Veff (∆∗ , ∆)
−1
= U ∆ − dξ
,
2
∗
2
∂∆
2
2π ω + ξ + ∆∆∗
R
R
Z
Z
∗
∂Veff (∆ , ∆)
ν̃(ξ)
dω
∆∗
−1 ∗
= U ∆ − dξ
,
∂∆
2
2π ω 2 + ξ 2 + ∆∆∗
R
(7.70a)
R
where
ν̃(ξ) := 2 ×
1 X
δ(ξ − ξK )
V K
(7.70b)
is the density of states per unit volume. Integration over frequencies is
done with the help of the residue theorem,
Z
∂Veff (∆∗ , ∆)
1
∆
ν̃(ξ)
−1
p
,
=U ∆ −
dξ
∗
∂∆
2
2
ξ 2 + ∆∆∗
R
(7.71)
Z
∂Veff (∆∗ , ∆)
1
ν̃(ξ)
∆∗
−1 ∗
p
=U ∆ −
dξ
.
∂∆
2
2
ξ 2 + ∆∆∗
R
Integration over energies is potentially divergent as it stands. A highenergy cutoff must be introduced. This cutoff could be the bandwidth
of some lattice regularization or it could be the Debye frequency above
7.
360SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
which repulsive Coulomb interactions dominate over the attractive interacting channel mediated by phonons. Historically, the Debye frequency was used. Thus,
+ωD
∂Veff (∆∗ , ∆)
1
→ U −1 ∆ −
∗
∂∆
2
Z
dξ
−ωD
∂Veff (∆∗ , ∆)
1
→ U −1 ∆∗ −
∂∆
2
ν̃(ξ)
∆
p
,
2
2
ξ + ∆∆∗
(7.72)
+ωD
ν̃(ξ)
∆∗
p
.
2
ξ 2 + ∆∆∗
Z
dξ
−ωD
The density of states can be Taylor expanded around the Fermi energy
and to lowest order in the ratio ωD /εF ,
+ωD
∂Veff (∆∗ , ∆)
1ν
≈ U −1 ∆ − F
∗
∂∆
2 2
Z
∆
dξ p
2
ξ + ∆∆∗
−ωD
(7.73)
+ωD
∗
∂Veff (∆ , ∆)
1ν
≈ U −1 ∆∗ − F
∂∆
2 2
,
∗
Z
∆
dξ p
2
ξ + ∆∆∗
−ωD
.
To leading order in |∆|/ωD ,
ω
D
+ |∆|
+ωD
Z
−ωD
1
dξ p 2
= 2
ξ + ∆∆∗
Z
dx √
0
1
x2
+1
ωD
|∆|
2 −2 !
4ωD
ωD
= ln
+O
(. 7.74)
∆∆∗
|∆|
= 2arcsinh
see Eq. 4.6.31 from Ref. [74]
Thus,
−2 !
∆∗ ∆
ωD
ωD
+O
,
,
2
4ωD
εF
|∆|
∗ −2 !
∂Veff (∆∗ , ∆)
1
ν
∆
∆
ω
ω
D
D
.
≈ U −1 ∆∗ + F ∆∗ ln
+O
,
2
∂∆
2 2
4ωD
εF
|∆|
(7.75)
∂Veff (∆∗ , ∆)
1ν
≈ U −1 ∆ + F ∆ ln
∗
∂∆
2 2
With F (x) = 12 x2 ln x − 14 x2 the primitive of f (x) = x ln x,
∗ 1 νF
∆∆
∗
∗
−1 ∗
∗
Veff (∆ , ∆) ≈ U ∆ ∆ +
∆ ∆ ln
− ∆ ∆ + A, (7.76a)
2
2 2
4ωD
7.3. TIME- AND SPACE-INDEPENDENT LANDAU-GINZBURG ACTION 361
up to an integration constant A ∈ C, provided the hierarchy of energy
scales
|∆| ωD εF
(7.76b)
holds. The effective potential only depends on the combination |∆|2
of the order parameters [recall Eqs. (7.34b-7.34a)]. In particular, the
effective potential does not depend on the phase φ of the order parameter ∆. The classical equation of motion for the effective potential
amounts to minimization, i.e.,
νF
|∆|
−1
0≈U +
ln
,
|∆| ωD εF ,
(7.77a)
2
2ωD
with the mean-field or saddle-point solution
2
|∆| ≈ 2ωD exp −
,
|∆| ωD εF .
U νF
(7.77b)
Equation (7.77b) is called the BCS gap equation. For Eq. (7.77b) to
hold, the weak coupling condition
U νF 1
(7.78)
must be satisfied. The mean-field solution only fixes the magnitude of
the order parameter ∆, not its phase.
7.3.2. Effective free energy in the vicinity of Tc . To probe
what happens when ∆∗ and ∆ become very small as a result of thermal
fluctuations, Eq. (7.50) is used,
1 XX
2
Fβ,µ (∆∗ , ∆) = −
log −ωn2 − ξK
βV ω K
n


X
X
1
1
1
 ∆∗ ∆
+ −
(7.79)
2
2
U
βV ω K ωn + ξK
n
+
+∞
X
j=2
j
(−1)j 1 X X
∆∗ ∆
.
2
j βV ω K ωn2 + ξK
n
As we did for the repulsive case, we can introduce the density of states
per unit volume
1 X
ν̃(ξ) := 2 ×
δ(ξ − ξK ),
(7.80a)
V K
in terms of which [compare with Eq. (7.52)]
∗
Fβ,µ (∆ , ∆) =
+∞
X
j=0
fj (∆∗ ∆)j ,
(7.80b)
7.
362SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
where the zeroth-order expansion coefficient is
Z+∞
1 X
f0 = −
dξ ν̃(ξ) log −ωn2 − ξ 2 ,
2β ω
n
(7.80c)
−∞
while the (j = 1, 2, · · · )-th-order expansion coefficient is
Z+∞
δj,1 (−1)j 1 X
ν̃(ξ)
fj =
+
dξ
2
U
2j β ω
(ωn + ξ 2 )j
n
−∞
Z+∞
δj,1 (−1)j 1 X
ν̃n (x)
−2j+1
dx
=
+
.
|ωn |
2 )j
U
2j β ω
(1
+
x
n
(7.80d)
−∞
Here,
ν̃n (x) := ν̃(x ωn ).
(7.80e)
The important consequence of dealing with an attractive interaction
is that the coefficient fj alternates in sign with j = 2, 3, · · · , i.e., f2j
is positive while f2j+1 is negative for j = 1, 2, · · · . The coefficient
f0 is only important insofar one is interested in the absolute scale of
Fβ,µ (∆∗ , ∆). The coefficient f1 is the most interesting one, since its
sign has the potential to change from positive to negative as the temperature is decreased. The putative change in the sign of f1 is yet
another signature of the instability of the Fermi-liquid ground state.
Above the transition temperature at which f1 vanishes, the Fermi sea
is a reasonable candidate for the ground state. Below the transition
temperature, the free energy Fβ,µ (∆∗ , ∆) favors condensation of the
order parameters ∆∗ and ∆. Since the Fermi sea is not compatible
with non-vanishing values ∆∗ and ∆ (the Fermi sea is built out of a
given number of electrons) this indicates that the ground state must
be fundamentally different from the Fermi sea.
An estimate of the transition temperature follows from application
of the residue theorem to compute f1 ,
Z+∞
1
1 X
ν̃ (x)
f1 =
−
|ωn |−1
dx n 2
U
2β ω
(1 + x )
n
=
−∞
1
1
2πi β X
−
×
×
|2n + 1|−1 ν̃n (i).
U
2β
2i
π n∈Z
(7.81)
The summation over fermionic Matsubara frequencies is divergent. To
regulate this divergence, introduce as a cutoff the Debye frequency
ωD > 0 above which the effective interaction is expected to become
repulsive. Let nD be the smallest positive integer with 0 < ωD < ωnD ,
π
nD := inf n = 0, 1, 2, · · · ωD < (2n + 1) ,
(7.82)
n
β
7.3. TIME- AND SPACE-INDEPENDENT LANDAU-GINZBURG ACTION 363
and substitute
|n|<n
1
νF XD
f1 → −
|2n + 1|−1
U
2 n∈Z
=
1
− νF
U
nD −1
X
(7.83)
|2n + 1|−1 .
n=0
Here, we used the Debye cutoff and assumed that the density of states
is a slowly varying function of energy on the scale of the Debye energy
whose Taylor expansion at the Fermi energy starts from a non-vanishing
value ν̃(0) ≡ νF , to replace the density of state ν̃n (i) = ν̃(iωn ) by its
value at the Fermi energy,
∂ ν̃(ξ) ν̃(ξ) = ν̃(0) +
ξ + ··· .
(7.84)
∂ξ 0
Since
β ωD
+ O(β 0 ),
(7.85)
2π
we can use the asymptotic formula from Eq. (0.132) of Ref. [57] where
γ = 0.5772 . . . is Euler’s constant and γ 0 := eγ to write
nD ∼
j
X
k=1
(2k − 1)−1 =
1
[ln j + ln (4γ 0 )] + O(j −2 )
2
(7.86)
at sufficiently low temperatures. Hence, (kB = 1)
νF
β ωD
β ωD
1
νF
2
0
ln
ln (4γ ) = 0 ⇐⇒ ln
− ln(4γ 0 )
f1 → −
−
=
U
2
2π
2
2π
U νF
π
⇐⇒ βc =
e+2/(νF U )
2γ 0 ωD
0
2γ ωD
e−2/(νF U ) ,
⇐⇒ Tc =
π
(7.87a)
i.e.,
T
f1 →
ln
.
(7.87b)
2
Tc
The transition temperature obtained from Eq. (7.87a) will be seen to
agree with the transition temperature (7.108).
The temperature dependence of fj , j = 2, 3, · · · , follows from writing
2j−1 X
∞
∞
X
β
−1
−2j+1
−1
β
ωn
(2n + 1)−2j+1
=β
π
n=0
n=0
(7.88)
2(j−1)
1
β
= ×
2−2j+1 ζ(2j − 1, 1/2).
π
π
ν F
7.
364SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
Here, the Riemann zeta function is defined by (see section 9.5 of Ref.
[57])
ζ(z, q) :=
+∞
X
(n + q)−z ,
Re z > 1,
q 6= 0, −1, −2, −3, · · · .
n=0
(7.89)
2
Since fj+1 /fj ∝ β , the expansion parameter is
β 2 ∆∗ ∆.
(7.90)
For the expansion of Fβ,µ (∆∗ , ∆) to be good,
∆∗ ∆
T2
(7.91)
must be small. This is trivially so when
0 = ∆∗ = ∆.
(7.92)
We know that the order parameters ∆∗ and ∆ saturate to non-vanishing
values upon approaching T = 0. Hence, the expansion that we chose
to perform on Fβ,µ (∆∗ , ∆) must break down arbitrarily close to T = 0.
The full temperature dependence of ∆∗ and ∆ follows instead from
requiring that Fβ,µ (∆∗ , ∆) be an extremum, i.e.,
0=
∂Fβ,µ (∆∗ , ∆)
,
∂∆∗
0=
∂Fβ,µ (∆∗ , ∆)
.
∂∆
(7.93)
These two (mean-field) equations can be linearized in the vicinity of Tc
if Fβ,µ (∆∗ , ∆) is truncated up to quadratic order in ∆∗ ∆, for ∆∗ ∆/T 2
is small near Tc . A solution for T . Tc to
0 ≈ f1 ∆ + 2f2 ∆∗ ∆2 ,
0 ≈ f1 ∆∗ + 2f2 (∆∗ )2 ∆,
(7.94a)
is
f
(∆ ∆)(T ) ≈ − 1 ∝ −T 2 ln
2f2
∗
T
Tc
.
(7.94b)
Thus,
∗
T −2 (∆ ∆)(T ) ≈


−(T − Tc )/Tc , T < Tc ,

0,
(7.95)
T > Tc ,
in the vicinity of Tc . Away from Tc , one cannot truncate the expansion
of Fβ,µ (∆∗ , ∆) in powers of T −2 (∆∗ ∆) to extract the dependence on T
of the order parameters.
7.4. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY
365
7.4. Mean-field theory of superconductivity
In this section, we derive the full temperature dependence of the
order parameters ∆∗ and ∆ that minimize the intensive grand canonical
potential defined by Eq. (7.67). Good references for this material can
be found in Refs. [9] and [75]. Starting point is the first line of Eq.
(7.49) with the substitution
i∆∗
√ 00 −→ ∆∗ ,
βV
i∆
√ 00 −→ ∆,
βV
(7.96)
and without the restriction on the summation over fermionic momenta,
1 ∗
1 XX
−iωn + ξK
∆
∗
Fβ,µ (∆ , ∆) = ∆ ∆−
log det
.
∆∗
−iωn − ξK
U
βV ω K
n
(7.97)
The dependence on temperature of the uniform and static order parameters ∆∗ and ∆ is obtained by requiring that Fβ,µ (∆∗ , ∆) is an
∗
extremum at the mean-field values ∆ and ∆ (below, we omit the
overline for notational simplicity). Thus, one must solve the saddle or
mean-field equations (7.93), i.e.,
∆
U XX
,
∆ =
2
2
βV ω K ωn + ξK + ∆∗ ∆
n
(7.98)
X
X
U
∆∗
∗
∆ =
.
2
βV ω K ωn2 + ξK
+ ∆∗ ∆
n
Define the quasiparticle excitation spectrum
2
2
EK
:= ξK
+ ∆∗ ∆.
(7.99)
The 2 × 2 particle-hole grading of the kernel entering the fermionic
determinant results in the existence of two branches of quasiparticle
excitations,
q
EK,± := ±
2
ξK
+ ∆∗ ∆ ≡ ±EK .
(7.100)
In terms of the quasiparticle excitation spectrum, the saddle-point
equations reduce to the single equation
X Z dz f˜ (z)
FD
2U
1 = (−1)
2
V K
2πi z 2 − EK
ΓK
U
=
V
XZ
K Γ
K
dz
f˜FD (z)
.
2πi (z − EK ) (z + EK )
Here, the Fermi-Dirac distribution function
1
f˜FD (z) = βz
,
z ∈ C,
e +1
(7.101a)
(7.101b)
7.
366SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
with its equidistant first-order poles at
zn = iωn ,
n ∈ Z,
(7.101c)
1
β
(7.101d)
with residues
Res f˜FD (z)
=−
iωn
was introduced (see Fig. 1). For any given K, ΓK is the path running
antiparallel to the imaginary axis infinitesimally close to its left and
parallel to the imaginary axis infinitesimally close to its right, i.e., it
goes around the imaginary axis in a counterclockwise fashion. Let
∂U±EK be two small circles centered about the quasiparticle excitation
energies ±EK , respectively. The path ΓK can be deformed into the two
closed path ∂U±EK of clockwise orientation without crossing any pole
in the complex plane z ∈ C. The saddle-point condition thus becomes
Z
U XX
dz
f˜FD (z)
1 =
V K ±
2πi (z − EK ) (z + EK )
∂U±E
K
U XX
= −
V K ±
f˜FD (z)
z ± EK
!
z=±EK
U X f˜FD (−EK ) − f˜FD (+EK )
= +
V K
2EK
U X tanh (βEK /2)
= +
V K
2EK
p
+ωD
Z
2
∗
ν̃(ξ) tanh β ξ + ∆ ∆/2
U
p
dξ
=
2
2
ξ 2 + ∆∗ ∆
−ωD
p
tanh β ξ 2 + ∆∗ ∆/2
p
dξ
.
ξ 2 + ∆∗ ∆
+ωD
≈ U
νF
2
Z
0
(7.102)
At zero temperature, Eq. (7.102) reduces to
+ωD
ν
1 ≈ U F
2
Z
1
dξ p
+ ∆∗ ∆
νF
ωD
= U
arcsinh
,
2
|∆|
ξ2
0
i.e.,
|∆(T = 0)| ≈
sinh
ωD
2
U νF
.
(7.103a)
(7.103b)
7.4. MEAN-FIELD THEORY OF SUPERCONDUCTIVITY
367
Im z
i!n
K
Re z
@U
@U+EK
EK
Figure 1. At non-vanishing temperatures, a summation over discrete fermionic Matsubara frequencies is converted into and integral along path ΓK in the complex
plane with the Fermi-Dirac distribution multiplying the
summand. Integration along ΓK can be performed by
deforming ΓK into ∂U−EK ∪ ∂U+EK if the summand has
poles at the quasiparticle energies ±EK .
In the so-called weak coupling limit,
U νF 1,
(7.104a)
2
|∆(T = 0)| ≈ 2ωD exp −
.
U νF
(7.104b)
Eq. (7.103b) reduces to
This expression coincides with Eqs. (7.63) and (7.77b) and resembles
the equation (7.87a) for Tc . 4
The self-consistent critical temperature at which
0 = ∆∗ (Tc ) = ∆(Tc )
(7.106)
4
The factor of 2 in the gap equation (7.104b) would not be present in Eq.
(7.77b) had we used the expansion (see Eq. 4.6.31 of [74])
1
1×3
1×3×5
−
+
+ ··· ,
2 × 2z 2
2 × 4 × 4z 4
2 × 4 × 6 × 6z 6
= log( z) + O z 0 ,
arcsinh(z) = log(2z) +
|z| 1,
(7.105)
to derive Eq. (7.77b). Clearly, the numerical prefactor to the exponential is not to
be taken seriously at this level (logarithmic accuracy) of approximation.
7.
368SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
is defined by
p
tanh βc ξ 2 /2
p
dξ ν̃(ξ)
ξ2
p
+ωD
Z
tanh βc ξ 2 /2
p
dξ
ξ2
+ωD
1=
U
4
Z
−ωD
≈
U νF
2
0
+βc ωD /2
U νF
=
2
Z
tanh x
x
0
0
U νF
2γ βc ωD
≈
ln
,
2
π
dx
γ 0 = eγ , γ Euler’s constant and
in the limit βc ωD 1. Thus [compare with Eq. (7.87a)],
0
2γ ωD
Tc ≈
e−2/(U νF ) .
π
2γ 0
≈ 1.13,
π
(7.107)
(7.108)
For 0 ≤ T ≤ Tc , the saddle-point equation (7.102) must be solved
numerically. Approaching Tc from below, Eq. (7.95) implies that
1/2
∆(T ) T
.
(7.109)
∆(T = 0) ∝ 1 − T
c
The exponent 1/2 is a trademark of the mean-field approximation. For
example, the dependence on temperature of the magnetization
q in the
mean-field approximation to the Ising model also behaves like 1 − TT
c
in the vicinity of the transition temperature from the ferromagnetic to
paramagnetic state. Above Tc no non-vanishing solution to the “gap
equation” (7.102) exists.
The single-particle Hamiltonian
 2

− ∇
+
µ
⊗
σ
∆(R)
⊗
(+iσ
)
0
2m
2
2 
HBdG := 
(7.110)
∇
∗
∆ (R) ⊗ (−iσ2 ) + 2m + µ ⊗ σ0
that enters the fermionic determinant (7.97) through
00
Sβ,µ
(∆∗ , ∆)
Zβ
:=
Z
dτ
0
d3 R
(∆∗ ∆)(R, τ ) 1
− log [Det (γ0 ⊗ σ0 ∂τ + HBdG )] ,
U
2
V
(7.111)
is called the Bogoliubov-de-Gennes Hamiltonian. Here, two gradings
are displayed explicitly. There is a particle-hole grading generated by
the Pauli matrices γ1 , γ2 , and γ3 together with the identity 2×2 matrix
γ0 , and the spin-1/2 grading generated by the Pauli matrices σ1 , σ2 ,
7.5. NAMBU-GORK’OV REPRESENTATION
369
and σ3 together with the identity 2 × 2 matrix σ0 . This is a redundant
representation. Correspondingly, there is a factor of 1/2 in front of the
logarithm of the determinant. The Bogoliubov-de-Gennes Hamiltonian
HBdG is of the general form
+K +D
HBdG :=
,
K = K† ,
D = −DT .
(7.112)
+D† −KT
When the superconducting order parameters ∆∗ (R) and ∆(R) that
00
enter in HBdG are chosen so as to minimize the effective action Sβ,µ
(∆∗ , ∆),
the single-particle eigenstates of HBdG are used to construct the meanfield superconducting ground state as well as excitations above it. The
single-particle eigenstates of HBdG are called quasiparticles since they
are not created by the original creation electron operators, but by linear
combinations of the original creation and annihilation electron operators (in the same spirit as with the Bogoliubov transformation from
section 2.4.1).
The spectrum of HBdG is characterized by the presence of a gap.
For an order parameter uniform or homogeneous in space,
∆∗ (R) = ∆∗ ,
∆(R) = ∆,
(7.113)
the gap is the same around the entire Fermi surface. For order parameters that vary in momentum space, the gap varies in magnitude and
even in sign around the Fermi surface.
Another unique property of HBdG is that its eigenvalues occur in
pairs of opposite sign. This is because HBdG , in addition to being
Hermitean,
HBdG = (HBdG )† ,
(7.114)
also obeys the transformation law
γ1 (HBdG )T γ1 = −HBdG .
(7.115)
The antiunitary transformation law on the left-hand side of Eq. (7.115)
defines a particle-hole transformation. This transformation implies a
spectral symmetry of the spectrum of HBdG by which the application
of the particle-hole transformation on any eigenstate of HBdG with the
non-vanishing single-particle energy ε delivers an eigenstate of HBdG
with the non-vanishing single-particle energy −ε.
7.5. Nambu-Gork’ov representation
Presuming an instability of the Fermi sea towards a ground state
acquiring a non-vanishing expectation value for the pairing fields
Φ∗↑↓ (R, τ ) := lim ψ↑∗ (r 1 , τ )ψ↓∗ (r 2 , τ ),
r 1 →r 2
Φ↑↓ (R, τ ) := lim ψ↓ (r 2 , τ )ψ↑ (r 1 , τ ),
r 1 →r 2
(7.116a)
whereby
R :=
r1 + r2
,
2
r := r 1 − r 2 ,
(7.116b)
7.
370SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
i.e., a superconducting instability, we chose in section 7.3 to decouple
an instantaneous and attractive contact two-body interaction
− U δ(r),
U > 0,
(7.117)
by presenting the partition function in the grand-canonical ensemble
according to Eq. (7.64). Taking note of the fact that Grassmann num∗
bers ψσKω
and ψσ0 K 0 ωn0 anticommute for all σ, K, ωn , σ 0 , K 0 , and ωn0 ,
n
0
the fermionic contribution S0 + SU0 to the action Sβ,µ
can equally well
be written
Zβ
Z
0
S0 + SU ≡ dτ d3 R (L0 + LU ) ,
(7.118a)
0
with the Lagrangian densities
V
5
∂τ − ∇2 − µ
ψ
(R,
τ
)
∆(R,
τ
)
↑
2m
ψ↓ (R, τ )
L0 +LU ≡
.
2
ψ↓∗ (R, τ )
+µ
∆∗ (R, τ )
∂τ + ∇
2m
(7.118b)
The 2 × 2 grading introduced in Eq. (7.118b) is called the particle-hole
grading. If one introduces the two independent Nambu spinors
(7.119a)
Ψ† (R, τ ) ≡ ψ↑∗ (R, τ ) ψ↓ (R, τ ) ,
ψ↑∗ (R, τ )
and
ψ↑ (R, τ )
Ψ(R, τ ) ≡
,
ψ↓∗ (R, τ )
then Eq. (7.118b) takes the compact form
L0 + LU = Ψ† (R, τ ) K∆∗ ,∆ Ψ(R, τ ).
(7.119b)
(7.119c)
Recalling that the polar decompositions of the pairing-order parameters
∆∗ and ∆ are
∆∗ (R, τ ) = |∆(R, τ )|e−iφ(R,τ ) ,
∆(R, τ ) = |∆(R, τ )|eiφ(R,τ ) ,
(7.120a)
respectively, the kernel K∆∗ ,∆ can be represented by
∇2
γ + iγ2
K∆∗ ,∆ = γ0 ∂τ + γ3 −
−µ + 1
|∆(R, τ )|eiφ(R,τ )
2m
2
(7.120b)
γ1 − iγ2
+
|∆(R, τ )|e−iφ(R,τ ) ,
2
in imaginary time τ , center-of-mass coordinates R, and with the matrices
1 0
0 1
0 −i
+1 0
γ0 :=
,
γ1 :=
,
γ2 :=
,
γ3 :=
,
0 1
1 0
+i 0
0 −1
(7.120c)
in the particle-hole grading.
5
The total derivatives that arise from the use of partial integration drop out
with the choice made for the boundary conditions.
7.6. EFFECTIVE ACTION FOR THE PAIRING-ORDER PARAMETER
S
HubbardStratonovich
,µ
S 0 ,µ
fermionic
integration
371
S 00,µ
Figure 2. Strategy used to construct effective action
for order parameter: (i) Start from pure fermionic action
with quartic and fermionic interaction. (ii) Introduce
order parameter by decoupling four-fermion interaction
through Hubbard-Stratonovich transformation. (iii) Integrate fermions in background of order parameter field.
The advantage of the Nambu representation, aside from its compactness, is that it displays explicitly the important property that all
the non-vanishing eigenvalues of the single-particle Hermitean Hamiltonian
H∆∗ ,∆ := K∆∗ ,∆ − γ0 ∂τ
(7.121)
come in pairs of opposite sign, because of Eq. (7.115). Moreover, because of the algebra obeyed by the Pauli matrices
γ = γ1 γ2 γ3 ,
γi γj = iijk γk ,
{γi , γj } = 2δij γ0 ,
i, j, k = 1, 2, 3,
(7.122)
the square of the Hamiltonian H∆∗ ,∆ takes the form
"
#
2
2
2
∇2
H∆∗ ,∆ ≡ K∆∗ ,∆ − γ0 ∂τ = γ0
−
− µ + |∆(R, τ )|2 .
2m
(7.123)
Since the square of an Hermitean operator is a positive operator, H 2 =
H † H = H H † (although not necessarily a positive definite one), the
2
explicit representation (7.123) of H∆∗ ,∆ allows to solve for the eigenvalues of H∆∗ ,∆ when the order parameter is independent of R and τ .
Indeed, if
∆∗ (R, τ ) = |∆| e−iφ ,
∆(R, τ ) = |∆| e+iφ ,
∀R, τ,
(7.124)
we then immediately recover the mean-field quasiparticle spectrum
q
2
EK,± := ± ξK
+ |∆|2 ≡ ±EK
(7.125)
of H∆∗ ,∆ that we derived in Eq. (7.100).
7.6. Effective action for the pairing-order parameter
We now repeat the strategy of section 6.4 that consists in the com00
putation of the effective action Sβ,µ
for the order parameter obtained
by integrating out all the fermions (see Fig. 2), i.e., we are after the
action Sfred in the effective action
00
Sβ,µ
= Scond + Sfred
(7.126a)
7.
372SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
from the partition function
Z
00
Zβ,µ ∝ D[∆∗ , ∆] exp −Sβ,µ
.
The fermionic determinant
exp (−Sfred ) := Det K∆∗ ,∆


2
−µ
∆(R, τ )
∂τ − ∇
2m

= Det 
∇2
∗
∆ (R, τ )
∂τ + 2m + µ
(7.126b)
(7.127)
is known as a Fredholm determinant in mathematics.
00
We have calculated Sβ,µ
exactly when the order parameter is space
and time independent [as in Eq. (7.124)] and provided the limit of infinite volume and vanishing temperature has been taken, see Eqs. (7.67)
00
and (7.76). Provided |∆| ωD εF holds, we found that Sβ,µ
only
∗
depends on the product ∆ ∆,
1
1 νF
|∆|
00
lim Sβ,µ ≈ lim βV
+
2 ln
− 1 |∆|2 .
β,V →∞
β,V →∞
U
2 2
2ωD
(7.128)
00
Consequently, Sβ,µ is independent of the phase φ of ∆ in this limit. The
00
classical equation of motion for the action Sβ,µ
yield the gap equation,
|∆| ≈ 2 ωD e−2/(U νF ) ,
(7.129)
in the weak coupling limit U νF 1. At non-vanishing temperature, we
00
solved for the saddle-point of Sβ,µ
and found a transition temperature
Tc ≈ 1.13 ωD e−2/(U νF ) ,
(7.130)
above which the magnitude of the pairing-order parameter vanishes.
00
Approaching the transition temperature from below, we expanded Sβ,µ
in powers of the squared magnitude of the pairing-order parameter,
thereby deriving the so-called Ginzburg-Landau free energy,
!
+∞
X
00
lim Sβ,µ
= lim βV
fj |∆|2j ,
(7.131)
V →∞
V →∞
j=0
where f1 ∼ ln(T /Tc ), fj ≥ 0, j = 2, 4, 6, · · · , and fj ≤ 0, j =
00
3, 5, 7, · · · . In the following, we are going to expand Sβ,µ
about the
mean-field solution |∆| to account for fluctuations of the pairing-order
parameter that vary in space and time in the vicinity of T = 0 and
T = Tc , respectively.
7.7. Effective theory in the vicinity of T = 0
Needed is the evaluation of the determinant (7.127) in the vicinity
of T = 0. The first question to address is how should we parametrize
the pairing-order parameter? To answer this question we shall rely on
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
373
the fact that Eqs. (7.128) and (7.129) do not depend on the phase φ
of the pairing-order parameter ∆ = |∆|eiφ . We shall thus choose the
non-linear parametrization [compare with Eq. (2.119)]
∆∗ (R, τ ) = |∆| e−2iθ(R,τ ) ,
∆(R, τ ) = |∆|e+2iθ(R,τ ) ,
(7.132)
of the manifold for the pairing-order parameter that minimizes Eq. (7.128).
By freezing the magnitude of the pairing-order parameter and only allowing space-time fluctuations θ(R, τ ) of the phase of the pairing-order
parameters around the mean-field value
∀R, τ,
θ(R, τ ) = θ̄ = 0,
(7.133)
it is insured that the contribution Scond is minimized. Indeed, the
probability weight for configuration (7.132) is only suppressed by the
factor


2
−µ
|∆|e+2iθ(R,τ )
∂τ − ∇
2m

Det 
2
Det K−2θ,+2θ
|∆|e−2iθ(R,τ )
∂τ + ∇
+µ
2m
≡


2
∇
Det K−2θ=0,+2θ=0
∂τ − 2m − µ
|∆|

Det 
|∆|
∂τ +
∇2
2m
+µ
(7.134)
compared to the probability weight of the saddle-point configuration.
To put it differently, collective excitations associated to space-time fluctuations of the magnitude of the pairing-order parameter cannot have
energies below the mean-field gap, whereas collective excitations associated to space-time fluctuations of the phase of the pairing-order parameter can. At energies well below the zero-temperature gap opened
by the superconducting order, we can neglect space-time fluctuations
in the magnitude of the pairing-order parameter, but we must account
for the space-time fluctuations of its phase. Before undertaking a direct
evaluation of Eq. (7.134), we look at some simpler limiting cases.
7.7.1. Spatial twist around |∆|. Choose in Eq. (7.132)
θ(R, τ ) = Qs · R,
b ≡ mV Q
b .
Qs = |Qs | Q
s
s
s
(7.135)
To evaluate

Det K−2θ,+2θ θ=Qs ·R
∂τ −
∇2
2m

|∆| exp (+2iQs · R)
,
−µ
:= Det 
2
∂τ + ∇
+µ
2m
(7.136)
|∆| exp (−2iQs · R)
perform the unitary (gauge) transformation
0
K−2θ,+2θ θ=Q ·R → K−2θ,+2θ
θ=Q
s
s ·R
,
(7.137a)
7.
374SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
where
0
K−2θ,+2θ
−1
K−2θ,+2θ U−θ,+θ := U−θ,+θ

2
s)
∂τ − (∇+iQ
−µ
2m

=
|∆|
θ=Qs ·R

|∆|
∂τ +
(∇−iQs )2
2m
(7.137b)

+µ
and
U−θ,+θ θ=Qs ·R
=
e+iQs ·R
0
.
0
e−iQs ·R
(7.137c)
The identity
Det K−2θ,+2θ θ=Q
s ·R
0
= Det K−2θ,+2θ
θ=Q
s ·R
(7.138)
has very important physical consequences. One might be tempted to
believe that this identity is a straightforward generalization of the invariance under a unitary transformation of the determinant of a matrix.
This is not so however. It is one feature of field theory, from which new
physics becomes possible, that an identity such as Eq. (7.138) is highly
non-trivial. Proving Eq. (7.138) amounts to proving the absence of the
chiral anomaly in a non-relativistic quantum field theory. We do not
provide a proof of Eq. (7.138) in this book. Instead, we assume that
Eq. (7.138) holds.
To proceed, we first make use of
(∇ ± iQs )2
∇2
Q · ∇ Q2s
∓ µ = ∂τ ∓
−i s
±
∓ µ.
(7.139)
2m
2m
m
2m
Second, we make use of
(∇ ± iQs )2
−i(K·R−ωn τ )
e
∓ µ e+i(K·R−ωn τ ) =
∂τ ∓
2m
K 2 Qs · K
Q2s
− iωn ±
+
±
∓ µ.
2m
m
2m
(7.140)
∂τ ∓
Third, we infer that, in a very large volume and at very low temperatures,
0
≈
log Det K−2θ,+2θ
Z 3 Z
2
dK
dω
−iω 0 + K
− µ0
|∆|
2m
βV
log det
,
2
(2π)3
2π
|∆|
−iω 0 − K
+ µ0
2m
V
R
(7.141a)
where we have introduced
− iω 0 = −iω +
Qs · K
,
m
−µ0 = −µ +
Q2s
.
2m
(7.141b)
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
375
Fourth, we assume that the integration over the fermionic Matsubara
frequency ω, whereby
Q ·K
Q ·K
iω = iω 0 + s
⇐⇒ ω = ω 0 − i s
,
(7.142)
m
m
can be done by shifting the path of integration in the complex-Matsubarafrequency plane from the real axis R to the horizontal line
Q ·K
∗
γK
:= R − i s
(7.143)
m
without encountering any singularity of the integrand (branch cuts).
This assumption is verified when (Λ is a momentum cutoff 6)
Qs · K (7.144a)
m < Vs Λ |∆|,
but breaks down, as we shall see after Eq. (7.152) in more details, when
Qs · K (7.144b)
m < Vs Λ / |∆|.
Here,
|Qs |
.
m
If so, Eqs. (7.127) and (7.138) can be used to deduce
0
Sfred |∆|e−2iQs ·R , |∆|e+2iQs ·R = Sfred
|∆|, |∆| ,
Vs ≡
(7.144c)
(7.145a)
whereby
0
0
Sfred
|∆|, |∆| := − log Det K−2θ,+2θ
θ=Qs ·R
!
2
2
Z 3 Z
dK
dω
K
log −ω 02 −
− µ0 − |∆|2
≈ − βV
3
(2π)
2π
2m
V
R
(7.145b)
for large volumes and low temperatures. Furthermore,
!
2
2
Z 3 Z
d
K
dω
K
0
Sfred
|∆|, |∆| ≈ − βV
log −ω 02 −
− µ0 − |∆|2
3
(2π)
2π
2m
∗
γK
V
Z
= − βV
V
d3 K
(2π)3
Z
dω 0
log −ω 02 −
2π
K2
− µ0
2m
2
!
− |∆|2
R
(7.146)
6
Remember that one should always impose a momentum cutoff, say Λ, that
arises from the band width or from the Debye frequency. Hence, |K| can be replaced by Λ in Eqs. (7.144a) and (7.144b), in which case the validity of shifting
the Matsubara frequency integral into the complex plane holds uniformly in the
fermionic momentum K.
,
7.
376SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
since we have assumed that no branch cuts has been crossed when
changing the integration variable in the complex plane of frequencies.
Taylor expansion in powers of µ0 − µ = −Q2s /2m then delivers
"
2 #
Q2s
∂Sfred Q2s
0
Sfred |∆|, |∆| ≈ Sfred |∆|, |∆| +
−
.
+O
2m
∂µ Qs =0
2m
(7.147)
If the definition of N = β −1 ∂µ ln Zβ,µ is used, the final result for
Eq. (7.145) becomes
"
#
2
2 2
Q
Q
N
s
s
0
Sfred
|∆|, |∆| ≈ Sfred |∆|, |∆| + βV
+O
. (7.148)
V 2m
2m
This is what is expected for a steady uniform flow with velocity Vs =
|Qs |/m of the entire pairing or superconducting condensate.
The physical interpretation of a spatial twist of the order parameter
is the following. Remember that the partition function Zβ,µ describes
a system in statistical equilibrium with a reservoir, i.e., the system exchanges energy and particle number with the reservoir whereby β and
µ determines the steady average energy and the steady average particle number stored in the system, respectively. Imagine the reservoir as
being the walls of a container within which the electrons are interacting. Electrons can go in and out of the container but on average there
are N electrons in the container at all times. Equations (7.132) and
(7.135) say that the center of mass of all paired electrons participating to the condensate has momentum 2Qs , i.e., paired electrons have
momenta Qs ± K, respectively. The unitary transformation (7.137c)
is nothing but a Galilean boost into the rest frame of the condensate
that is moving with respect to the walls of the reservoir. 7 Whereas
unpaired electrons have Matsubara frequencies ω and energies K 2 /2m
in the frame of reference of the container, Eq. (7.142) tells that, in the
rest frame of the condensate, mean-field quasiparticles have Dopplerb · K and Eq. (7.146) tells that
shifted Matsubara frequencies ω + iVs Q
s
7 A Galilean transformation is a transformation to a new frame of reference
that leaves the time difference |t1 − t2 | and space separation |x1 − x2 | unchanged
as well as the equation of motion mẍ = 0 of a free particle form invariant. It is
given by
t0 = ±t + a,
x0 = Ox + vt + w,
v, w ∈ R3 ,
O ∈ O(3).
(7.149a)
The transformation law of momentum p ≡ mẋ and kinetic energy Ekin ≡ (1/2)mẋ2
under a Galilean transformation are
p0 = Op + mv,
a ∈ R,
0
Ekin
= Ekin + m (Oẋ) · v + (1/2)m v 2 .
(7.149b)
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
quasiparticles have energies
q
0
0
EK,± := ± (K 2 /2m − µ0 )2 + |∆|2 ≡ ±EK
,
377
(7.150)
respectively. The second term on the right-hand side of Eq. (7.147), i.e.,
the use of the chemical potential µ, says that statistical equilibrium is
defined with respect to the stationary walls of the container and that
it is the quasiparticle states with negative energies as seen from the
container that are occupied [recall Eq. (7.141b) and use Eq. (7.153)
with Qs = 0].
As soon as the speed Vs of the condensate with respect to the walls
of the container reaches the critical value set by the quasiparticle gap
|∆|, breaking of some electron pairs in the moving condensate takes
place. These unpaired electrons are said to be “normal” and they do
not flow along with the condensate. Phenomenologically, the second
term on the right-hand side of Eq. (7.148) should then become
βV
N Q2s
N Q2
N Q2s
−→ βV s s < βV
.
V 2m
V 2m
V 2m
(7.151)
The fermionic density N/V has been replaced by the smaller density
Ns /V with Ns < N the depleted number of electrons that still participate to the pairing condensate.
The justification of Eq. (7.151) is based on the observation that,
as soon as |Qs |Λ/m is larger than the quasiparticle gap |∆|, the tran∗
sition (7.146) is not legal anymore as the shift R → γK
of path of
integration in the complex-valued frequency plane encounters branch
cuts of the logarithm in the integrand. A related difficulty also arises if
we represent the sum over the fermionic Matsubara frequencies [which
at low temperatures becomes an integral as in Eq. (7.145b)] as an integral in the complex-frequency plane that picks up the residues of the
Fermi-Dirac distribution, see Fig. 1. Indeed, in doing so, the right-hand
side of Eq. (7.145b) becomes
X Z dz
XX 02
02
02
ln ωn + EK =
β
−
log (−iz)2 + EK
f˜FD (z)
2πi
K ω
K
n
ΓK
X Z dz
0
0
[log (EK
− z) + log (EK
+ z)] f˜FD (z).
=β
2πi
K
ΓK
(7.152)
For a given K in the sum on the right-hand side, the integrand in
0
the complex z-plane has branch cuts whenever |Re z| > EK
. Hence,
it is not permissible to deform the path ΓK into semi-circles enclosing
isolated poles as is done in Fig. 1. What can be done, however, is
to perform a Taylor expansion in powers of Q2s /(2m) of the integrand
that converts the branch cuts into isolated poles. In doing so, one
7.
378SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
replaces the term
2
2
EK
= ξK
+ |∆|2 ,
XX
K
−
∂Sfred
∂µ
ξK =
2
∂µ ln ωn02 + EK
K2
2m
in Eq. (7.147) by (remember that
Qs =0
− µ)
=
XX
=
XX
ωn
ωn
K
ωn
K
=
X
Z
β
K
ΓK
2
∂µ EK
2
ωn02 + EK
(−2)ξK
2
Q ·K
2
ωn + i sm
+ EK
(−1)2 2ξK f˜FD (z)
dz
2
2πi
Q ·K
2
−iz + i sm
+ EK
dz (−1)3 2ξK f˜FD (z)
2
2πi
Q ·K
2
K
z − sm
− EK
ΓK
Z
X
dz
f˜FD (z)
= β (−1)3 2ξK
2πi z − Qs ·K − |E | z −
K
K
m
Γ
=
X
Z
β
K
= β
X
(−1)4 ξK
K
Qs ·K
m
+ |EK |
X f˜FD ( Qs ·K ± |EK |)
m
(±)|E
K|
±
≡ βNs .
(7.153)
Relative to the limit Qs → 0, the Fermi-Dirac distribution in Eq. (7.153)
removes from the summation over K all those contributions such that
Qs · K
− |EK | > 0,
(7.154a)
m
while it adds to the summation over K all those contributions such
that
Qs · K
+ |EK | < 0.
(7.154b)
m
The net result when |V s | = |Qs |/m is above the critical velocity set
by the mean-field gap is the depletion N − Ns > 0 in the number of
electrons participating to the condensate.
7.7.2. Time twist around |∆|. Choose in Eq. (7.132)
θ(R, τ ) = −Ωτ.
(7.155)
To evaluate
2
∂τ − ∇
−µ
|∆| exp (−2iΩτ )
2m
:= Det
,
2
θ=−Ωτ
|∆| exp (+2iΩτ )
∂τ + ∇
+µ
2m
(7.156)
perform the unitary (gauge) transformation
0
K−2θ,+2θ θ=−Ωτ → K−2θ,+2θ
,
(7.157a)
θ=−Ωτ
Det K−2θ,+2θ 7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
379
where
0
−1
K−2θ,+2θ
:=
(U
)
K
U
−θ,+θ
−2θ,+2θ −θ,+θ θ=−Ωτ
θ=−Ωτ
2
−
µ
|∆|
∂τ − iΩ − ∇
2m
,
=
2
∂τ + iΩ + ∇
+µ
|∆|
2m
(7.157b)
and
U−θ,+θ θ=−Ωτ
−iΩτ
e
0
=
.
0
e+iΩτ
As before, it can be shown that
0
Det K−2θ,+2θ θ=−Ωτ = Det K−2θ,+2θ
θ=−Ωτ
(7.157c)
(7.158)
holds. Thus, in a very large volume and at very low temperatures
0
log Det K−2θ,+2θ = log Det K−2θ,+2θ
Z 3 Z
2
dK
dω
− µ0
|∆|
−iω + K
2m
≈ βV
log det
,
2
(2π)3
2π
|∆|
−iω − K
+ µ0
2m
V
R
(7.159a)
where we have introduced
µ0 = µ + iΩ.
0
|∆|, |∆| with
If so, − log Det K−2θ,+2θ θ=−Ωτ = Sfred
(7.159b)
0
0
Sfred
|∆|, |∆| := − log Det K−2θ,+2θ
θ=−Ωτ
!
2
2
Z 3 Z
dK
dω
K
≈ −βV
log −ω 2 −
− µ0 − |∆|2
(2π)3
2π
2m
V
R
∂Sfred = Sfred |∆|, |∆| − iΩ −
+ O Ω2
∂µ
θ=0
N
= Sfred |∆|, |∆| − iβV
Ω + O Ω2 .
(7.160)
V
The last equality follows from the definition N = β −1 ∂µ ln Zβ,µ .
7.7.3. Conjectured low-energy action for the phase of the
condensate. Choose in Eq. (7.132)
1
θ(R, τ ) = φ(R, τ ).
2
(7.161)
7.
380SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
Combination of Eqs. (7.148) and (7.160) yields at low temperatures
and in a very large volume
Sfred |∆|e+iφ(R,τ ) , |∆|e−iφ(R,τ ) − Sfred |∆|, |∆| ≈
)
(
2
Z
Zβ
∂
φ
1
N
∇φ
N
τ
+
d3 R dτ i
+ O (∂τ φ)2 , (∇φ · ∇φ)2 .
V
2
2m V
2
V
0
(7.162)
Analytical continuation
τ = it,
t ∈ R,
(7.163)
gives
− lim Sfred |∆|e+iφ(R,τ ) , |∆|e−iφ(R,τ ) − Sfred |∆|, |∆| ≈
τ →it
Z
i
V
)
2
Z−iβ (
N
∂
φ
N
∇φ
1
t
dt −
d3 R
−
+ O (∂t φ)2 , (∇φ · ∇φ)2 .
V
2
2m V
2
0
(7.164)
Observe that Eq. (7.164) is left unchanged if
1 2
φ(R, it) −→ φ(R, it) + 2 Qs · R −
Q t .
2m s
(7.165)
(The boundary term vanishes because of the periodic boundary conditions.) This transformation is a quantum counterpart to the classical
Galilean transformation in footnote 7 [see Eqs. (7.149a) and (7.149b)].
Let us relax the assumption that the electronic density is frozen to
the value N/V . It is then tempting to conjecture the Madelung action
defined by
Z
Z
3
SMad [φ, ρ|A0 , A] := d R dt LMad ,
V
LMad = −ρ
∂t φ
− e A0
2
R
1
−
ρ
2m
∇φ
+ eA
2
2
λ
−
8
2
N
ρ−
,
V
(7.166)
for the effective low-energy action describing space- and time-dependent
fluctuations of the phase φ(R, t) of the pairing-order parameter ∆(R, t) =
|∆|e+iφ(R,t) , as well as space- and time-dependent fluctuations of the
electronic density ρ(R, t). Here, A0 (R, t) and A(R, t) are the scalar
and vector potentials of electrodymanics, respectively. They should
be understood as playing the role of classical external sources obeying
Maxwell equations at this stage. The convention e > 0 is chosen for the
electric charge. The positive coupling λ with the dimension of energy
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
381
times volume (in units with ~ = c = 1) freezes the density ρ(R, t) to
the value N/V in the limit λ → ∞.
By defining 8
r
ρ(R, t)
Ξ(R, t) :=
exp iφ(R, t) ,
e∗ := 2e,
m∗ = 2m,
2
(7.167)
Eq. (7.166) can be brought to (we have reinstated Planck’s constant ~
and the speed of light c)
Z
Z
0
3
0
S [Ξ|A0 , A ] = d R dt L,
V
∗
R
∗
L = Ξ (i~ ∂t − (−e
A00 = A0 −
2
2
1 ~
(−e∗ ) 0
λ
1
N
2
−
∇−
A Ξ −
|Ξ| −
,
2m∗ i
c
2
2V
i~ c
A0 = A + ∗ ∇ ln ρ.
2e
(7.168)
) A00 ) Ξ
i~
∂ ln ρ,
2e∗ t
If we ignore the dependence of the gauge fields A00 and A0 on ρ, the
classical equation of motion for Ξ that follows from Eq. (7.168),
2
ie∗
1N
~2
ie∗
2
i~ ∂t −
A Ξ=− ∗ ∇+
A Ξ + λ |Ξ| −
Ξ,
~ 0
2m
~c
2V
(7.169)
is known as the Gross-Pitaevskii non-linear Schrödinger equation for
Ξ. The Gross-Pitaevskii equation is often used as a model for the
motion of the superconductor (superfluid) condensate. Revival in the
interest for the Gross-Pitaevskii equation has ensued the experimental
observation of Bose-Einstein condensation in vapors of rubidium. A recent review on the Gross-Pitaevskii equation can be found in Ref. [76].
Our “derivation” of the Gross-Pitaevskii equation could be misleading
in that we have only been considering space-time dependent fluctuations of the phase of the superconducting order parameter whereas the
Gross-Pitaevskii equation allows for fluctuations of the amplitude of Ξ
as well. A rigorous justification for the Gross-Pitaevskii equation in
the context of superfluidity in three- (two-) dimensional position space
can be found in Ref. [77] ([78]).
Invariance of Eq. (7.168) under any global gauge transformation
Ξ(R, t) → exp(iα) Ξ(R, t),
α ∈ R,
(7.170)
implies the continuity equation
0 = ∂t J0 + ∇ · J ,
8
∆.
(7.171a)
It is important to stress that Ξ is not the same as the pairing-order parameter
7.
382SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
whereby the density and currents are
J0 := Ξ∗ Ξ,
(7.171b)
and
ie∗ 0
~
∗
J :=
Ξ ∇+
A Ξ − c.c. ,
(7.171c)
2m∗ i
~c
respectively. Alternatively, the density J0 and current J are
ρ
ρ
~
e∗
J0 = ,
J = V,
V := ∗ ∇φ +
A ,
(7.172)
2
2
m
~c
if the so-called Madelung transformation (7.167) is undone. 9
When the vector field ∇φ is rotation free, one says that there are
no vortex singularities in Ξ. If so, the vorticity of the velocity field V
is entirely determined by the magnetic field
B = ∇ ∧ A0 = ∇ ∧ A
(7.173)
through
∇∧V =+
e∗
e∗
B
⇐⇒
∇
∧
V
−
B = 0.
m∗ c
m∗ c
(7.174)
The condition (7.174) for the absence of vortices supplemented with
(∇ρ) ∧ V = 0
10
(7.179)
can be combined with Maxwell’s equations
4πe∗
∇∧B =
J,
c
∇ · B = 0,
(7.180)
9
Alternatively, variation of action (7.166) with respect to φ gives the continuity
equation.
10 In a classical fluid, conservation of particle number implies the continuity
equation
dρ
+ ρ (∇ · v) .
dt
In a steady fluid all ∂t vanish, i.e., the continuity equation reduces to
0 = ∂t ρ + ∇ · (ρv) = [∂t ρ + (∇ρ) · v] + ρ (∇ · v) ≡
0 = ∇ · (ρv) .
(7.175)
(7.176)
An incompressible fluid is defined by the conditions that
dρ
,
dt
in which case the continuity equation reduces to
0 = ∂t ρ,
0=
0 = ∇ · v.
(7.177)
(7.178)
Neglecting fluctuations in the magnitude of the pairing-order parameter as was
done to derive Eq. (7.164) implies that the condensate is taken as incompressible,
i.e., ρ is constant.
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
383
and the definition (7.172) of the current to give the equation of motion
4πe∗
0 = ∇∧ ∇∧B−
J
c
ρ 4πe∗
2
= ∇ (∇ · B) − ∇ B −
∇∧
V
c
2
4πe∗ 2 (ρ/2)
2
= − ∇ −
B
(7.181)
m∗ c2
obeyed by a static magnetic field. The length
−1/2
4πe∗ 2 (ρ/2)
λLondon :=
m∗ c2
(7.182)
is called the London penetration depth. The London penetration depth
does not depend on ~. Quantum mechanics enters through ~ only if
vortices are present as is implied by the modification to (7.174) brought
upon by vortices. The London penetration depth controls the exponential decay of a solution to Eq. (7.181). This property that a static
magnetic field becomes massive inside a superconductor is called the
Meissner effect. We are now going to see that the combined effects
of the Meissner effect and of the ability of the condensate to screen
static charges is to provide the photon with an effective mass inside a
superconductor.
7.7.4. Polarization tensor for a BCS superconductor. The
goal of this section is to further substantiate a low-energy effective
theory for the phase φ = 2θ of the pairing-order parameter. To this
end, we modify the partition function (7.64) by coupling the fermions
to the classical gauge fields of electromagnetism (ϕ, A) through the
minimal coupling (in imaginary time)
∇
∇
→
− (−e) A(R, τ ). (7.183)
i
i
We choose the conventions e > 0 for the unit of electric charge, and ~ =
1 and c = 1 for Planck’s constant and the speed of light, respectively.
We are going to use the Nambu-Gork’ov representation introduced in
section 7.5. This is to say that we need the Bogoliubov-de-Gennes
Hamiltonian (7.112), whereby we make the identification
"
#
2
1
∇
+K →+
+ e A(R, τ ) − µ − e ϕ(R, τ ) σ0 ,
2m i
"
#
2
(7.184)
1
∇
− KT → −
− e A(R, τ ) − µ − e ϕ(R, τ ) σ0 ,
2m i
− µ → −µ + (−e) ϕ(R, τ ),
+ D → |∆|e+2iθ(R,τ ) (iσ2 ) .
7.
384SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
Because we chose a pairing-order parameter that is a singlet in SU (2)
spin space, we might as well ignore the spin-1/2 grading to work with
the single-particle Bogoliubov-de-Gennes Hamiltonian [compare with
Eq. (7.110)]
!
)]2
+2iθ(R,τ )
− [∇+ieA(R,τ
−
µ
−
e
ϕ(R,
τ
)
|∆|e
2m
HBdG :=
.
)]2
|∆|e−2iθ(R,τ )
+ [∇−ieA(R,τ
+ µ + e ϕ(R, τ )
2m
(7.185)
Needed is the evaluation to Gaussian order in θ, ϕ, and A in the
zero temperature limit T → 0 of the fermionic determinant
−Sfred [θ,ϕ,A]
e
Z
≡
D[ψ ∗ , ψ] e−(S0 +SU )[θ,ϕ,A]
0
(7.186)
:= Det (∂τ γ0 + HBdG ) .
However, because we would like to compare the Gaussian approximaRPA
tion to the effective theory Sfred [θ, ϕ, A] to the effective action Sβ,µ
defined in Eq. (6.38), we shall perform the analytical continuation
ϕ(R, τ ) ∈ R → −iϕ(R, τ ) ∈ iR
(7.187)
that allows us to interpret ϕ(R, τ ) as the Hubbard-Stratonovich field
that decouples an instantaneous Coulomb interaction. Thus, we shall
perform, at vanishing temperature, the Gaussian approximation of
!
|∆| e+2iθ(R,τ )
Det
A(R,τ )]2
|∆| e
∂τ + [∇−ie 2m
+ µ − ie ϕ(R, τ )
(7.188a)
in the background of the pairing-order parameter
∂τ −
[∇+ie A(R,τ )]2
−µ+
2m
−2iθ(R,τ )
ie ϕ(R, τ )
∆∗ (R, τ ) = |∆|e−2iθ(R,τ ) ,
∆(R, τ ) = |∆|e+2iθ(R,τ ) ,
(7.188b)
and of the Euclidean electromagnetic potentials −iϕ(R, τ ) and A(R, τ ).
The short-hand notation
x ≡ (R, τ ),
K ≡ (K, ωn ),
ωn =
π
(2n + 1),
β
n ∈ Z,
(7.188c)
will be used from now on.
The lesson learned in sections 7.7.1 and 7.7.2 is that the local gauge
transformation
ψσ∗ (x) = ψσ∗0 (x) e−iθ(x) ,
ψσ (x) = e+iθ(x) ψσ0 (x),
(7.189)
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
385
is advantageous. It turns the action S0 + SU0 into the action (S0 + SU0 )0
whereby
Z
0 0
(S0 + SU ) = d4 x L0 + L1,1 + L1,2 + L2 ,
∇2
|∆|
†0 ∂τ − 2m − µ
Ψ0 ,
L0 = Ψ
∇2
|∆|
∂τ + 2m + µ
0
†0 +(i∂τ θ) + ie ϕ
Ψ0 ,
L1,1 = Ψ
0
−(i∂τ θ) − ie ϕ
(7.190)
 
+ (∇θ)+eA
·∇
0
2mi
†0 
 Ψ0 ,
L1,2 = Ψ
(∇θ)+eA
·∇
0
+
2mi
1
2
0
†0 + 2m [(∇θ) + eA]
L2 = Ψ
Ψ0 .
1
0
− 2m
[(∇θ) + eA]2
Here, we are using the notation
f (· · · ∇) g ≡ f (· · · ∇g) − (· · · ∇f )g
(7.191)
and we made use of the following identities. First, if δ denotes a differential while f , g = eiθ , and h denote three functions, then
h∗ g −1 (δ + if )2 (gh) = ih∗ (δ 2 θ)h − h∗ (δθ)2 h + 2ih∗ (δθ)(δh) + h∗ (δ 2 h)
| {z } | {z } |
{z
} | {z }
#1,2
#2
#1,2
#0
+ ih∗ (δf )h − 2h∗ (δθ)f h + 2ih∗ f (δh) − h∗ f 2 h,
| {z } | {z } | {z } | {z }
#1,2
#2
#1,2
#2
(7.192)
as
(δ + if )2 (gh) = δ 2 + i(δf ) + 2if δ − f 2 (gh),
δ(gh) = (δg)h + g(δh)
= eiθ [i(δθ)h + (δh)] ,
2
2
(7.193)
2
δ (gh) = (δ g)h + 2(δg)(δh) + g(δ h)
= eiθ i(δ 2 θ)h − (δθ)2 h + 2i(δθ)(δh) + (δ 2 h) .
Second, we converted the terms #1, 2 into the total differentials
ih∗ (δ 2 θ)h + 2ih∗ (δθ)(δh) = iδ (h∗ (δθ)h) − i(δh∗ )(δθ)h + ih∗ (δθ)(δh)
(7.194)
and
ih∗ (δf )h + 2ih∗ f (δh) = iδ (h∗ f h) − i(δh∗ )f h + ih∗ f (δh)
(7.195)
respectively. Third, the periodic boundary conditions obeyed by the
functions f , g, and h allow to drop any total derivatives after integration.
7.
386SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
The kernel in L0 defines the unperturbed Green function
G0 := − γ0 ∂τ + γ3
−1
∇2
−
− µ + γ1 |∆|
.
2m
(7.196)
To first order in θ, ϕ, and A, there are two contributions
V1,1 := γ3 [(i∂τ θ) + ie ϕ] ,
(7.197)
and
V1,2 := γ0
1
[(∇θ) + eA] · ∇ ,
2mi
(7.198)
in the kernels of L1,1 and L1,2 , respectively. Observe that (i∂τ θ) couples
to the electronic density in the same way as the scalar potential does
and thus is proportional to γ3 in the particle-hole grading. On the
other hand, (∇θ) couples to the paramagnetic current
J p :=
1 X ∗
[ψσ (∇ψσ ) − (∇ψσ∗ ) ψσ ]
2mi σ=↑,↓
(7.199)
as the vector potential does and is thus proportional to γ0 in the
particle-hole grading. To second order in θ or A, there is a single
contribution
1
V2 := γ3
[(∇θ) + eA]2 ,
(7.200)
2m
in the kernel of L2 , i.e., ∇θ contributes to the diamagnetic current
!
1 X ∗
J d := −
ψ ψ [(∇θ) + eA] ,
(7.201)
m σ=↑,↓ σ σ
as the vector potential does. The matrix elements of G0 , V1,1 , V1,2 , and
2
V2 in reciprocal space are, given the notation ξK = K
− µ,
2m
−1
G0K δK 0 ,K = − −iγ0 ωn + γ3 ξK + γ1 | ∆|
δK 0 ,K
1
=− 2
+iγ
ω
+
γ
ξ
+
γ
|
∆|
δK 0 ,K ,
0
n
3
K
1
2
ωn + ξK
+ |∆|2
1 V1,1 K 0 ,K = γ3 √
(ωn − ωn0 ) θK−K 0 + ie ϕK−K 0 ,
βV
(K − K 0 ) · (K + K 0 )
(K + K 0 )
1
V1,2 K 0 ,K = γ0 √
−
θK−K 0 + i
· eAK−K 0 ,
2mi
2mi
βV
1 1 V2K 0 ,K = γ3 √
[(∇θ) + eA]2 K−K 0 ,
βV 2m
(7.202)
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
387
respectively. Here, we are using the symmetric convention [x ≡ (R, τ ),
K ≡ (K, ωn ), and Q ≡ (Q, $l )]
Z
1 X
1
+iK x
f (x) = √
fK e
,
fK = √
d4 K f (x) e−iK x ,
βV K
βV
βV
Z
X
1
1
gQ e+iQ x ,
gQ = √
g(x) = √
d4 Q g(x) e−iQ x ,
βV Q
βV
βV
1
(g1 g2 )(x) = √
βV
X
(g1 g2 )Q e+iQ x ,
Q
1 X
(g1 g2 )Q = √
(g ) (g )
,
βV K 1 +K 2 Q−K
(7.203)
for the Fourier transforms of the Grassmann-valued f and the complexvalued g and g1 g2 , respectively.
As in section 6.4, we need to approximate a fermionic determinant
of the form
Tr ln M := Tr ln(M0 + M1 )
= Tr ln M0 1 + M0−1 M1
= Tr ln M0 + Tr ln 1 +
M0−1 M1
(7.204)
.
We thus perform the expansion
Tr ln M = Tr ln(−G−1
0 + M1 )
= Tr ln −G−1
+ Tr ln (1 − G0 M1 )
0
∞
X
1
−1
= Tr ln(−G0 ) −
Tr (G0 M1 )n
n
n=1
(7.205)
to the desired order. The unperturbed Green function G0 = −(M0 )−1
and the perturbation M1 = V1,1 + V1,2 + V2 are defined in Eq. (7.202).
To quadratic order in the fields θ, ϕ, and A, we must thus evaluate
Tr ln M = Tr ln(−G−1
0 )
− Tr G0 V1,1 − Tr G0 V1,2
1
1
− Tr (G0 V2 ) − Tr G0 V1,1 G0 V1,1 − Tr G0 V1,2 G0 V1,2 − Tr G0 V1,1 G0 V1,2
2
2
+ ··· .
(7.206)
7.7.4.1. First-order contributions.
If we impose the condition for
charge neutrality, i.e., ϕq q=0 = 0 and Aq q=0 = 0, there is no firstorder contributions to the expansion of the fermionic determinant.
7.7.4.2. Second-order contributions. There are two contributions,
one to order n = 1 and one to order n = 2 in the expansion of the
7.
388SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
series (7.205). These two contributions are given by
X −Tr (G0 V2 ) = −
tr G0K δK,K 0 V2 K 0 ,K
K,K 0
=−
ρ0 X +iQ θ+Q + e A+Q · −iQ θ−Q + e A−Q
2m Q
(7.207)
and
1
1
− Tr G0 V1,1 G0 V1,1 − Tr G0 V1,2 G0 V1,2 − Tr G0 V1,1 G0 V1,2 ,
2
2
(7.208)
respectively. We have used Eqs. (7.202) and (7.203) and introduced
the mean-field electronic density [compare with Eq. (6.21e)]
1
ρ0 :=
βV
Zβ
dτ
0
=
Z
3
dr
V
XZ
∗
D[ψ , ψ]
σ=↑,↓
e−
R
d4 x L0
Z0
(ψσ∗ ψσ )(r, τ )
1 X
tr (G0K γ3 )
βV K
(7.209)
to reach the second line of Eq. (7.207). The cross term in Eq. (7.208)
vanishes since the ground state, here an isotropic gapped Fermi sphere,
preserves the rotational invariance of the Hamiltonian. In other words,
ρ(x) J p (x) Gapped FS = 0.
(7.210)
We now introduce the Fourier components Πµνq of the polarization
tensor Πµν , µ, ν = 0, 1, 2, 3 through
k
Π00q ≡ Π00q
1 X :=
tr G0(k+q) γ3 G0k γ3 ,
βV k
Πijq ≡ Π⊥
ijq
:=
1 X (ki + qi /2) (kj + qj /2) tr G0(k+q) γ0 G0k γ0 ,
2
βV k
m
Πµνq ≡ 0,
i, j = 1, 2, 3,
for µ = 0, ν = 1, 2, 3 or µ = 1, 2, 3, ν = 0.
(7.211)
In terms of the polarization tensor, the contributions (7.208) to the expansion of the fermionic determinant to second order in the background
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
389
fields are given by
1
− Tr G0 V1,1 G0 V1,1 =
2
k
(7.212)
1 X
−
+$l θ+q + ie ϕ+q Π00q −$l θ−q + ie ϕ−q ,
2
q=($l ,q)
and
1
− Tr G0 V1,2 G0 V1,2 =
2
3
X X
21
(−1)
−qi θ+q + ie Ai(+q) Π⊥
ijq +qj θ−q + ie Aj(−q) ,
2
i,j=1
q=($l ,q)
(7.213)
respectively. Collecting all terms in the expansion to Gaussian order
of the fermionic determinant (7.186) gives the RPA partition function
RPA
Zβ,µ
Z
RPA
D[θ] e−Sβ,µ ,
∝
RPA
Sβ,µ
=
1
2
k
+Ωl θ+Q + ie ϕ+Q Π00Q −Ωl θ−Q + ie ϕ−Q
X
Q=(Ωl ,Q)
1
+
2
+
X
3
X
Qi θ+Q − ie Ai(+Q) Π⊥
ijQ Qj θ−Q + ie Aj(−Q)
Q=(Ωl ,Q) i,j=1
ρ0
2m
X
+iQ θ+Q + eA+Q · −iQ θ−Q + e A−Q .
Q=(Ωl ,Q)
(7.214)
The RPA action should be compared with action (7.166).
It is time to evaluate the BCS polarization tensor by performing a
Taylor expansion in powers of the four-momentum transfer q about q =
0. This expansion is well-defined owing to the presence of a gap that
removes any potential infrared singularities. For simplicity, we work
at vanishing temperature. We begin with the longitudinal component
k
Π00Q defined by
k
Π00q =
1 X
tr G0(k+q) γ3 G0k γ3
βV k
2 X X |∆|2 − ξk ξk+q + ωn (ωn + $l )
=−
,
2
βV k ω (ωn2 + Ek2 ) [(ωn + $l )2 + Ek+q
]
n
(7.215)
7.
390SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
where we recall that Ek2 := ξk2 + |∆|2 . When β = ∞ and q = (q, $l ) =
(0, 0),
Z
Z
d3 k
2
dω |∆|2 − ξk2 + ω 2
k
Π00q = −
βV
(2π)3 /V
2π/β (ω 2 + Ek2 )2
R
Z
Z
dω |∆|2 − ε2 + ω 2
≈ −νF dε
.
(7.216)
2π (ω 2 + ε2 + |∆|2 )2
R
R
To reach the last line, we have extended the range of the integration
over the single-particle energies to all real numbers, as the integral
remains well defined. In doing so the error can be estimated. The
frequency integration with the measure dω
gives the residue
2π
2
2
2
|∆| − ε + ω
iRes . (7.217)
2 2 q
q
2
2
2
2
ω − i ε + |∆|
ω + i ε + |∆|
√
ω=i
With the help of Resz=a
k
Π00q
f (z)
(z−a)n
=
1
(n−1)!
Z
≈ −νF
dε i
R
ν
= − F
2
ν
= − F
2
Z
dε
dn−1 f
dz n−1
dx
ε2 +|∆|2
, we conclude that,
z=a
|∆|2 − ε2 + ε2 + |∆|2
q
3
4
ε2 + |∆|2 i
|∆|2
ε2 + |∆|2
R
Z
3/2
1
(1 + x2 )3/2
R
+∞
νF
x
√
= −
2
1 + x2 −∞
= −νF
(7.218)
when β = ∞ and q = (q, $l ) = (0, 0). This is the same result as
obtained from evaluating the RPA polarization function at q = 0 in
the jellium model, see Eqs. (6.58), (6.65), and (6.78).
Next, we turn our attention to the transversal components Π⊥
ijq of
the BCS polarization tensor, which are defined by
1 X (ki + qi /2) (kj + qj /2)
tr
G
γ
G
γ
Π⊥
:=
ijq
0(k+q) 0 0k 0
βV k
m2
2 X (ki + qi /2) (kj + qj /2) |∆|2 + ξk ξk+q − ωn (ωn + $l )
=
,
2
βV k
m2
(ωn2 + Ek2 )[(ωn + $l )2 + Ek+q
]
(7.219)
7.7. EFFECTIVE THEORY IN THE VICINITY OF T = 0
391
where we recall that Ek2 := ξk2 + |∆|2 . When β = ∞ and q = (q, $l ) =
(0, 0),
Π⊥
ijq
Z
= +2
d3 k ki kj
(2π)3 m2
Z
dω |∆|2 + ξk2 − ω 2
2π (ω 2 + Ek2 )2
R
Z
= +2
3
d k ki kj 1
|∆|2 + ξk2 − ξk2 − |∆|2
2πi
q
3
(2π)3 m2 2π
2
4
ξk + |∆|2 i
= 0.
(7.220)
Comparison of Eqs. (7.215) and (7.219) shows that the Matsubara summations agree when ∆ = 0 but differ as soon as ∆ 6= 0. In other words,
the transversal components of the Fermi-liquid polarization tensor do
not vanish when T = 0 and q = 0. The fact that the Matsubara summation vanishes when T = 0 and q = 0 in the transversal components
of the BCS polarization function can thus be ascribed unambiguously
to a macroscopic property of the superconducting ground state. The
phase of the superconducting ground state is so “stiff” that application
of an external perturbation of the vector-gauge type does not induce a
paramagnetic current response at very low energies and very long wavelengths. The superconducting ground state is said to be incompressible
with respect to a vector-gauge perturbation. The only non-vanishing
response to the external vector-gauge perturbation comes from the diamagnetic current response [term proportional to ρ0 in Eq. (7.214)] at
very low energies and very long wavelengths in the superconducting
ground state. At non-vanishing temperature, particle-hole excitations
with energies larger than the single-particle gap 2|∆| induce a nonvanishing paramagnetic response, i.e.,
lim Π⊥
ijq 6= 0
q→0
for T > 0.
(7.221)
7.
392SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
To gain more insights into the response of the superconducting order
to the insertion of a test charge, define the partition function
Z
Zβ,µ := D[θ, ϕ] e−S ,
S :=
X Q2
ϕ+Q ϕ−Q
8π
Q
k
1X
+
+Ωl θ+Q + ie ϕ+Q Π00Q −Ωl θ−Q + ie ϕ−Q
2 Q
(7.222)
3
1XX
+
Qi θ+Q Π⊥
ijQ Qj θ−Q
2 Q i,j=1
ρ X
+ 0
+iQ θ+Q · −iQ θ−Q .
2m Q
Here, we have used the RPA approximation (7.214) after switching off
the external vector potential. We also added a kinetic term to the scalar
potential, i.e., endowed the scalar potential with its own dynamics.
We chose the kinetic term corresponding to a Coulomb interaction,
although we could have chosen a short-range potential instead. To
lowest order in an expansion of the BCS polarization tensor in powers
of Ωl /|∆| and |Q|/|∆|, the effective action in Eq. (7.222) simplifies to
S≈
X Q2
Q
νF
ρ0 2
ϕ ϕ +
e ϕ+Q − iΩl θ+Q e ϕ−Q + iΩl θ−Q +
Q θ+Q θ−Q .
8π +Q −Q
2
2m
(7.223)
Integration over the scalar potential ϕ then gives the approximate lowenergy effective action
!
X
[i(νF /2) e Ωl ]2
ρ
S≈
+ Q2
+ (νF /2) Ω2l + 0 Q2 θ+Q θ−Q
2
2m
+ (νF /2) e
Q
8π
!
(7.224)
2
X
(νF /2) Q
ρ
8π
=
Ω2l + 0 Q2 θ+Q θ−Q .
Q2
2
2m
+ (νF /2) e
Q
8π
After analytical continuation to real time τ = it, the zero’s of the kernel
for the phase θ of the superconducting order parameter give access to
some collective excitations in the superconducting state. The kernel
does not possess zero’s for arbitrary small frequencies
e l := iΩl ,
Ω
(7.225)
since Q2 can be factorized from the kernel. Had we replaced the
Coulomb potential
4π
VCB Q := 2
(7.226)
Q
7.8. EFFECTIVE THEORY IN THE VICINITY OF T = Tc
393
by a short-range potential with a non-diverging and non-vanishing limit
as Q → 0, we would have found a branch of collective excitations with
a dispersion relation
e
Ω(Q)
∝ |Q|.
(7.227)
The presence of a such a branch of excitation is an example of Goldstone
theorem and its absence as a result of the long-range nature of the
Coulomb interaction is an example of the Anderson-Higgs mechanism
by which “would be Goldstone bosons” are eaten up by a gauge degree
of freedom and photons acquire an effective mass.
The “would be Goldstone bosons” can be found at the energy scale
of the plasma frequency. At this energy scale approximation (7.224)
is not reliable anymore. To access the dynamics at the energy scale
of the plasma frequency, we must integrate over the superconducting
phase in Eq. (7.222), whereby we can ignore the contributions from
the transversal components of the BCS polarization tensor. (We are
taking the limit Ωl → 0 before taking the limit Q → 0.) This gives the
effective action for the scalar potential

2 
k
1
Ω (ie) Π00Q
X  Q2 1
2 l

2 k
S≈
+
(ie)
Π
+

 ϕ+Q ϕ−Q
00Q
ρ0
k
2
1
2
8π
2
−
Π
Ω
+
Q
00Q l
Q
2
2m
2


Ω2
ρ0 e
ρ0
k
+ 8πl + 16πm
Q2
X − 21 Π00Q 2m

 Q2 ϕ+Q ϕ−Q .
=
ρ0
k
2
1
2
− 2 Π00Q Ωl + 2m Q
Q
(7.228)
To leading order in the transfer momentum Q, the kernel for the scalar
potential is given by
r
Ω2P
Q2
4π ρ0 e2
1 + 2 ϕ+Q ϕ−Q ,
ΩP :=
.
(7.229)
8π
Ωl
m
This kernel only differs from the one for the jellium model by the replacement of the electron density by ρ0 defined in Eq. (7.209). Thus,
the physics at high energies (∼ ΩP |∆|) is largely unaffected by the
superconducting ground state.
7.8. Effective theory in the vicinity of T = Tc
Define the dimensionless grand canonical potential [which is also
called the Landau-Ginzburg free energy, be aware that we keep the
same notation as for the intensive grand canonical potential (7.67) and
7.
394SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
work in units for which ~ = c = kB = 1]
Zβ
∗
F [∆ , ∆] :=
Z
1
d r |∆|2 (r, τ )−log Det
U
3
dτ
0
V
2
∂τ − ∇
−µ
∆(r, τ )
2m
2
∆∗ (r, τ )
∂τ + ∇
+µ
2m
(7.230)
We have performed an expansion of this functional with the amplitude |∆(r, τ )| of the superconducting order parameter frozen to a uniform value |∆| in powers of a space-time fluctuating phase θ(r, τ ) =
arg ∆(r, τ ) in section 7.7. In the vicinity of Tc , it makes no sense
to distinguish between amplitude and phase fluctuations of the order
parameter and it is more natural to perform the expansion
β
∗
F [∆ , ∆] :=
∞ Z
X
j=0 0
Z
3
d rF
dτ
(j)
∂
∂
i , −i
|∆|2j (r, τ ).
∂r
∂τ
(7.231)
V
In the theory of classical continuous phase transitions, an expansion
in powers of the order parameter is called a Landau-Ginzburg theory. Symmetry dictates which powers of the order parameter enter the
expansion (7.231). Here, the U (1) gauge symmetry only allows even
powers 2j.
In this section, we give a microscopic derivation to the time-independent
Landau-Ginzburg functional
Z
h
∗
F [∆ , ∆] = β V × constant + β × d3 r a(T ) |∆|2 (r) + b(T ) |∆|4 (r)
V
∗
+ c(T ) (∇∆ ) · (∇∆) (r) + · · ·
i
(7.232)
by relating the temperature dependent coefficients a(T ), b(T ), and c(T )
to the microscopic coupling constants (m, µ, U ) entering the BCS
Hamiltonian. We show that
T
a(T ) = (νF /2) ln
,
(7.233a)
Tc
(ν /2)
1
ζ(3, 1/2) F 2 ,
(7.233b)
b(T ) =
2
16π
T
1
µ(νF /2)
1
(vF )2 (νF /2)
c(T ) =
ζ(3,
1/2)
=
ζ(3,
1/2)
(7.233c)
,
24π 2
mT2
48π 2
T2
the ζ-function being defined by
ζ(x, y) :=
+∞
X
n=0
1
.
(n + y)x
(7.233d)
Before deriving Eqs. (7.232) and (7.233) observe that by replacing the
gradient term ∇ in Eq. (7.230) by the covariant derivative ∇ + ieA
.
7.8. EFFECTIVE THEORY IN THE VICINITY OF T = Tc
395
that couples fermions carrying the electric charge −e < 0 to an external static vector potential A(r), one can derive the time-independent
Landau-Ginzburg functional
Z
h
∗
F [∆ , ∆, A] = β V × constant + β × d3 r a(T ) |∆|2 (r) + b(T ) |∆|4 (r)
V
i
+ c(T ) (D∆) · (D∆) (r) + · · · ,
∗
(7.234a)
where
D := ∇ + i(2e)A,
(7.234b)
by a straightforward extension of the computation to follow. For a
recent review on the use of Landau-Ginzburg functional for superconductors see Ref. [79].
We organize the expansion of the fermionic determinant around the
unperturbed Green function
−1
∇2
G0 := − γ0 ∂τ + γ3 −
−µ
,
(7.235)
2m
i.e., we need to perform the expansion
Tr log − (G0 )−1 + ∆(r, τ )γ+ + ∆∗ (r, τ )γ− =
−1
Tr log − (G0 )
−
+∞
X
1
j=1
j
Tr
∗
G0 ∆(r, τ )γ+ + ∆ (r, τ )γ−
j ,
(7.236)
where the matrices
γ+ :=
0 1
0 0
,
γ− :=
0 0
1 0
(7.237)
in the particle-hole grading have been introduced. The contribution
const := Tr log −(G0 )−1
(7.238)
induces a renormalization of the Landau-Ginzburg free energy through
a mere constant.
First-order contributions (j = 1)
The contributions to first order in the superconducting order parameter vanish because of the algebra obeyed by the Pauli matrices.
Second-order contributions (j = 2)
The four traces
0
a 0
0 1
a 0
0 1
0 a
0 a0
tr
= tr
= 0,
0 b
0 0
0 b0
0 0
0 0
0 0
(7.239a)
7.
396SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
tr
tr
a 0
0 b
a 0
0 b
0 0
1 0
0 1
0 0
a0 0
0 b0
a0 0
0 b0
a0 0
0 b0
0 0
1 0
0 0
1 0
0 0
b 0
0 0
= 0,
b0 0
(7.239b)
0 a
0 0
0 0
= ab0 ,
b0 0
(7.239c)
= tr
= tr
0 a0
tr
= tr
= ba0 ,
0 0
(7.239d)
are needed to evaluate all contributions to the expansion of the Fredholm determinant that are of quadratic order in the superconducting
order parameter.
Introduce the short-hand notation [q = (q, $l )]
a 0
0 b
F (2) [∆∗ , ∆] :=
0 0
1 0
0 1
0 0
0 0
b 0
X 1
2 1 ∆∗q ∆q + Tr
G0 ∆(r, τ )γ+ + ∆∗ (r, τ )γ−
.
U
2
q
(7.240)
In reciprocal space, the trace
F (2) [∆∗ , ∆] =
X 1
∆∗q ∆q
U
q
1 XXX +
tr (G0 )k+q γ+ (G0 )k γ− ∆+q ∆∗+q
2βV q ω k
n
X
X
X 1
tr (G0 )k+q γ− (G0 )k γ+ ∆∗−q ∆−q
+
2βV q ω k
n
(7.241)
simplifies greatly, for
F (2) [∆∗ , ∆] =
X 1
∆∗q ∆q
U
q
1 XXX
1
+
∆ ∆∗
2βV q ω k −iωn − i$l + ξk+q (−iωn − ξk ) +q +q
n
1 XXX
1
+
∆∗ ∆ .
2βV q ω k −iωn − i$l − ξk+q (−iωn + ξk ) −q −q
n
(7.242)
7.8. EFFECTIVE THEORY IN THE VICINITY OF T = Tc
397
It is convenient to change k to k − q/2
X 1
F (2) [∆∗ , ∆] =
∆∗q ∆q
U
q
+
+
∆+q ∆∗+q
1 X
2βV q,ω ,k −iω − i$ /2 + ξ
−iω
+
i$
/2
−
ξ
n
n
l
l
n
k+q/2
k−q/2
∆∗−q ∆−q
1 X
.
2βV q,ω ,k −iω − i$ /2 − ξ
−iωn + i$l /2 + ξk−q/2
n
l
n
k+q/2
(7.243)
Finally, if we change q to −q on the last line, we obtain
X
F (2) [∆∗ , ∆] ≡
Kq(2) ∆∗q ∆q ,
(7.244a)
q
where
1 X
1
1
.
+
U βV ω ,k −iω − i$ /2 + ξ
−iω
+
i$
/2
−
ξ
n
n
l
l
n
k+q/2
k−q/2
(7.244b)
Contribution (7.244) resembles the contribution (6.58) for the polarization of the Jellium model except for one important difference.
The Jellium model has no particle-hole grading. Energy eigenvalues
of the non-interacting Fermi gas are always subtracted from the Matsubara frequencies in the denominator. Here, the particle-hole grading
implies that the energy eigenvalues of the non-interacting Fermi gas
are subtracted in the particle-particle channel [factor a in Eq. (7.239c)]
and subtracted in the hole-hole channel [factor b0 in Eq. (7.239c)].
The fermionic Matsubara sum can be replaced by an integral over
the Fermi-Dirac distribution f˜FD (z) = 1/[exp(βz) + 1] = 1 − f˜FD (−z),
see Fig. 1,
Z
1
1 X
dz
f˜FD (z)
(2)
Kq
=
−
U
V k
2πi z + i$ /2 − ξ
z − i$l /2 + ξk−q/2
l
k+q/2
Γ
Kq(2) :=
k
1
1 X
=
+ (−1)2
U
V k
f˜FD (+i$l /2 − ξk−q/2 )
+i$l − ξk−q/2 − ξk+q/2
+
f˜FD (−i$l /2 + ξk+q/2 )
!
−i$l + ξk+q/2 + ξk−q/2
X f˜FD (+i$l /2 − ξk−q/2 ) − f˜FD (−i$l /2 + ξk+q/2 )
1
2 1
=
+ (−1)
U
V k
+i$l − ξk−q/2 − ξk+q/2
=
1
1 X 1 − f˜FD (−i$l /2 + ξk−q/2 ) − f˜FD (−i$l /2 + ξk+q/2 )
+ (−1)2
.
U
V k
+i$l − ξk−q/2 − ξk+q/2
(7.245)
7.
398SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
The denominator on the right-hand side of Eq. (7.245) differs from the
denominator (6.65) in the polarization function for the Jellium model,
as it is the sum instead of the difference of the energy eigenvalues of
the non-interacting Fermi gas that now appear.
(2)
In the static limit q = (q, $l ) → (q, 0), the kernel Kq simplifies
to
1
1 XX
1
(2)
Kq,0 = +
(7.246a)
U
βV ω k (iωn + ξk ) iωn − ξk+q
n
or, equivalently,
(2)
Kq,0 =
1 X 1 − f˜FD (+ξk ) − f˜FD (+ξk+q )
1
−
.
U
V k
ξk + ξk+q
(7.246b)
Representation (7.246a) is the one that we choose to perform the gradient expansion
(2)
1 XX
1
1
1
+
eq·∂k
U
βV ω k iωn + ξk
iωn − ξk
n
1 XX
1
1
= −
U
βV ω k ωn2 + ξk2
Kq,0 =
n
1 XX
∂ξk
1
1
2
+
(−1) q ·
βV ω k iωn + ξk (iωn − ξk )2
∂k
n
1 XX
1
1
1
+
(q · ∂k )2
βV ω k iωn + ξk 2
iωn − ξk
(7.247)
n
3
+ O(q ).
The first line on the last equality of the right-hand side was evaluated
in Eq. (7.87b). It is given by (νF /2) ln(T /Tc ). The second line vanishes
under the assumption that the inversion symmetry
ξ+k = ξ−k
(7.248)
holds for the non-interacting fermion gas. The third line can be evaluated with the help of the identities
1
∂
∂
∂
2
q·
q·
k = q·
(q · k) = q 2 ,(7.249a)
2
∂k
∂k
∂k
∂
∂
·
k2 = 2 × 3 = 6,
(7.249b)
∂k ∂k
2
∂ξk
k2
2
= 2 =
(ξ + µ) .
(7.249c)
∂k
m
m k
7.8. EFFECTIVE THEORY IN THE VICINITY OF T = Tc
399
Thus,
(2)
Kq,0
q2 X X
1
1
(∂k )2
+ O(q 3 )
6βV ω k iωn + ξk
iωn − ξk
n
2 XX
T
q
1
1
∂k
ln
−
· ∂k
+ O(q 3 )
Tc
6βV ω k
iωn + ξk
iωn − ξk
n
T
q2 X X
ξk + µ
3
).
ln
+
2
2 + O(q (7.250)
Tc
3mβV ω k (iωn + ξk ) (iωn − ξk )
ν
= F ln
2
=
νF
2
=
νF
2
T
Tc
+
n
The summation over momenta is converted into an integral over the
density of states ν̃(ξ), whereby the definition of the density of states
accounts for the spin-1/2 degree of freedom,
Z
νF
T
q2 X
ν̃(ξ) ξ + µ
(2)
Kq,0 =
ln
+
+ O(q 3 )
dξ
2
Tc
3mβ ω
2 (ωn2 + ξ 2 )2
n R
(7.251)
Z
νF
T
q2 X
ν̃(ξ)
µ
3
=
ln
+ O(q ).
+
dξ
2
Tc
3mβ ω
2 (ωn2 + ξ 2 )2
n
R
The last equality follows for any density of states that obeys ν̃(ξ) =
ν̃(−ξ). For a density of state that is not even under ξ → −ξ but is
non-vanishing at the Fermi energy, one has the approximation
Z
νF
T
q 2 µ νF X
1
(2)
Kq,0 ≈
ln
+ O(q 3 )
+
dξ
2
Tc
3mβ 2 ω
(ωn2 + ξ 2 )2
n R
2
X
νF
T
1
q µ νF
=
ln
2πiRes
+
+ O(q 3 ).
2
2
2
2
Tc
3mβ 2 ω
(ωn + ξ ) ξ=+i|ωn |
n
(7.252)
The residue of
1
2 +ξ 2 )2
(ωn
at ξ = +i|ωn | is the expansion coefficient
(−2)(2i|ωn |)−3 = (−1)2
1
1
= h
+3
4i|ωn |
4i π
β
1
(2n + 1)
i+3
(7.253)
that enters the simple pole in the Laurent series expansion
(ξ + i|ωn |)−2
(2i|ωn |)−2
(−2)(2i|ωn |)−3
1
=
=
+
+ ··· .
(ωn2 + ξ 2 )2
(ξ − i|ωn |)+2
(ξ − i|ωn |)+2
(ξ − i|ωn |)+1
(7.254)
This gives
+∞
T
q 2 µβ 2 νF X
1
νF
(2)
3
+
Kq,0 ≈
ln
+ O(q )
2
2
Tc
24π m 2 n=0 n + 1 3
2
(7.255)
νF
T
q 2 µβ 2 νF
=
ln
+
ζ(3, 1/2) + O(q 3 ).
2
Tc
24π 2 m 2
7.
400SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
The first term in the kernel gives the coefficient (7.233a) whereas the
second term gives the coefficient (7.233c).
Third-order contribution (j = 3)
They vanish because of the Pauli algebra.
Fourth-order contribution (j = 4)
The coefficient (7.233b) [as was the coefficient (7.233a)] can be read
from the expansion of the Fredholm determinant about a space- and
time-independent |∆|2 performed in section 7.3.2. In the ratio
c(T )
2
µ
= × ,
b(T )
3 m
(7.256)
the factor 1/3 comes from Eq. (7.249b), the factor 2µ/m from Eq.
(7.249c).
7.9. Problems
7.9.1. BCS variational method to superconductivity.
Introduction. The mean-field and RPA approximations to superconductivity were treated using path-integral techniques. It is also
valuable to derive the mean-field approximation following Bardeen,
Cooper, and Schrieffer in their seminal papers. [80, 81, 82]
Definitions. Consider the interacting Hamiltonian
Ĥ := Ĥ0 + Ĥ1 ,
XX
ξk ĉ†kσ ĉkσ ,
Ĥ0 :=
k
Ĥ1 :=
(7.257b)
σ=↑,↓
X
k,k
(7.257a)
Vk,k0 ĉ†+k↑ ĉ†−k↓ ĉ−k0 ↓ ĉ+k0 ↑ ,
(7.257c)
0
~2 k
− µ.
(7.257d)
2m
The interaction matrix elements obey Vk,k0 = Vk∗0 ,k . The BardeenCooper-Schrieffer (BCS) variational wave function is defined by
Y
†
iϕ †
|ϕi :=
uk + vk e ĉ+k↑ ĉ−k↓ |0i ,
(7.258)
ξk :=
k
in terms of the electron creation and annihilation operators. This BCS
wave function can be thought of as a coherent state for Cooper pairs
(see Ref. [80]) with vk (uk ) the amplitude (not) to have a Cooper
pair with relative momentum k. The numbers uk , vk ∈ R obey the
normalization conditions
u2k + vk2 = 1
and ϕ ∈ [0, 2π[ is a global phase.
Exercise 1.1:
(7.259)
7.9. PROBLEMS
401
(a) Express the expectation value in the variational state |ϕi of
the kinetic energy hϕ| Ĥ0 |ϕi in terms of the parameters uk ,
vk , and ϕ.
(b) Express the expectation value in the variational state |ϕi of
the interacting energy hϕ| Ĥ1 |ϕi in terms of the parameters
uk , vk , and ϕ.
(c) Does hϕ| Ĥ |ϕi depend on the global phase ϕ?
From now on, we assume that the matrix elements of the interaction
potential take the reduced from
Vk,k0 = −V.
Define the complex-valued parameter
X
∆ := V
uk vk .
(7.260)
(7.261)
k
Exercise 1.2:
(a) Express hϕ| Ĥ |ϕi in terms of vk only.
(b) Minimize hϕ| Ĥ |ϕi with respect to vk to show that
1
ξk
1
ξk
2
2
vk =
1−
,
uk =
1+
,
(7.262a)
2
Ek
2
Ek
where
q
Ek = + ξk2 + ∆2 .
(7.262b)
(c) Express hϕ| Ĥ |ϕi and Eq. (7.261) in terms of ξk and Ek .
(d) Consider the subspaces with the quantum numbers k ↑ and
−k ↓. Show that the states
uk + vk ĉ†k↑ ĉ†−k↓ |0i ,
ĉ†k↑ |0i ,
ĉ†−k↓ |0i ,
vk − uk ĉ†k↑ ĉ†−k↓ |0i
(7.263)
are orthogonal to each other and normalized to one.
(e) Consider the state |2, ki which is defined as
|2, ki := vk −
uk ĉ†k↑ ĉ†−k↓
0
6=k kY
k
uk0 + vk0 ĉ†k0 ↑ ĉ†−k0 ↓ |0i .
(7.264)
0
Show that
h2, k| Ĥ |2, ki − hϕ| Ĥ |ϕi ≈ 2 Ek ,
(7.265)
p
where Ek := ∆2 + ξk2 . What terms have been dropped here?
Exercise 1.3: We now go back to the Hamiltonian (7.257). Show
that the BCS wave function (7.258) is obtained from minimizing the
7.
402SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
z
Figure 3. A hollow cylindrical superconductor.
energy hϕk , ϕ| Ĥ |ϕk , ϕi of the trial wave function
i ϕk
|ϕk , ϕi = uk + vk e
ĉ†k↑ ĉ†−k↓
0
6=k kY
uk0 + vk0 ei ϕ ĉ†k0 ↑ ĉ†−k0 ↓ |0i ,
k0
(7.266)
where ϕ, ϕk ∈ R and under the condition
Vk,k0 = Vk0 ,k
(7.267)
in Eq. (7.257c).
7.9.2. Flux quantization in a superconductor. Consider a superconducting hollow cylinder. Assume that the superconducting current density j s (x, t) is defined by
j s (x, t) := −e∗ ns v(x, t),
(7.268)
where ns is the superfluid density, v the average speed of the charge
carriers, and −e∗ < 0 the charge of one charge carrier.
Exercise 1.1:
(a) Show that, in the classical limit, the electric field E and the
current density j s are related by
1 dj
e∗
E=
.
m
ns e∗ d t
(7.269)
Consider a closed path C in the x-y plane surrounding the hole shown
in Fig. 3. The fluxoid φM is defined by
I
Z
mc
dl · j.
(7.270)
φM := ds · B +
ns e∗ 2
Ω
Note that φ =
R
C
ds · B is just the ordinary magnetic flux through the
Ω
area Ω with the oriented (anticlockwise) boundary C = ∂Ω.
7.9. PROBLEMS
403
(b) Use Maxwell equations and Eq. (7.269) to show that the fluxoid φM is conserved in time
∂ φM
= 0.
(7.271)
∂t
∗
(c) Using the identity m v = p + ec A, show that the fluxoid can
be expressed as
I
c
dl · p.
(7.272)
φM = − ∗
e
C
(d) Applying the Bohr-Sommerfeld quantum condition conclude
that the fluxoid is quantized
hc
n,
n ∈ Z.
(7.273)
e∗
Early experiments on superconductivity indicated that e∗ = 2 e, which
was a big puzzle at that time. Today we know that e∗ is in fact the
charge of a Cooper pair.
φM =
7.9.3. Collective excitations within the RPA approximation.
Introduction. Equation (7.214) is the main result of chapter 7. It
is a dynamical effective theory for the low-energy and long-wavelength
degrees of freedom in a superconducting phase with a nodeless meanfield gap that preserves the symmetries under spin rotations and reversal of time. These low-energies and long-wavelength degrees of freedom
are represented by a real-valued scalar field θ, the phase of the superconducting order parameter, that couples to external (source) electromagnetic gauge fields A with the component A0 in imaginary time
(µ = 0) and the three components A in three-dimensional position
space (µ = 1, 2, 3). The dynamical effective theory is an approximate
one, for only terms quadratic in θ and the gauge fields A have been
kept in a gradient expansion (an expansion in powers of the mean-field
Green function). It is the existence of the mean-field superconducting
gap ∆ that justifies this expansion. It applies to fields that vary slowly
on the characteristic length scale
~ vF
ξ=
(7.274)
∆
where vF is the Fermi velocity. The length ξ is called the superconducting coherence length. It diverges upon approaching the transition
at which the mean-field gap vanishes. Hence, the effective field theory (7.214) applies deep in the superconducting phase. Equation (7.214)
is of the generic form
Z
Z[A] := D[θ] e−S[θ,A]
(7.275a)
7.
404SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
with the Euclidean action
Z
Z
1
4
S[θ, A] := +
d x d4 y ∂µ θ − q Aµ (x) Πµν (x − y) (∂ν θ − q Aν ) (y)
2
∗
1X
iQµ θQ − q Aµ Q Πµν
iQν θQ − q Aν Q ,
=+
Q
2 Q
(7.275b)
where q is the electromagnetic gauge charge (not necessarily the charge
−e < 0 of the electron), summation is implied over repeated labels
µ, ν = 0, 1, 2, 3 in position space and imaginary time, the kernel Πµν (x−
y) = [Πνµ (y−x)]∗ is Hermitean, and the Fourier conventions are defined
by Eq. (7.203).
Gauge invariance. Exercise 1.1: Verify that the Euclidean action
and the partition function (7.275) are invariant under the local gauge
transformation
θ → θ + q ϕ,
Aµ → Aµ + ∂µ ϕ.
(7.276)
Current-current correlation functions. Define the current functional
δS
µ
j [θ, A] := +
[θ, A]
(7.277)
δAµ
and the susceptibility functional
µν
υ [θ, A] := +
δ2S
δAµ δAν
[θ, A],
(7.278)
for µ, ν = 0, · · · , 3. They represent the slope and the curvature of the
action S at [θ, A], respectively. Define the current functional
δ ln Z
µ
J [A] :=
[A]
(7.279)
δAµ
and the susceptibility functional
µν
Υ [A] :=
δ 2 ln Z
δAµ δAν
[A],
(7.280)
for µ, ν = 0, · · · , 3. They represent the slope and the curvature of the
action −F := ln Z at A, respectively.
Exercise 2.1:
(a) Show that
J µ [A] = − hj µ i [A]
(7.281a)
and
Υµν [A] = + hj µ j ν i [A] − (J µ J ν ) [A] − hυ µν i [A],
(7.281b)
where
R
D[θ] e−S[θ,A] (· · · )
h(· · · )i [A] := R
.
D[θ] e−S[θ,A]
(7.281c)
7.9. PROBLEMS
405
(b) Compute j µ [θ, A] and υ µν [θ, A] for the generic action (7.275)
both in (position) space and (imaginary) time and in four momentum space.
Collective excitations without gauge invariance. We break the gauge
invariance under the transformation (7.276) by defining the effective
low-energy and long-wavelength theory of the generic form
Z
Z := D[θ] e−S[θ]
(7.282a)
with the Euclidean action
Z
Z
1
4
S[θ] := +
d x d4 y ∂µ θ (x) Πµν (x − y) (∂ν θ) (y)
2
∗
1X
Qµ θQ Πµν
Qν θQ .
=+
Q
2 Q
(7.282b)
Exercise 3.1:
(a) Under what generic conditions does the phase field θ support
excitations that disperse at vanishing temperature? Hint: Do
a Taylor expansion of Πµν
Q in powers of the four-momentum Q
at vanishing temperature.
(b) Write down the kernel for the phase field θ to leading order in
a gradient expansion of Πµν for the polarization tensor in section 7.7.4 at zero temperature and show that there are gapless
excitations. Compare this conclusion with the discussion that
follows Eq. (7.227).
7.9.4. The Hall conductivity in a superconductor and gauge
invariance.
Introduction. What is the Hall conductivity of a superconductor at
vanishing temperature? At the mean-field level, we immediately encounter a difficulty with the fact that charge is not anymore a good
quantum number. Instances of quantization of the Hall conductivity can only make sense if charge is a good quantum number. This
suggests that the Hall conductivity cannot be quantized in a superconductor. Whatever value it takes, it cannot be universal. However,
one might object that this conclusion is an artifact of the mean-field
approximation. Hence, it would be desirable to reach this conclusion
using a more general argument. This we try by revisiting Laughlin flux
insertion argument from section 5.5.3.
Definition. We imagine a two-dimensional superconductor with a
mean-field nodeless gap confined to the Corbino geometry of section
5.5.3.
Exercise 1.1: Does assumption L1 from section 5.5.3 hold? Answer this question in the mean-field approximation first and then in
the RPA approximation presented in section 7.7.4.
7.
406SUPERCONDUCTIVITY IN THE MEAN-FIELD AND RANDOM-PHASE APPROXIMATIONS
Exercise 1.2: Does assumption L2 from section 5.5.3 hold? Answer this question in the mean-field approximation first and then in the
RPA approximation presented in section 7.7.4. Discuss the role played
by gauge invariance or lack thereof. Hint: Explain why it suffices to
use the effective action defined by Eq. (7.224) in order to answer this
question.
Exercise 1.3: How does flux quantization affect the reasoning
from section 5.5.3 that constrains the Hall conductivity to be a rational
number in units of e2 /h.
CHAPTER 8
A single dissipative Josephson junction
Outline
A phenomenological model for a Josephson junction is presented.
The DC and AC Josephson effects are described. A model for a dissipative Josephson junction is given both at the classical and quantum
levels. At the quantum level, a dissipative Josephson junction is shown
to realize the Caldeira-Leggett model of dissipative quantum mechanics. The method of instantons in quantum mechanics is reviewed. The
phase diagram of a dissipative Josephson junction is discussed using
renormalization-group methods and a duality transformation. The existence of lines of weak- and strong-coupling fixed points is established.
8.1. Phenomenological model of a Josephson junction
We have derived in section 7.7 a low-energy effective action for
space and time fluctuations of the superconducting order parameter.
We have argued that the most important (collective) degree of freedom
that needs to be accounted for is the phase φ(x) = 2θ(x), x ≡ (R, τ )
of the superconducting order parameter,
∆(x) = |∆| exp + 2iθ(x) ≡ |∆| exp + iφ(x) .
(8.1)
Fluctuations in the magnitude of the pairing-order parameter about
the mean-field value |∆| were argued to account for collective excitations with characteristic energy of the order of the mean-field gap
∝ |∆|. This is why such fluctuations can be neglected at temperatures
well below the mean-field gap. On the one hand, we argued that the
imaginary-time derivative of θ(x) couples to electrons with the electrical charge −e < 0 through the electronic density
X
ρ(x) =
ψσ∗ (x) ψσ (x)
(8.2)
σ=↑,↓
in the same way as the scalar potential ϕ(x) that conveys the electronic
Coulomb interaction does,
Z
i d4 x ρ(x) [(∂τ θ)(x) + e ϕ(x)] .
(8.3)
βV
407
408
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
On the other hand, we argued that the space-derivative of θ(x) couples
to electrons through the paramagnetic and diamagnetic currents
1 X ∗
Jp (x) =
[ψ (x) (∇ψσ )(x) − (∇ψσ∗ )(x) ψσ (x)] ,
2 mi σ=↑,↓ σ
!
(8.4)
1 X ∗
Jd (x) = +
ψ (x) ψσ (x) [(∇θ) (x) + e A(x)] ,
m σ=↑,↓ σ
respectively, in the same way as the vector potential A(x) does [recall
Eq. (7.190)],
Z
1
4
d x Jp (x) · [(∇θ)(x) + e A(x)] + Jd (x) · [(∇θ)(x) + e A(x)] .
2
βV
(8.5)
Absent from our perturbative calculations of the effective action for θ
is the fact that θ is defined modulo π, i.e., that the effective action for θ
must be periodic in θ and that, in particular, θ can support singularities
called vortices.
The phenomenological model of a Josephson junction attempts to
describe the coupling between two superconducting metals in close
proximity by postulating the validity of Eqs. (8.3) and (8.5) for each
superconductor and by proposing an interaction between the two superconductors that is periodic in the phase mismatch between the phase
θ1 of the pairing-order parameter in superconductor 1 and the phase θ2
of the pairing-order parameter in superconductor 2. The microscopic
mechanism that motivates this choice of a coupling is coherent tunneling of paired electrons between the two superconductors, a highly
controversial idea at the time (see Ref. [83] for a historical perspective).
A first simplifying assumption is that the phases θ1 and θ2 of each superconductor are taken to be constant in space. Correspondingly, the
vector potentials Aα are taken to be vanishing whereas the scalar potentials ϕα only vary in imaginary time, α = 1, 2. In other words,
the non-interacting contribution to the action describing a Josephson
junction is simply
S0 := i
β
X Z
dτ Nα (τ ) [(∂τ θα ) (τ ) + e ϕα (τ )] ,
(8.6)
α=1,2 0
whereby Nα (τ ), φα (τ ) = 2θα (τ ), and ϕα (τ ), are the number of electrons, the phase of the pairing-order parameter, and the applied potential in superconductor α at imaginary time τ , respectively. The
effective action (8.6) follows from Eq. (7.190) by ignoring the quasiparticle Lagrangian density L0 while assuming that L1,2 and L2 vanish,
leaving L1,1 as the sole contribution.
8.1. PHENOMENOLOGICAL MODEL OF A JOSEPHSON JUNCTION
409
It is customary to perform the change of variables
N± := N1 ± N2 ,
θ± := θ1 ± θ2 ,
ϕ± := ϕ1 ± ϕ2 ,
(8.7a)
when coupling two “levels”, in which case
β
Z
i X
S0 =
dτ Nα (τ ) [(∂τ θα ) (τ ) + e ϕα (τ )] .
2 α=±
(8.7b)
0
A second assumption consists in considering the case when N+ is time
independent, i.e., both superconductors form a closed system in which
the total number of electrons N+ is conserved at all imaginary times.
Consequently,
S0
i
i
=
N+ θ+ (β) − θ+ (0) + e β ϕ+ +
2
2
Zβ
dτ N− (τ )
∂τ θ− (τ ) + e ϕ− (τ )
0
=
i
2
Zβ
dτ N− (τ )
∂τ θ− (τ ) + e ϕ− (τ ) ,
(8.8a)
0
if l = 0 and ϕ+ = 0 are chosen in
θ+ (β) − θ+ (0) = π l,
l ∈ Z,
1
ϕ+ :=
β
Zβ
dτ ϕ+ (τ ).
(8.8b)
0
The contribution to the Josephson junction action arising from the
interaction between the two superconductors is taken as the simplest
function of φ− := φ1 − φ2 that is periodic with periodicity 2 π and that
penalizes a non-vanishing phase difference between the two superconductors, 1
Zβ
SUJ := −2 UJ
Zβ
dτ cos φ− = −2 UJ
0
dτ cos(2θ− ),
UJ > 0.
0
(8.10)
1
Expansion in powers of φ− of the interacting Lagrangian
LUJ := −2 UJ cos φ− = −2 UJ + UJ φ2− + O φ4− ,
(8.9)
shows that our Gaussian approximation for the effective action obeyed by φ in
chapter 7, say Eq. (7.166), is simply obtained by identifying φ− with ∇φ.
410
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
The quantum model for a Josephson junction is then defined by the
partition function
Z
Zβ := D[N− , φ− ] exp −Sβ ,
Zβ
Sβ :=
dτ
L0 + LUJ ,
(8.11)
0
1
L0 := N− i∂τ φ− + ie∗ ϕ− ,
4
LUJ := −2 UJ cos φ− ,
φ− (τ ) = φ− (τ + β),
e∗ = 2 e,
ϕ− (τ ) = ϕ− (τ + β),
in the background of the scalar potential ϕ− . Quantum mechanics
comes about from the integration over the measure D[N− , φ− ] for the
bosonic coherent states φ− and N− (see appendix A.1.2).
The classical equations of motion for a Josephson junction follow
from
0 = δSβ
i
=
4
Zβ
dτ δN−
∂τ φ− + e∗ ϕ−
0
i
+
4
Zβ
dτ −∂τ N−
δφ−
0
Zβ
+ 2 UJ
dτ
δφ− sin φ− .
0
(8.12a)
They are
i ∂τ φ− = −ie∗ ϕ− ,
i ∂τ N− = +8 UJ sin φ− ,
in imaginary time. Analytical continuation to real time
τ = it,
ϕ− (τ ) = +iϕ− (t),
(8.12b)
2
(8.13a)
yields
∂t φ− (t) = +e∗ ϕ− (t),
∂t N− (t) = +8 UJ sin φ− (t) .
(8.13b)
Had we chosen canonical quantization instead of a path integral
formulation, we would have elevated N− and φ− to the level of operators
N̂− and φ̂− obeying the equal-time commutation relation
[N̂− , φ̂− ] = i,
2
We are undoing Eq. (7.187).
[N̂− , N̂− ] = [φ̂− , φ̂− ] = 0,
(8.14)
8.1. PHENOMENOLOGICAL MODEL OF A JOSEPHSON JUNCTION
(a)
(b)
N1 ,
N1 ,
1,
1,
'1
'1
N2 ,
UJ
2,
411
'2
N2 ,
2,
'2
Figure 1. (a) Two superconductors are separated by
a thin non-superconducting layer. (b) At zero temperature the thin non-superconducting layer acts like a tunnel
barrier to paired electrons (Cooper pairs). Quantum tunneling of Cooper pairs can be driven by application of a
voltage difference VJ ≡ ϕ1 − ϕ2 between superconductors
1 and 2. The Josephson equations model macroscopically
the current flow driven by quantum tunneling of Cooper
pairs across the non-superconducting layer.
and used the representation
Zβ ∝ Tr e−β Ĥ ,
Ĥ := −e∗ N̂− ϕ− − 8 UJ cos φ̂− ,
(8.15)
of the partition function together with the equations of motion
i∂t φ̂− = [φ̂− , Ĥ],
(8.16)
i∂t N̂− = [N̂− , Ĥ],
to recover Eq. (8.13).
To bring Eq. (8.13) to the canonical representation of the Josephson
equations,
ϕ− (t) = ϕ1 (t) − ϕ2 (t) =: VJ (t)
(8.17)
is reinterpreted as the voltage difference VJ (t) between superconductors
1 and 2, respectively, at real time t. Correspondingly, the electrical
current at real time t that flows between superconductors 2 and 1,
owing to the negative charge −e < 0 of the electron, is
1
IJ (t) := (−e) (∂t N1 ) (t) = (−e) ∂t N− (t).
(8.18)
2
Here, conservation of the total charge was used,
0 = ∂t N+ = ∂t N1 + ∂t N2 =⇒ ∂t N− = +2∂t N1 .
(8.19)
Now, the classical equations of motion (8.13) in real time are rewritten
e∗ VJ (t)
∂t φ− (t) = +
,
~
2 e∗ UJ
IJ (t) = −
sin φ− (t) .
~
(8.20)
412
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
N1 ,
1,
'1
N2 ,
2,
'2
VJ (t) = 0
IJ (t) = IJ
A
time independent
Ra =
Va
I
Figure 2. Setup for the DC-Josephson effect. The
Josephson current IJ is time independent. The Josephson voltage VJ ≡ ϕ1 − ϕ2 vanishes.
Here, ~ has been reinstated and it is customary to define the unit of
current 3
2 e∗ UJ
4 e UJ
I0 :=
=
.
(8.21)
~
~
Equations (8.20) and (8.21) define the “classical” Josephson equations
of motion (see Fig. 1). These equations are “classical” in the sense that
they follow from a variational Ansatz on the action of the full quantum
mechanical partition function
Z
Z−iβ := D[N− , φ− ] exp +iS−iβ ,
S−iβ
Z−iβ dt L0 + LUJ ,
:=
0
e∗ VJ
1
L0 := N− − ∂t φ− +
,
e∗ = 2 e,
4
~
2U
LUJ := + J cos φ− ,
~
φ− (t) = φ− (t − iβ),
ϕ− (t) = ϕ− (t − iβ).
(8.22)
8.2. DC Josephson effect
The so-called DC-Josephson effect is derived from Eq. (8.20) by
assuming that the current IJ (t) = IJ in the circuit of Fig. 2 is constant
3
As a check of units, the potential difference V has the units of energy per
unit charge and the coupling constant UJ has the units of energy. Hence, e∗ V /~
has the units of inverse time and I0 = 2 e∗ UJ /~ has the units of charge per unit
time.
8.3. AC JOSEPHSON EFFECT
N1 ,
1,
'1
N2 ,
2,
413
'2
VJ (t) = VJ
IJ (t) = IJ
has periodicity
h
e ⇤ VJ
A
Ra =
Va
I
Figure 3. Setup for the AC-Josephson effect. The
Josephson voltage VJ ≡ ϕ1 − ϕ2 is time independent and
non-vanishing. The Josephson current JJ is periodic in
time with period h/(e∗ VJ ).
in time, in which case
I
sin φ− (t) = − J
I0
(8.23a)
is time independent and
VJ (t) = +
~
∂
φ
(t) = 0.
t
−
e∗
(8.23b)
The resistance
VJ (t)
=0
(8.24)
IJ (t)
of the Josephson junction vanishes when the current passing through
the circuit of Fig. 2 is time independent. This is the DC-Josephson
effect.
RJ (t) :=
8.3. AC Josephson effect
The so-called AC-Josephson effect is derived from Eq. (8.20) by
assuming that the voltage VJ (t) = VJ 6= 0 in the circuit of Fig. 3 is
constant in time and non-vanishing, in which case
φ− (t) = φ− (t = 0) +
e∗ V J
t
~
(8.25a)
and
e∗ VJ
IJ (t) = −I0 sin φ− (t = 0) +
t .
(8.25b)
~
When the voltage drop at the Josephson junction is time independent
and non-vanishing, the current in the circuit of Fig. 3 is periodic with
414
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
period
h
.
e∗ VJ
(8.26)
This is the AC-Josephson effect that allows a measurement of the
charge of the Cooper pair
e∗ ≡ 2 e.
(8.27)
8.4. Dissipative Josephson junction
8.4.1. Classical. Consider the setup in Fig. 4 that defines a “classical” dissipative Josephson junction. There are three additive contributions to the current I(t) flowing from superconductor 2 to superconductor 1:
(1) A capacitive current
IC (t) := −C (∂t VJ ) (t),
(C has units of squared charge per energy)
(8.28)
where VJ (t) = ϕ− (t) is the voltage difference between superconductors 1 and 2 and the proportionality constant C is time
independent and called the capacitance.
(2) A Josephson current
IJ (t) = −
2 e∗ UJ
sin φ− (t) ≡ −I0 sin φ− (t) ,
~
(8.29)
where UJ is the characteristic interaction strength between superconductors 1 and 2.
(3) An Ohmic current
Is (t) := −
VJ (t)
Rs
(Rs has units of energy × time per squared charge).
(8.30)
We shall shortly see that this current is dissipative. Here,
the time-independent Ohmic conductance 1/Rs measures the
strength of the dissipation. Dissipation is maximal in the limit
of an infinite Ohmic conductance (a vanishing Ohmic resistance Rs = 0) by which the entire current from superconductor 2 to superconductor 1 is carried by Is (t), i.e., in the
limit IC (t)/Is (t) = IJ (t)/Is (t) = 0. In the opposite limit of
a vanishing Ohmic conductance (an infinite Ohmic resistance
Rs = ∞) by which the entire current from superconductor 2
to superconductor 1 is carried by IC (t) + IJ (t), i.e., in the limit
Is (t)/IC (t) = Is (t)/IJ (t) = 0, there is no dissipative contribution to the current from superconductor 2 to superconductor
1.
8.4. DISSIPATIVE JOSEPHSON JUNCTION
415
C
N1 ,
1,
'1
UJ
N2 ,
2,
'2
Rs
Figure 4. Pictorial view of a dissipative Josephson
junction made of superconductors 1 and 2 separated by a
thin non-superconducting layer. The thin layer between
the two superconductors is modeled macroscopically by a
capacitance, a Josephson coupling, and a resistor in parallel. The (dimensionful) coupling constants C, UJ , and
Rs are the capacitance, Josephson coupling, and Ohmic
resistance, respectively (according to footnote 3, C has
dimensions of squared charge per energy, UJ has dimensions of energy, and Rs has dimensions of energy×time
per squared charge).
The “classical” equations of motion defining this dissipative Josephson
junction are
e∗
∂t φ− (t) = + VJ (t),
(8.31a)
~
I(t) = IC (t) + IJ (t) + Is (t)
V (t)
= −C (∂t VJ ) (t) − I0 sin φ− (t) − J . (8.31b)
Rs
Insertion of (8.31a) into (8.31b) yields the second order differential
equation
0=C
~
1 ~
φ̈− +
φ̇ + I0 sin φ− + I,
∗
e
Rs e∗ −
φ̇− ≡ ∂t φ− .
(8.32)
We will take Eq. (8.32) as our mathematical definition of a classical
dissipative Josephson junction. This equation can be reinterpreted as
follows. Equation (8.32) describes a classical spinless point particle
with the mass
~
(e∗ = 2 e),
(8.33)
C ∗
e
moving on a circle of unit radius with coordinate φ− subjected to: 4
4
A particle of mass m obeys Newton equation
m ẍ = −c ẋ − V 0 (x) + Fext (t),
(8.34)
when subjected to a damping proportional to its speed ẋ, an energy conserving
potential V (x), and an external force Fext (t).
416
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
• A driving time-dependent force
− I(t).
(8.35)
• A time-independent (conservative) potential
− I0 cos φ− .
(8.36)
• A damping force
1 ~
φ̇ .
(8.37)
Rs e∗ −
When the forcing term I(t) is time independent, the potential
V (φ− ) := −I0 cos φ− + Iφ− ,
−V 0 (φ− ) = −I0 sin φ− − I
(8.38)
is sometimes called a washboard potential.
8.4.2. Caldeira-Leggett model. In their ground breaking paper
[84], Caldeira and Leggett motivated, defined, and resolved the question: What is the effect of dissipation on quantum tunneling? As we
shall see, their analysis can be recast in the context of a dissipative
Josephson junction as the motivation, definition, and resolution to the
question: What is the effect of dissipation on quantum coherence? To
stress the conceptual difference between quantum tunneling and quantum coherence, we will first review the starting point of Caldeira and
Leggett.
Caldeira and Leggett consider first an isolated and non-dissipative
quantum system that has been initially prepared to be in the close
vicinity of the unique metastable minimum of the cubic potential 5
1
1
V (q) = M $02 q 2 − λ2 q 3
2
3
2 27
q
q
=
V
1−
,
(8.39)
4 0 q0
q0
3 M $02
V0 ≡ V (q)| M $02 ,
q0 ≡
,
0 < $0 , λ ∈ R.
q= 2
2 λ2
λ
Here, q can be interpreted as the coordinate on the real line R of a
spinless point particle of mass M with the classical Lagrangian 6
1
L0 = M q̇ 2 − V (q).
(8.40)
2
5
The assumption that V (q) is cubic is done without loss of generality as long
as V (q) is sufficiently smooth and has the general shape of a cubic potential, i.e.,
has a single local minimum.
6 In most applications of quantum tunneling, however, q is not a geometrical
coordinate as would be the case if one wants to describe the tunneling of an alpha
particle out of a nucleus or of an electron out of an atom in a strong electric field,
but a macroscopic or collective degree of freedom. For example, in the case of a
SQUID ring, q is the magnetic flux trapped in the ring. SQUID stands for (rf ) superconducting quantum interference device, i.e., a superconducting ring interrupted
by a Josephson junction (see section 6.3 of Ref. [75] and chapter 7 of Ref. [85]).
8.4. DISSIPATIVE JOSEPHSON JUNCTION
417
Although a classical particle sitting in the metastable minimum of
V (q) cannot escape this local minimum, a quantum particle can decay
through the potential barrier V0 into a continuum of states. Within the
WKB approximation, the probability per unit time Γ0 for the particle
to escape a generic metastable potential well is given by
Γ0 = A0 e−
B0
~
[1 + O(~ $0 /V0 )] ,
r
B0
,
A 0 = C0 $0
2π~
Zq0 p
B0 = 2 dq 2 M V (q),
(8.41)
0
whereby, for our cubic potential,
B0 =
36 V0
,
5 $0
C0 =
√
60.
(8.42)
To model dissipation at the classical level, Caldeira and Leggett
choose the simplest possible model in which the classical equations of
motion of the isolated system
dV
0
0
0 = M q̈ + V (q),
V (q) ≡
(q),
(8.43)
dq
are modified by the addition of damping and forcing terms,
0 = M q̈ + η q̇ + V 0 (q) − Fext (t),
η ∈ R.
(8.44)
In practice, Eq. (8.44) should be thought of as an equation whose validity is empirical, i.e., the phenomenological parameters M and η together with the potential V (q) and the external force Fext (t) have been
measured experimentally. 7 Caldeira and Leggett then ask:
(1) How to construct a quantum mechanical system that yields
the equation of motion (8.44) in the classical limit?
(2) Is the effect of dissipation on the tunneling probability (8.41)
uniquely determined for a given potential V (q) by the friction
coefficient η or is it model dependent?
(3) Does dissipation increase or decrease the tunneling probability
(8.41)?
7
Of course, dissipation can manifest itself in much more complicated ways. (i)
Fourier transformation of Eq. (8.44) with respect to time holds only for a restricted
range of frequencies. (ii) The dissipative term η q̇ is replaced by f (q) q̇ with the
function f (q) not constant in q. (iii) The dissipative term η q̇ is replaced by higher
order time derivatives f (q) ∂tn q, n odd. (iv) The dissipative term η q̇ is replaced by
the time convolution η ∗ q̇. (v) Dissipation is non-linear, i.e., q̇ enters in a non-linear
way in the Lagrangian. Such generalizations are discussed in Ref. [84].
418
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
The answers given by Caldeira and Leggett to questions 3 and 2 are that
dissipation decreases the tunneling probability and that this effect is
indeed uniquely determined by the macroscopic parameter η. We now
turn to the answers to questions 1 and 2 that will be directly relevant
to the issue of quantum coherence in a dissipative Josephson junction.
8.4.2.1. Modeling dissipation in quantum mechanics. To model dissipation at the quantum level, one can imagine coupling the isolated
system to some environment. There are then two possible routes. In
the first one, the environment is modeled by turning the deterministic
Hamiltonian of the isolated system into a statistical ensemble of random Hamiltonians. An alternative approach followed by Caldeira and
Leggett is to envision the “universe” made up of the environment and
the system of interest as a deterministic one, i.e., endowed with Hamiltonian dynamics, to assume that the environment is made of infinitely
many degrees of freedom, and to assume that the coupling between the
environment and the system of interest is weak. Assuming that the
environment is in the thermodynamic limit allows to consider the limit
of strong dissipation without relaxing the condition that the coupling
between the environment and the system is weak. A weak coupling
between the environment and the system makes plausible the modeling of the environment as a collection or bath of non-interacting harmonic oscillators. The classical Lagrangian of the “universe” is taken
by Caldeira and Leggett to be
L = L0 + Lbath + Lint + Ladia + Lext (t),
(8.45a)
where the Lagrangian L0 describes a classical spinless point particle of
mass M > 0 in a conservative potential V ,
1
(8.45b)
L0 := M q̇ 2 − V (q),
2
the Lagrangian Lbath describes a family of classical harmonic oscillators
with the masses mι > 0 and harmonic frequencies $ι > 0 labeled by
the index ι,
1X
Lbath :=
mι ẋ2ι − $ι2 x2ι ,
(8.45c)
2 ι
the Lagrangian Lint describes the linear coupling (with the coupling
constants cι ) between the point particle with mass M and coordinate
q and the (harmonic) bath,
X
Lint := +q
cι x ι ,
(8.45d)
ι
the Lagrangian Ladia is included for convenience,
1
Ladia := − M ($adia )2 q 2 ,
2
X c2
1
ι
M ($adia )2 :=
, (8.45e)
2
2
2
m
$
ι
ι
ι
8.4. DISSIPATIVE JOSEPHSON JUNCTION
419
and Lext (t) describes a non-conservative force,
Lext (t) := +Fext (t) q(t).
(8.45f)
The first question to address is how to choose the masses mι > 0
and frequencies $ι > 0 for the bath and how to choose the (linear)
coupling constants cι ∈ R between the bath and the coordinate q so as
to reproduce the phenomenological equation of motion (8.44).
To answer this question one must compare the Fourier transform
with respect to time of (8.44), i.e.,
0 = −M $2 q$ − iη $ q$ + [V 0 (q)]$ − Fext $ ,
(8.46)
with the Fourier transform with respect to time of the coupled equations of motion for q and the bath
X
0 = M q̈ + V 0 (q) −
cι xι + M ($adia )2 q − Fext (t),
ι
(8.47)
2
0 = mι ẍι + mι $ι xι − cι q,
∀ι,
i.e.,
0 = −M $2 q$ + [V 0 (q)]$ −
X
cι xι$ +
ι
X c2 1
ι
2
m
ι $ι
ι
!
q$ − Fext $ ,
(8.48a)
and
0 = −mι $2 xι$ + mι $ι2 xι$ − cι q$ ,
∀ι.
(8.48b)
Insertion of Eq. (8.48b),
xι$ =
1
cι
q ,
2
mι $ι − $ 2 $
$ 6= $ι ∀ι,
(8.49)
into Eq. (8.48a) yields
0 = −M $2 q$ + [V 0 (q)]$ − K$ q$ − Fext $ ,
X c2 1
1
ι
K$ := −
− 2
,
$ 6= $ι ∀ι.
2
2
m
$
$
−
$
ι
ι
ι
ι
(8.50)
Equation (8.50) provides two insights. First, the equation of motion
obeyed by the coefficient q$=0 is unaffected by the coupling to the
bath. This is so because the contribution Ladia is precisely chosen to
prevent a renormalization of the coefficient [V 0 (q)]$=0 in the Fourier
expansion with respect of time of V 0 (q). Second, the bath parameters
{mι > 0, $ι > 0} and the linear coupling constants {cι ∈ R} to the
bath must be chosen in the following way. First, we do the algebraic
420
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
manipulation
Im K$
!
X c2 1
1
ι
= Im −
− 2
2
m
$
$ι − $2
ι
ι
ι
!
2
X c2
$
ι
.
= Im (−1)2
2 ($ 2 − $ 2 )
m
$
ι
ι
ι
ι
(8.51)
Second, we do the analytic and algebraic manipulations


∞
X c2 Z
1
ι

Im K$ = Im $2
d$
e δ($
e − $ι ) 2
2 − $2 )
m
$
e
(
$
e
ι
ι
0


!
∞
Z
X c2
1
ι

δ($
e − $ι )
= Im $2
d$
e
2
2)
m
$
e
$
e
(
$
e
−
$
ι
ι
0


!
∞
2 Z
2
X
2$
π
cι
1
.
= Im 
d$
e
δ($
e − $ι )
(8.52)
2
π
2 ι mι $
e
$
e ($
e − $2 )
0
Third, we introduce the spectral function
π X c2ι
δ($ − $ι )
J($) :=
2 ι mι $
(8.53)
to absorb the underlined factor,


∞
2 Z
2$
1

Im K$ =Im 
d$
e J($)
e
2
π
$
e ($
e − $2 )
0


∞
h
i
2 Z
2$
J($)
e
1
= Im 
d$
e
P 1 + sgn($) iπ δ($
e − |$|)  .
$−|$|
e
π
$
e $
e + |$|
0
(8.54)
The last line defines how to regularize the first-order pole when $ =
$ι > 0. To this end, the principal-value distribution P has been introduced and we have chosen to move the pole away from the real axis in
the $-complex
e
plane according to the rule $ → $ + i0+ . Finally, we
impose the condition
Im K$ = η $.
(8.55)
Hence, we infer that the choice
J($) = η $ Θ($),
(8.56)
where Θ($) denotes the Heaviside step function, satisfies Eq. (8.55).
This choice requires that the eigenfrequencies {$ι > 0} are densely
distributed on the positive real axis, for the function J($) would not
8.4. DISSIPATIVE JOSEPHSON JUNCTION
421
be continuous had there been a discrete component to the spectrum
{$ι > 0}. The real part of K$ is then of the order
$
η$×
,
(8.57)
$bath
where $bath is some characteristic frequency in the bath. If $/$bath is
typically small, then one can work with Eq. (8.46). Otherwise we must
allow for a complex friction coefficient (admittance) in Eq. (8.46). 8
One might naively expect that the characteristic frequency entering
the tunneling rate is [see Eq. (8.39) for the definition of $0 ]
$0
if M $02 η $0
(8.58)
in the lightly damped regime. Similarly, one might naively expect that
the characteristic frequency entering the tunneling rate is
$0
$0 ×
$0
if M $02 η $0
(8.59)
η/M
in the heavily damped regime. If so, the approximation of neglecting
the real part of the kernel K$ will remain good in the heavily damped
regime if it is a good approximation in the lightly damped regime.
With these preliminary considerations in hand, we are ready to
define the quantum dissipative model through the partition function
(~ = 1)
Z
Z
Zβ := D[q] D[x] exp −Sβ
(8.60a)
with the action
Sβ = S0 + Sbath + Sint + Sadia + Sext
(8.60b)
and the Lagrangian
Zβ
=
dτ (L0 + Lbath + Lint + Ladia + Lext ) ,
(8.60c)
0
whereby
M
(∂τ q)2 + V (q)
(8.60d)
2
is the Lagrangian for a spinless point particle of mass M in the conservative potential V ,
Xm
ι
Lbath =
(∂τ xι )2 + $ι2 x2ι
(8.60e)
2
ι
L0 =
8
Starting from Ohm’s law V (t) = (R ∗ I)(t) (∗ denotes a convolution), Fourier
transformation with respect to time defines the complex impedance V$ = z$ I$
whereby Re z$ is called the resistance and Im z$ is called the reactance. The
admittance is 1/z$ with Re 1/z$ the conductance and Im 1/z$ the susceptance.
422
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
is the Lagrangian for a family of harmonic oscillators,
Lint = −q
X
cι x ι
(8.60f)
ι
is the Lagrangian that couples the spinless point particle of mass M in
the conservative potential V to the bath,
X c2
1
ι
M ($adia )2 =
,
2
2
2
m
$
ι
ι
ι
1
Ladia = + M ($adia )2 q 2 ,
2
(8.60g)
is included for convenience, and
Lext (τ ) := −Fext (τ ) q(τ )
(8.60h)
is the Lagrangian for a driving force. We are imposing the periodic
boundary conditions
q(τ ) = +q(τ + β),
xι (τ ) = +xι (τ + β),
∀ι.
(8.60i)
Analytical continuation τ = it has been performed to go from the
Lagrangians in Eq. (8.45) to those in Eq. (8.60). The bosonic measures
are best defined after performing a Fourier transformation with respect
to imaginary time. In the bath,
1 X
xι (τ ) = √
x
e−i$l τ ,
β $ ι $l
l
xι $ l
1
=√
β
Zβ
(8.61)
dτ xι (τ ) e+i$l τ ,
2π
$l =
l,
β
l ∈ Z,
0
and
Z
∞ Z
YY
dRe xι $l Z dIm xι $l
√
√
D[x] ≡
.
2π
2π
ι l=0
R
(8.62)
R
Fourier transform with respect to imaginary time and the measure of q
are defined similarly. Fourier transformation with respect to imaginary
8.4. DISSIPATIVE JOSEPHSON JUNCTION
423
time gives the representation
Sβ = S0 + Sbath + Sint + Sadia + Sext
X
=
(L0 + Lbath + Lint + Ladia + Lext ) ,
$l
p
M 2
$l q(+$l ) q(−$l ) + β [V (q)]$ δ$l ,0 ,
l
2
Xm
ι
2
2
Lbath =
$l + $ι xι (+$l ) xι (−$l ) ,
2
ι
X 1
Lint = −
cι
q(+$l ) xι (−$l ) + q(−$l ) xι (+$l ) ,
2
ι
L0 =
1
Ladia = + M ($adia )2 q(+$l ) q(−$l ) ,
2
p
Lext = − β [Fext q]$ δ$l ,0 ,
X c2
1
ι
M ($adia )2 =
,
2
2
2
m
$
ι
ι
ι
l
(8.63)
of the action (8.60b).
The strategy that we are going to follow is to integrate the degrees
of freedom from the bath. To this end, observe that completing the
square of Lbath + Lint with respect to q±$l is achieved by adding and
subtracting to Lbath + Lint
X
c2ι
q
.
(8.64)
q
2 mι ($l2 + $ι2 ) (+$l ) (−$l )
ι
Hence,
Lbath + Lint
cι
=
q
+
xι (+$l ) −
2
mι ($l2 + $ι2 ) (+$l )
ι
cι
× xι (−$l ) −
q
mι ($l2 + $ι2 ) (−$l )
X
c2ι
−
q
q
.
2
2 ) (+$l ) (−$l )
2
m
($
+
$
ι
ι
l
ι
(8.65)
Xm
ι
$l2
$ι2
By changing bath integration variables to
cι
q
,
xι (+$l ) = x̃ι (+$l ) +
mι ($l2 + $ι2 ) (+$l )
∀ι, $l ,
(8.66)
one can decouple the bath from q. Integration over the bath degrees
of freedom {x̃ι (+$ ) } produces an overall constant, the bosonic deterl
minant
∞
YY
1
Nbath :=
,
(8.67)
2
2)
m
($
+
$
ι
ι
l
ι l=0
424
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
while turning the partition function without damping
Z
Zβ 0 := D[q] exp (−S0 )
(8.68)
into the partition function (we set Fext = 0 for simplicity)
Z
Zβ = Nbath D[q] exp (−S0 − S1 ) ,
X
S0 + S1 =
(L0 + L1 ) ,
$l
p
M 2
$l q(+$l ) q(−$l ) + β [V (q)]$ δ$l ,0 ,
l
2
L1 = K$l q(+$l ) q(−$l ) ,
X c2 1
1
ι
,
K$ =
−
2 mι $ι2 $ι2 + $2
ι
L0 =
(8.69)
in the presence of damping. The kernel K$ is related to the kernel in
Eq. (8.50) by analytical continuation
1
K$ = − × lim Kω .
2 ω→+i$
(8.70)
In terms of the spectral function (8.53),
1
K$ =
π
Z∞
d$
e J($)
e
0
1
$
e
− 2
$
e
$
e + $2
.
(8.71)
By undoing the Fourier transformation to Matsubara frequencies, the
effective action
Sβ0 = S0 + S1
(8.72)
induced by integrating over the degrees of freedom from the bath is
represented by
Zβ
S0 =
dτ
1
2
M (∂τ q) + V (q) ,
2
0
Zβ
Zβ
S1 =
dτ
0
(8.73)
dτ 0 q(τ ) K(τ − τ 0 ) q(τ 0 ),
0
where the non-local kernel in imaginary time induced by damping is
defined by
1X
K(τ ) :=
K$l e−i$l τ ,
β $
l
Zβ
K$ l =
0
dτ K(τ ) e+i$l τ .
(8.74)
8.4. DISSIPATIVE JOSEPHSON JUNCTION
425
It is shown in appendix H.1 that, when the spectral function is chosen
as in Eq. (8.56), the kernel K(τ ) can be written in the form
Xη
S1 =
|$l | q(+$l ) q(−$l )
2
$
l
2
Z+∞ Zβ
q(τ ) − q(τ 0 )
0 η
=
dτ dτ
4π
τ − τ0
−∞
(8.75)
0
by which it is explicitly seen to be positive definite.
Equation (8.75) is the central result of this section. Whereas Eq. (8.69)
answers question 1, Eq. (8.75) answers question 2. Insofar as all of what
is needed of the environment is that the spectral function (8.53) satisfies
Eq. (8.56) in some appropriate range of frequencies, 9 the phenomenology of linear damping encoded by Eq. (8.46) is independent of the
microscopic details defining the environment, say the choice mι > 0,
$ι > 0, and cι ∈ R. Moreover, Eq. (8.75) gives us a strong hint to the
answer to question 3. Indeed, Eq. (8.75) tells us that
S0 + S1 > S0 .
(8.78)
It is then very suggestive to conjecture that a WKB-like estimate would
replace the tunneling rate in Eq. (8.41) by
B0
Γ0 = A0 e− ~ [1 + O(~ $0 /V0 )] ,
(8.79)
whereby
S0 + S1 > S0 =⇒ B 0 > B0 ,
(8.80)
i.e., dissipation would decrease the tunneling rate of a particle initially
prepared in a metastable minimum of the potential V (q).
It is essential to observe that Eqs. (8.69) and (8.75) hold not only
for a cubic potential but for any smooth potential which is a C-number
function of q, say the washboard potential (8.38). However, asking
about the tunneling rate out of a metastable minimum only makes
sense for a potential V (q) of the cubic type. For a potential V (q) with
two or more degenerate absolute minima the issue of tunneling rate
out of these minima is meaningless as such since the quantum particle can always tunnel back to its initial position at the bottom of one
of the wells. For a potential with degenerate minima, we know that
quantum eigenstates, as opposed to their classical counterparts, are
9
For example,
η$
(8.76)
1 + $2 τR2
reduces to Eq. (8.56) when $ is smaller than the characteristic frequency $0 of the
cubic potential which itself is much smaller than the inverse of the relaxation time
τR ,
$2 τR2 < $02 τR2 1.
(8.77)
JR ($) :=
426
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
delocalized in the absence of damping. For a symmetric double well
potential, the probability to find the particle in the ground state at
the bottom of the left well equals the probability to find the particle
at the bottom of the right well if the system is isolated. For a cosine
potential, eigenstates are Bloch waves, i.e., plane waves with wave vector commensurate with the periodicity of the potential, if the system is
isolated. The meaningful question to ask when the potential V (q) has
several degenerate absolute minima, say is periodic, is if delocalized
states of Zβ 0 remain delocalized in the presence of damping, i.e., if a
state prepared initially to be delocalized remains for ever delocalized
as time evolves. The terminology of quantum coherence is also used in
the literature as a synonymous to delocalization. The notion of quantum coherence emphasizes the wave-like nature of delocalized states.
Quantum coherence can be detected by interference effects. Whether
we are after the effects of dissipation on quantum tunneling or dissipation on quantum coherence, a tool is needed to evaluate Zβ in some
approximation. Instantons techniques will be the tools that we choose.
8.5. Instantons in quantum mechanics
8.5.1. Introduction. Consider the classical Lagrangian ( ˙ denotes t derivative)
L :=
1 2
ẋ − V (x; g),
2
V (x; g) =
1
F (g x).
g2
(8.81)
Here, the mass of the particle moving on the real line with the coordinate x has been set to one and the analytic function F has a zero of
order 2 at the origin. The classical equation of motion
ẍ = −
1 dF (g x)
dF (g x)
⇐⇒
gẍ
=
−
g2
dx
d(g x)
(8.82)
is independent of the coupling constant g since the coupling constant
g factorizes under the rescaling y := g x:
1 1 2
d F (y)
L= 2
ẏ − F (y) =⇒ ÿ = −
.
(8.83)
g
2
dy
If one can solve the classical equation of motion for g = 1, we know
the solution for all g’s. This, however, is not true anymore after quantization as we know that ~ (or ~ g 2 after rescaling) plays a crucial role
in the combination L/~ [or L/(~ g 2 ) after rescaling] that appears in
the path integral description of the quantum theory. For example, the
amplitude |T (E)| for transmission through a potential barrier of an incoming plane wave with energy E is given by [compare with Eq. (8.41)
8.5. INSTANTONS IN QUANTUM MECHANICS
427
where Γ0 ∝ |T (E)|2 ]

 1
|T (E)| = exp −
~
Zx2

p

dx 2(V − E) [1 + O(~)] ,
(8.84)
x1
where x1 and x2 are the classical turning points from the left and right,
respectively.
What about performing perturbation theory for small g? Hereto,
the classical and quantum theory differ. For the classical theory, one
would expect perturbation theory around g = 0 to be valid. However,
this is certainly not true for the quantum theory as is illustrated by
Eq. (8.84). A result like Eq. (8.84) that is non-perturbative in ~ is
usually derived by matching solutions of Schrödinger equation in different regions of space (WKB method). This method is difficult to
extend beyond one dimension and/or one particle. The method of instantons that relies on the path integral representation of the quantum
theory can also reproduce Eq. (8.84). Moreover, it has the advantage
of extending to higher dimensions and/or field theory. As with onedimensional quantum mechanics, instantons techniques give access to
phenomena that are intrinsically non-perturbative in the interaction
potential of the field theory.
8.5.2. Semi-classical approximation within the Euclideanpath-integral representation of quantum mechanics. Consider
the quantum Hamiltonian
Ĥ =
p̂2
+ V (x̂),
2
[x̂, p̂] = i~,
(8.85)
that describes the motion of a spinless point particle of unit mass on
the real line with the position operator x̂. Instanton techniques rely
not on the operator representation of the quantum theory but on the
Euclidean path integral representation,
Z
−ĤT /~
hxf |e
|xi i ∝ D[x] e−S/~ .
(8.86)
On the left-hand side of Eq. (8.86), |xi i and |xf i are the initial and
final position eigenstates and T is a positive number with dimension
of time. The left-hand side is of interest since it can be expanded in
terms of the exact eigenstates |ni of Ĥ,
X
hxf |e−ĤT /~ |xi i =
e−εn T /~ hxf |ni hn|xi i,
(8.87)
n
so that, for large T , the leading term in this expansion gives the ground
state and its energy.
428
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
On the right-hand side of Eq. (8.86), S stands for the Euclidean
action ( ˙ denotes τ derivative)
+T
Z /2
S :=
dτ
ẋ2
+ V (x) ,
2
(8.88)
−T /2
and D[x] denotes the measure over all functions x that obey the boundary conditions
x(−T /2) = xi ,
x(+T /2) = xf .
(8.89)
Explicit construction of the measure D[x] proceeds as follows. If x̄(τ )
is some arbitrary function obeying the boundary conditions, then any
given function x(τ ) that obeys the same boundary conditions can be
expanded in terms of some chosen set of complete, real, and orthonormal functions xn (τ ) that vanish at ±T /2,
X
x(τ ) = x̄(τ )+
cn xn (τ ),
n
+T
Z /2
dτ xm (τ ) xn (τ ) = δm,n ,
xn (±T /2) = 0,
−T /2
(8.90)
and the (normalized) measure is now given by
D[x] =
Y dc
√ n .
2π~
n
(8.91)
Observe that the measure does not depend on x̄(τ ). The right-hand side
is of interest because it can readily be evaluated in the semi-classical
(small ~) limit through a saddle-point approximation of the argument
in the exponential (Boltzmann) weight.
The idea behind the saddle-point approximation relies on the assumption that if the prefactor of the action [here 1/~ or 1/(~ g 2 ) when
V has the form given in Eq. (8.81)] is extremely large the dominant
contribution to the path integral will come from all the paths (there
might be more than one) that are global minima of the action. Assume
that S has a minimum x̄(τ ) that obeys the boundary condition. Taylor
expansion around this minimum of the action yields ( ˙ and 0 denote τ
8.5. INSTANTONS IN QUANTUM MECHANICS
429
and x derivatives, respectively)
+T
Z /2
S[x̄ + y] =
dτ
1 2
x̄˙ + V (x̄)
2
−T /2
+T
Z /2
dτ [−x̄¨ + V 0 (x̄)] y
+
(8.92)
−T /2
+T
Z /2
+
dτ
1
y [−ÿ + V 00 (x̄) y]
2
−T /2
+ ··· .
Here, y(±T /2) = 0 to accommodate the boundary conditions. This is
the reason for which all boundary terms vanish after partial integration.
By assumption, the second line vanishes and if we truncate the Taylor
expansion up to second order in y, we find
+T
Z /2
S[x̄ + y] =
dτ
x̄˙ 2
+ V (x̄)
2
−T /2
1
+
2
+T
Z /2
−T /2
(8.93a)
d2 V d2
dτ y − 2 +
y
dτ
dx2 x̄
+ ··· ,
whereby x̄ is the solution to the differential equation
− x̄¨ + V 0 (x̄) = 0,
x̄(−T /2) = xi ,
x̄(+T /2) = xf .
(8.93b)
The Taylor expansion Eq. (8.93a) suggests that the path y be expanded
in terms of the orthonormal eigenfunctions xn of the Hermitean operator −∂τ2 + V 00 (x̄)
y(τ ) =
X
cn xn (τ ),
2
−∂τ + V 00 (x̄) xn = λn xn ,
xn (±T /2) = 0.
n
(8.94)
If x̄ is truly a minimum, all eigenvalues λn must be larger or equal to
zero. Thus, if we insert the expansion (8.94) in terms of the orthonormal modes of the kernel on the second line of the right-hand side of
430
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
Eq. (8.93a) into Eq. (8.86), we obtain
Z
−ĤT /~
hxf |e
|xi i ∝
D[x] e−S[x]/~
− ~1
= e
+T
R /2
dτ
−T /2
x̄˙ 2
+V
2
(x̄)

×

+∞
YZ
n −∞
λn
1
dc
√ n e− 2 cn ~ cn 
2π~
× [1 + O(~)]
!
Y 1
S[x̄]
− ~
p
= e
×
× [1 + O(~)]
λn
n
= p
exp (−S[x̄]/~)
Det [−∂τ2 + V 00 (x̄)]
[1 + O(~)] .
(8.95)
In going to the second line, we made use of the orthonormality of the
eigenmodes xn and we assumed that the non-Gaussian contributions
are of order ~. In going to the third line, we assumed that λn > 0, in
which case it is reassuring to observe that the Gaussian contribution
that results from integrating over cn is of order zero in powers of ~.
Equation (8.95) encodes the semi-classical approximation to quantum
mechanics within the path integral formalism.
From a technical point of view, the semi-classical approximation of
quantum mechanics is reduced to:
(1) Solving the differential equation
x̄¨ = −[−V 0 (x̄)],
x̄(−T /2) = xi ,
x̄(+T /2) = xf ,
(8.96)
that represents Newton’s equation in the potential −V . Observe that
1
˙ 2 − V (x̄)
E := (x̄)
(8.97)
2
is a constant of the motion.
(2) Calculating the determinant
Det −∂τ2 + V 00 (x̄) ,
−T /2 ≤ τ ≤ +T /2,
(8.98)
with hard-wall boundary conditions.
8.5.3. Application to a parabolic potential well. As a first
example, we apply the semi-classical approximation to the case when
the potential V in Eq. (8.85) has a non-degenerate absolute minimum
at the origin (see Fig. 5), i.e.,
1 2 2
ω x + O(x4 ).
2
Initial and final positions are chosen to be
V (x) =
xi = xf = 0.
(8.99)
(8.100)
8.5. INSTANTONS IN QUANTUM MECHANICS
431
V (x)
x
Figure 5. Potential well V (x) with a single nondegenerate minimum at x = 0.
The unique solution to Eq. (8.93b) is x̄ = 0 for which S[x̄] = 0 and
Det [−∂τ2 + V 00 (x̄)] = Det (−∂τ2 + ω 2 ). The amplitude for the particle
to remain at the origin after “time” T is
1
hx = 0|e−ĤT /~ |x = 0i ∝ p
[1 + O(~)] .
(8.101)
Det (−∂τ2 + ω 2 )
Needed is the determinant of the Hermitean operator −∂τ2 + ω 2 .
Observe that the wave function
1
ψ0 (x) = sinh ω [τ + (T /2)]
(8.102)
ω
obeys
−∂τ2 + ω 2 ψ0 (τ ) = 0,
ψ0 (−T /2) = 0,
(∂τ ψ0 )(−T /2) = 1.
(8.103)
10
Furthermore, it can be shown that
Det −∂τ2 + ω 2 ∝ ψ0 (+T /2),
(8.104)
whereby the proportionality constant is independent of ω. Hence,
1
[1 + O(~)]
hx = 0|e−ĤT /~ |x = 0i
∝p
Det (−∂τ2 + ω 2 )
r
ω
[1 + O(~)] .
(8.105)
∝
sinh (ωT )
From the asymptotic limit T → ∞,
X
hx = 0|e−ĤT /~ |x = 0i
=
|hn|x = 0i|2 e−εn T /~
n
∝
√
ω e−ωT /2 1 + O(e−2 ω T ) [1 + O(~)]
(8.106)
,
we conclude that the energy of the ground state n = 0 is
~ω
ε0 =
,
2
10
Appendix 1 in chapter 7 of Ref. [86].
(8.107)
432
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
V (x)
(a)
+a
a
x
V (x)
+a
a
x
(b)
Figure 6. (a) Double potential well. (b) Inverted double potential well.
whereas the probability for the particle to be at the origin is
√
|hn = 0|x = 0i|2 ∝ ω [1 + O(~)] .
(8.108)
These are properties of a quantum harmonic oscillator whose ground
state wave function is (after reinstating the mass m of the particle)
m ω 41
mω 2
e− 2~ x .
(8.109)
π~
8.5.4. Application to the double well potential. Next, we
apply the semi-classical approximation to the case when the potential
V in Eq. (8.85) has doubly-degenerate absolute minima at ±a (see Fig.
6), i.e.,
V (x) = V (−x),
V (a + x) =
1 2 2
ω x + O(x4 ).
2
(8.110)
Initial and final positions are:
• Case I : xi = +xf = ∓a.
• Case II: xi = −xf = ∓a.
A solution to Eq. (8.93b) in case I is x̄ = ∓a for which S[x̄] = 0 and
Det [−∂τ2 + V 00 (x̄)] = Det (−∂τ2 + ω 2 ). The amplitude for the particle
to remain at ∓a after “time” T in the “trivial background” x̄ = ∓a is
1
h∓a|e−ĤT /~ | ∓ ai ∝ p
Det (−∂τ2 + ω 2 )
[1 + O(~)] .
(8.111)
8.5. INSTANTONS IN QUANTUM MECHANICS
433
x
+a
⌧
⌧0
a
/ 1/!
Figure 7. Sketch of a single-instanton profile.
This solution is unique if T is finite. However, in the limit T → ∞,
whereby the particle initially starts at −a, say, reaches +a at some
intermediate time, only to come back to its initial position at +T /2.
Evidently this process can repeat itself an arbitrary number of times.
We thus expect an infinity of non-equivalent solutions x̄2n labeled by
the even integer 2n = 2, 4, · · · , to Newton’s equation in the inverted
double well potential. Such solutions x̄2n , when they exist, are called
instantons whenever n is non-vanishing. The “trivial” solution x̄ = ∓a
is denoted x̄0 .
To construct x̄2n as well as to evaluate S[x̄2n ] and the Gaussian
determinant Det[−∂τ2 + V 00 (x̄2n )], we turn our attention to case II and
consider the solution x̄1 with an infinitesimally small constant of motion
(energy)
1
0+ = (x̄˙ 1 )2 − V (x̄1 ).
(8.112)
2
The trajectory x̄1 describes the particle starting at x = −a, say, and
reaching the top of the opposite hill at +a with an infinitesimally small
velocity. Hence, x̄1 is a strictly increasing function of −T /2 < τ <
+T /2, x̄˙ 1 > 0. This solution is called a single instanton and it satisfies
x̄1
Z
√
dx
˙x̄1 = + 2 V ⇐⇒ τ = τ0 + p
.
2 V (x)
(8.113)
0
The integration constant τ0 is the time at which x̄1 = 0. As we shall
see shortly, τ0 can be interpreted as the position of the instanton. The
first important property of Eq. (8.113) is that the solution x̄1 is a
function of the combination τ − τ0 . The solution obtained by reversing
time τ → −τ is called a single anti-instanton and is denoted x̄−1 .
The trajectory x̄−1 now describes the particle starting at x = +a and
reaching the top of the opposite hill at −a with an infinitesimally small
velocity after time T . Strictly speaking, an (anti-)instanton can only
be constructed in the asymptotic limit T → ∞ if it is to have a smooth
velocity.
The second important property of x̄1 follows from the fact that
Eq. (8.113) is, to a very good approximation for large times τ 1/ω
434
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
√
(remember that x̄˙ 1 is strictly positive), given by (ω ≡ + ω 2 > 0)
x̄˙ 1 = ω (a − x̄1 ) + O[(a − x̄1 )3 ]
⇐⇒ ∂τ (a − x̄1 ) = −ω (a − x̄1 ) + O[(a − x̄1 )3 ]
1
(8.114)
⇐⇒ (a − x̄1 )(τ ) ∝ c e−ω τ + · · · ,
τ
ω
1
⇐⇒ x̄1 (τ ) ∝ a − c e−ω τ + · · · ,
τ ,
ω
where the constant c is fixed by the boundary condition on x̄1 as τ →
∞. The larger the curvature ω 2 of V at the two degenerate minima
∓a, the steeper the valley between the two degenerate maxima of −V
and the longer the time a spinless point particle with unit mass on the
real line will be close to a for τ > 0 relative to the time it will be close
to the origin. The characteristic time scale of the instanton 1/ω tells
us when a spinless point particle with unit mass on the real line has a
non-negligible speed. The interpretation of the exponential dependence
on time in Eq. (8.114) is that a single instanton is well localized in time
around τ0 (see Fig. 7). There are field theories in which instantons can
be constructed but for which it is not possible to assign a characteristic
scale, say if the instanton is scale invariant. When this happens the
semi-classical method outlined here fails to extend to a field theoretical
context.
The fact that a single instanton is exponentially localized in time
around its “center” τ0 suggests that it behaves like a point particle
located at τ0 in the limit ω −1 /T → 0, ω −1 held fixed. Consequently,
the single instanton should really be denoted x̄1;τ0 . However, invariance
under time translation as T → ∞ of Ĥ and of the boundary conditions
implies that τ0 can be arbitrarily chosen, i.e., S[x̄1;τ0 ] is independent
of τ0 . 11 Taken together, these properties suggest that a saddle point
x̄n describing a trajectory that starts at −a at time −T /2, crosses
the origin n times at the successive times τ1 τ2 · · · τn−1 τn , and reaches +a when n is odd (or −a when n is even) at time
T /2 ω −1 , can be construed as a string of single instantons and
anti-instantons beginning with a single instanton x̄1;τ1 , followed by a
single anti-instanton x̄−1;τ2 , and so on. Except for the ordering of the
instantons center, τ1 τ2 · · · τn−1 τn , the action in this
instanton background is independent of the centers due to invariance
under time translation.
In summary, motivated by the existence of single instantons, we
have assumed the existence of instanton configurations x̄n , n ∈ Z,
in terms of which the transition amplitude between the initial states
xi = ±a and the final states xf = ±a can be formally written in the
11
Technically, one can always write x̄1;τ0 = f (τ − τ0 ) and perform the change
of variable τ − τ0 = τ 0 in the limit T → ∞.
8.5. INSTANTONS IN QUANTUM MECHANICS
435
semi-classical approximation as
!
∞ Z
X
−1/2
h−a|e−ĤT /~ | − ai ∝
D[x̄2n ] e−S[x̄2n ]/~ Det0 −∂τ2 + V 00 (x̄2n )
n=0
× [1 + O(~)] ,
!
∞ Z
X
−1/2
h+a|e−ĤT /~ | − ai ∝
D[x̄2n+1 ]e−S[x̄2n+1 ]/~ Det0 −∂τ2 + V 00 (x̄2n+1 )
n=0
× [1 + O(~)] .
(8.115)
This expression is formal as neither x̄n nor the measure D[x̄n ] were
explicitly constructed. The meaning of the prime over the functional
determinants also needs explanation. The construction of the instanton configurations x̄n and their measure D[x̄n ] is performed in the limit
T ω −1 whereby ω 2 := V 00 (±a). In this limit, the instanton configuration x̄n is thought of as an ordered string of single instanton/antiinstanton located at τ1 τ2 · · · τn−1 τn . Pictorially, the
trajectory x̄n is a sequence of sharp jumps (on the scale of T ω −1 )
between the values −a and +a at time τ2i+1 for a single instanton and
between the values +a and −a at time τ2(i+1) for a single anti-instanton.
The width ∼ T /(n + 1) of the plateaus at ±a is much larger than the
width ω −1 of the jumps. This is tantamount to assuming that the single instantons behave like a dilute gas of hardcore point-like particles.
If so, it is reasonable to write
+T
Z /2
Z
D[x̄n ] ≈
dτn
−T /2
n
→
T
n!
τn−1
Zτn
−T /2
Zτ3
Z
dτn−2 · · ·
dτn−1
−T /2
Zτ2
dτ2
−T /2
dτ1
−T /2
by time translation invariance of the integrand
(8.116a)
for the integral over the measure of the instanton x̄n ,
e−S[x̄n ]/~ ≈ e−n S[x̄1 ]/~
≡ e−n S1 /~ ,
+T
Z /2
Z+a
p
dτ (x̄˙ 1 )2 = dx̄1 2 V (x̄1 ),
S1 =
−T /2
−a
(8.116b)
for the Boltzmann weight of the instanton x̄n , and
!
!
n
n
Y
Y
dczero-mode
1
Kn
j
√
p
p
=:
dτj
2π~
Det (−∂τ2 + ω 2 )
Det0 [−∂τ2 + V 00 (x̄n )]
j=1
j=1
(8.116c)
436
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
for the measure of the zero modes from the kernel [−∂τ2 + V 00 (x̄n )] of
the instanton x̄n . The right hand side of Eq. (8.116c) defines the number K that measures the effect on the determinant of “how much” (or
rather how little) V 00 (x̄n ) deviates from ω 2 as is explained in Eqs. (2.26)
and (2.27) from Ref. [86]. Also, K is implicitly assumed to be nonvanishing, positive, and independent of the center of the instanton. To
see this requires a careful definition and evaluation of Det [−∂τ2 + V 00 (x̄1 )]
in account of time translation invariance that we postpone for the time
being. We will see that there always exist an eigenvalue zero of −∂τ2 +
V 00 (x̄1 ) and that this eigenvalue must be removed from Det [−∂τ2 + V 00 (x̄n )]
as is indicated by the prime in Det0 [−∂τ2 + V 00 (x̄n )]. We will then see
that the integration over the instantons coordinates in Eq. (8.116a) is
just what is needed to account for the integration over the measure
of the zero modes (the eigenfunctions with vanishing eigenvalue). Finally, observe that although it had been assumed that all instantons
center are very (infinitely) far from each others, Eq. (8.116a) breaks
this assumption. We will verify below that the error thus committed
is negligible in the limit T → ∞.
The outcome of this discussion is that, within the dilute instanton
gas approximation, the semi-classical approximation for the amplitude
of a spinless point particle with unit mass on the real line to propagate
between the minima of a double well potential V is
−ĤT /~
h−a|e
2n
∞
X
K e−S1 /~ T
| − ai ∝ p
× [1 + O(~)]
(2n)!
Det (−∂τ2 + ω 2 ) n=0
√
∝ ω e−ωT /2 cosh K e−S1 /~ T × [1 + O(~)]
(8.117a)
1
for the case when initial and final positions are the position −a at which
V is minimal and
−ĤT /~
h+a|e
2n+1
∞
X
K e−S1 /~ T
× [1 + O(~)]
| − ai ∝ p
(2n + 1)!
Det (−∂τ2 + ω 2 ) n=0
√
∝ ω e−ωT /2 sinh K e−S1 /~ T × [1 + O(~)]
(8.117b)
1
for the case when the initial position is the position −a at which V is
minimal while the final position is the position +a at which V is also
minimal. By comparison with the exact eigenstate expansion
hxf |e−Ĥ T /~ |xi i =
XX
m σ=±
hxf |m; σi hm; σ|xi i e−εm;σ T /~ ,
(8.118)
8.5. INSTANTONS IN QUANTUM MECHANICS
437
where m labels all the energy eigenstates in a single potential well
whereas σ = ± labels (naively) the “bonding” and “anti-bonding” linear combinations, we conclude that, within the semi-classical approximation,
1
ε0;− = ~ ω − ~ K e−S1 /~
(8.119a)
2
is the (lowest) energy of the bonding state with the squared amplitude
1√
|hx = −a|0; −i|2 ∝ +
ω
(8.119b)
2
and the overlap
1√
ω
(8.119c)
hx = +a|0; −i h0; −|x = −ai ∝ +
2
on the one hand, while
1
ε0;+ = ~ ω + ~ K e−S1 /~
(8.120a)
2
is the (first excited) energy of the anti-bonding state with the squared
amplitude
1√
|hx = −a|0; +i|2 ∝ +
ω
(8.120b)
2
and the overlap
1√
hx = +a|0; +i h0; +|x = −ai ∝ −
ω
(8.120c)
2
on the other hand. The difference in the energy of the bonding and
anti-bonding states is proportional to exp(−S1 /~). It vanishes as the
+a
p
√
R
surface S1 = dx̄1 2 V (x̄1 ) underneath the “tunneling barrier 2 V ”
−a
diverges. In this limit of an infinitely high potential barrier, bonding
and anti-bonding states are degenerate in energy and the single potential well result is recovered up to a degeneracy of two. This degeneracy
is broken by barrier penetration which is exponentially small [strictly
speaking only ε0;− − ε0;+ should be expanded semi-classically since the
correction of order ~ beyond the semi-classical approximation of the
individual energies ε0;∓ is already much larger than the exponentially
small lifting of the degeneracy in the limit S1 → ∞ resulting from the
symmetry under x → −x of V (±a + x) ≈ 21 ω 2 x2 ].
Is our assumption of a dilute gas of instanton self-consistent? For
any fixed value of
z := K e−S1 /~ T ,
(8.121)
P m
the terms in the exponential series
z /m! grow with m until m is
m
of order of z. After this point the terms will decrease rapidly with m.
The important terms in the instanton gas expansion are thus those for
which
m
m ≤ K e−S1 /~ T ⇐⇒
≤ K e−S1 /~ .
(8.122)
T
438
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
For small ~, the important terms in the dilute gas expansion are those
for which the gas density m/T is exponentially small. The average
separation T /m between instantons is therefore exponentially large
and independent of T for sufficiently large T (as the dependence of
S1 on T becomes negligible for T → ∞) provided we can show that
K is also independent of T as T → ∞. We conclude that the error
committed in Eq. (8.116a) is inconsequential.
It is time to return to the evaluation of the determinant Det [−∂τ2 + V 00 (x̄1 )].
Recall that we are seeking the eigenfunctions and eigenvalues of −∂τ2 +
V 00 (x̄1 ) obeying hard-wall boundary conditions when τ = ±T /2. We
also recall that x̄1 is a solution to Newton’s equation −ẍ + V 0 (x) = 0
with vanishing constant of motion (energy) 12 (x̄˙ 1 )2 − V (x̄1 ) = 0 that,
without loss of generality, represents a single instanton centered at τ0
(the time at which x̄1 vanishes). In other words, the trajectory x̄1 is a
strictly increasing function of τ − τ0 that interpolates
between −a at
p
˙
−T /2 and +a at +T /2 with a velocity x̄1 = 2 V (x̄1 ) which is strictly
positive if −T /2 < τ < +T /2 and vanishes at ∓T /2.
The first important observation is that the velocity x̄˙ 1 is itself an
eigenfunction of −∂τ2 + V 00 (x̄1 ) with vanishing eigenvalue that obeys
the hard-wall boundary conditions. To see this, we note that x̄˙ 1 does
vanish at the initial and final times ∓T /2. Moreover,
2
−∂τ + V 00 (x̄1 ) x̄˙ 1 = ∂τ [−x̄¨1 + V 0 (x̄1 )] = 0
(8.123)
since x̄1 was constructed to obey Newton’s equation in the inverted
potential −V . The normalized eigenfunction of −∂τ2 + V 00 (x̄1 ) with
vanishing eigenvalue that obeys hard-wall boundary conditions will be
denoted [compare with Eq. (8.94) and make use of Eq. (8.116b)]
1
x1 := p x̄˙ 1 ,
S1
+T
Z /2
Z+a
dτ (x̄˙ 1 )2 = dx̄1 x̄˙ 1 = S1 .
(8.124)
−a
−T /2
Since −∂τ2 + V 00 (x̄1 ) with hard-wall boundary conditions defines a Hermitean Hamiltonian for a spinless point particle of unit mass in one
dimension, and since the eigenfunction x1 is nodeless on −T /2 < τ <
+T /2, the eigenvalue λ1 = 0 of x1 must be the lowest in the spectrum: All remaining eigenfunctions xn , n = 2, 3, · · · , must have nodes
on −T /2 < τ < +T /2, i.e., strictly positive eigenvalues λn . Assuming
the expansion
y = c1 x 1 +
∞
X
cn x n
(8.125)
n=2
for a small deviation y around x̄1 , we define the restricted functional
determinant Det0 [−∂τ2 + V 00 (x̄1 )] to be the functional determinant of
8.5. INSTANTONS IN QUANTUM MECHANICS
439
[−∂τ2 + V 00 (x̄1 )] with the omission of its vanishing eigenvalue λ1
1
0
p
Det
[−∂τ2
+V
00 (x̄
1 )]
:=
∞
Y
1
p .
λn
n=2
With the help of the Gaussian identity
Z+∞
1
1
dc
p =
√ n e− 2~ cn λn cn
λn
2π~
(8.126)
(8.127)
−∞
and the orthonormality of the eigenfunctions xn in the mode expansion (8.125), we may write


+T
∞
+∞
R /2
P
1
∞
∞ Z
− 2~
ck cl
dτ xk [−∂τ2 +V 00 (x̄1 )] xl
Y
Y
1
dc
n
k,l=2
−T
/2
e
p =
√
.
λ
2
π
~
n
n=2
n=2
−∞
(8.128)
Since λ1 = 0, we can extend the lower bound on the sum in the argument of the exponential to include k, l = 1,


+T
∞
+∞
R /2
P
Z
1
∞
∞
− 2~
ck cl
dτ xk [−∂τ2 +V 00 (x̄1 )] xl
Y
Y 1
dc
−T /2
p =
√ n  e k,l=1
.
λ
2
π
~
n
n=2
n=2
−∞
(8.129)
Finally, we make another use of the mode expansion (8.125) [see also
Eqs. (8.93a) and (8.95)],


+T
+∞
R /2
1
∞ Z
− 2~
dτ y [−∂τ2 +V 00 (x̄1 )] y
Y
1
dc
n 
−T /2

p
√
=
e
.
2π~
Det0 [−∂τ2 + V 00 (x̄1 )]
n=2
−∞
(8.130)
The existence of the eigenvalue λ1 = 0 has the disastrous consequence
that Det [−∂τ2 + V 00 (x̄1 )] vanishes. The eigenfunction x1 is called a zero
mode. It originates in the fact that the center τ0 of x̄1 can always be
chosen to be zero with the help of the change of variable τ − τ0 = τ 0 in
the limit T → ∞. As a corollary Det0 [−∂τ2 + V 00 (x̄1 )] is independent
of τ0 and so is K. The association of instantons to zero modes is not
particular to this example but is a generic feature of instanton physics.
It is crucial to avoid integrating too early over the measure of zero
modes: Had we formally integrated over the measure dc1 in Eq. (8.130)
we would have encountered a divergence. The strategy that we will use
instead is to treat the zero mode separately from all other eigenmodes.
Treating the zero√modes separately requires an explicit construction
of the measure dc1 / 2 π ~ in terms of the instanton x̄1 . Fortunately,
this can be done without any detailed knowledge on the potential other
than the existence of the two degenerate minima of V . On the one
hand, a small change dτ0 in the center τ0 of the instanton induces the
440
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
V (x)
(a)
2a
a
(b)
0
+a
+2a
x
+a
+2a
x
V (x)
2a
a
Figure 8. (a) Periodic potential. (b) Inverted periodic potential.
small change x̄˙ 1 dτ0 in the instanton. Since this small change vanishes
when τ = ∓T /2, we can ascribe it to a change
dy = x̄˙ 1 dτ0
(8.131)
of the small Gaussian fluctuations obeying hard-wall boundary conditions about the instanton. On the other hand, a small change dc1 in
the expansion coefficient of the zero-mode contributes a change
dy = x1 dc1
(8.132)
to the same small Gaussian fluctuations obeying hard-wall boundary
conditions about the instanton.
We have thus derived a relation be√
tween the measure dc1 / 2 π ~ of the zero mode and the arbitrariness
in the choice of the instanton center,
1
dc
dy
√ 1
= √
2π~
2 π ~ x1
r
S1 dy
By Eq. (8.124)
=
2 π ~ x̄˙ 1
r
S1
By Eq. (8.131)
=
dτ .
2π~ 0
By Eq. (8.132)
(8.133)
We finally arrive at the result for K defined in Eq. (8.116c)
s
s
r
dc
√ 1
2
2
Det (−∂τ + ω )
S1
Det (−∂τ2 + ω 2 )
K := 2 π ~ ×
=
.
dτ0
Det0 [−∂τ2 + V 00 (x̄1 )]
2 π ~ Det0 [−∂τ2 + V 00 (x̄1 )]
(8.134)
√
Observe that K is proportional to 1/ ~.
8.5.5. Application to the periodic potential. We apply the
semi-classical approximation to the case when the potential V in Eq. (8.85)
is periodic, i.e., has infinitely many degenerate minima at n a, n ∈ Z
8.5. INSTANTONS IN QUANTUM MECHANICS
441
[see Figs. 8(a) and 8(b)],
V (x) = V (x + n a),
V (n a + x) =
1 2 2
ω x + O(x4 ).
2
(8.135)
Initial and final states are
xi = ji a,
xf = jf a.
(8.136)
We can borrow the complete analysis of the double well potential except
for one restriction present before and absent here. Owing to the periodicity of the minima, it is not necessary anymore to alternate instantons
and anti-instantons in time. To see this draw all the minima of the potential on the line, thus defining a one-dimensional lattice with lattice
spacing a. For the double well potential, the lattice is made of two sites.
For the periodic potential, the lattice is made of infinitely many sites.
Picture all the instanton configurations as a time ordered sequence of
nearest-neighbor jumps taking place at times τ1 τ2 · · · , in such
a way that initial and final states are reached after a time T . A jump
to the right (left) at time τm represents a single (anti-) instanton centered at τm . For the double well problem, jumping can only take place
between the same two sites and thus a jump to the right is necessarily
followed by a jump to the left. In contrast, for the periodic potential,
the only condition on the sequence of nearest neighbor jumps at the
ordered times τ1 τ2 · · · τn−1 τn is that the number nr of
nearest-neighbor jumps to the right minus the number nl of nearest
neighbor-jumps to the left equals jf − ji . In particular, for nr and nl
given, it does not matter whether jumps to the right alternate with
jumps to the left. This implies that the integration over the instanton
centers is


τnr
τ3
τ2
+T
/2
Z
Z
Z
Z
T nr +nl


=
dτnr
dτnr −1 · · ·
dτ2
dτ1 
nr ! nl !
−T /2

−T /2
+T
Z /2

×
−T /2
τ̄n
Z
−T /2
Zτ̄3
l
dτ̄nl −1 · · ·
dτ̄nl
−T /2
−T /2
−T /2
Zτ̄2
dτ̄2

(8.137)

dτ̄1  .
−T /2
We conclude that the amplitude for the initial state xi = ji a to evolve
into the final state xf = jf a is, as T → ∞, given by the semi-classical
approximation
−ĤT /~
hjf a|e
|ji ai ≈
√
−ωT /2
ωe
∞
X
(K e−S1 /~ T )n+n̄
δn−n̄,jf −ji .
n! n̄!
n,n̄=0
(8.138)
442
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
If we represent the Kronecker delta on the right-hand side by an integral, we may then write
−ĤT /~
hjf a|e
√
−ωT /2
|ji ai ≈ ω e
Z2 π
∞
X
dθ +iθ(n−n̄−jf +ji )
(K e−S1 /~ T )n+n̄
.
e
n!
n̄!
2
π
n,n̄=0
0
(8.139)
After interchanging the order between the summations over the integers
n and n̄ and the integral over θ, we reach the desired representation
−ĤT /~
hjf a|e
√
−ωT /2
Z2 π
|ji ai ≈ ω e
dθ −iθ(jf −ji )
e
2π
0
∞
X
n 1
n̄
1
×
K e−S1 /~ T e+iθ
K e−S1 /~ T e−iθ
n!
n̄!
n,n̄=0
=
√
−ωT /2
Z2 π
ωe
dθ −iθ(jf −ji )
exp 2 K e−S1 /~ T cos θ .
e
2π
0
(8.140)
The interpretation of this result is that the infinite degeneracy of the
harmonic oscillator mode in the limit of infinite potential barrier has
been lifted by tunneling. A band of low lying states has emerged with
the energy
1
ε(θ) = + ~ ω − 2~ K e−S1 /~ cos θ,
2
0 ≤ θ ≤ 2 π,
(8.141)
and the overlap
1
hθ|x = ji ai ∝ √
ω 1/4 e+iθji .
2π
(8.142)
Within the semi-classical approximation, we have recovered the tightbinding band of states of the Hamiltonian
X †
X †
1
Ĥtb := ~ ω
ĉj ĉj + ~ K e−S1 /~
ĉj ĉj+1 + h.c. .
(8.143)
2
j∈Z
j∈Z
The band width 4 ~ K e−S1 /~ is twice that for the double well potential.
8.5.6. The case of an unbounded potential of the cubic
type. So far, we only considered potentials V (x) that have absolute
minima. When the potential has a unique global minimum, the classical path x̄ describing a spinless point particle with unit mass on the
real line stuck at the bottom of the potential minimizes the Euclidean
8.5. INSTANTONS IN QUANTUM MECHANICS
(a)
(b)
V (x)
443
V (x)
x0
V0
V0
x
x0
x
Figure 9. (a) Potential well V (x) with a metastable
minimum at x = 0. (b) Inverted potential well V (x) with
a metastable minimum at x = 0.
action. Indeed, for any trajectory x(τ ),
+T
Z /2
S[x] =
dτ
1 2
ẋ + V (x)
2
−T /2
≥ T V (x̄)
= S[x̄].
(8.144)
By construction, x̄ obeys Newton equation in the inverted potential
−V , i.e.,
x̄¨ = V 0 (x̄),
(8.145)
since x̄¨ = 0 and V 0 (x̄) = 0. The trajectory x̄ is also characterized by
the constant of motion
1 2
E[x̄] =
x̄˙ − V (x̄)
2
= −V (x̄).
(8.146)
When the potential has several global minima ι, all trajectories
x̄ι for which a spinless point particle with unit mass on the real line
is stuck at the ι-th bottom of the potential are global minima of the
Euclidean action S[x] with the action
S[x̄ι ] = T V (x̄ι )
(8.147)
and the constant of motion
E[x̄ι ] = −V (x̄ι ).
(8.148)
In addition, we constructed instanton trajectories x̄n that are local
minima of the action S[x]. Instantons can be visualized as trajectories by which a spinless point particle with unit mass on the real line
444
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
rolls n-times away from the absolute maxima of the inverted potential −V (x) with the constant of motion [compare with Eq. (8.112) for
which V (x̄ι ) = 0+ ]
1 2
x̄˙ − V (x̄n ) = −V (x̄ι ) = E[x̄ι ],
2 n
Correspondingly, instanton x̄n has the action
E[x̄n ] :=
+T
Z /2
S[x̄n ] =
dτ
∀ι.
(8.149)
1 2
x̄˙ + V (x̄n )
2 n
−T /2
+T
Z /2
=
dτ
1 2
x̄˙ − V (x̄n ) + 2 V (x̄n )
2 n
−T /2
+T
Z /2
=
dτ [−V (x̄ι ) + 2 V (x̄n )]
−T /2
≥ T V (x̄ι )
= S[x̄ι ],
∀ι.
(8.150)
Potentials with degenerate absolute minima are relevant to a dissipative Josephson junction but not to the Caldeira-Leggett model of
quantum tunneling for which the potential is of the cubic type [see
Eq. (8.39)], i.e., unbounded from below with a single metastable minimum (see Fig. 9). For concreteness, we take the classical potential to
be
1
1
V (x) = ω 2 x2 − λ2 x3
2
3
2 27
x
x
=
V
1−
,
(8.151)
4 0 x0
x0
3 ω2
V0 ≡ V (x)|x= ω2 ,
x0 ≡
,
0 < ω, λ ∈ R.
2 λ2
λ2
Classically, a spinless point particle with unit mass on the real line that
initially sits at the origin x = 0 will remain for ever in the metastable
minimum of the cubic potential (8.151). If x̄ denotes this trajectory,
+T
Z /2
S[x̄] =
dτ
1 2
x̄˙ + V (x̄)
2
= T V (0) = 0
(8.152)
−T /2
is a local minimum of the action with the constant of the motion
1 2
E[x̄] = x̄˙ − V (x̄) = −V (0) = 0.
(8.153)
2
8.5. INSTANTONS IN QUANTUM MECHANICS
445
There are trajectories called bounces and denoted x̄n (τ ) that share
the same energy (8.153) with x̄(τ ) ≡ 0, are local extrema of the action,
and are built out of the single bounce x̄1 (τ ) by which a spinless point
particle with unit mass on the real line rolls along the constant energy
curve
1
0 = (x̄˙ 1 )2 − V (x̄1 )
(8.154)
2
from the top of the hill at x = 0 to the classical turning point x0 of the
inverted potential −V (x) to come back to the top of the hill at x = 0.
The action of a single bounce x̄1 (τ ) is larger than that of x̄(τ ),
+T
Z /2
S[x̄1 ] =
dτ
1
2
˙
(x̄ ) + V (x̄1 )
2 1
−T /2
+T
Z /2
=
dτ 2 V (x̄1 )
−T /2
Zx0
=
0
dτ
dx̄1
2 V (x̄1 ) +
dx̄1
Zx0
= 2
dx̄1 p
0
Zx0
dx̄1
= 2
1
2 V (x̄1 )
Z0
dx̄1
x0
dτ
2 V (x̄1 )
dx̄1
2 V (x̄1 )
p
2 V (x̄1 )
0
≡ S1 > 0.
(8.155)
One difference with having a potential with an absolute minimum is
that the bounce is not a local minimum but a saddle point. To see
this, for any positive energy E consider a classical path x̄E (τ ) obeying
the classical equation of motion
T
T
x̄¨E = V 0 (x̄E ),
− ≤τ ≤+
(8.156)
2
2
with the constant energy
1
0 < E = (x̄˙ E )2 − V (x̄E )
(8.157)
2
whereby the particle sits at x = 0 at time −T /2 with the positive
kinetic energy E, reaches a turning point xE larger than the classical
turning point x0 , and return to x = 0 at time +T /2. The dependence
on the energy E > 0 of the action S[x̄E ] must be unbounded from
below as E → ∞ since this limit corresponds to a particle provided
with enough kinetic energy to spend more and more time into the
446
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
classically forbidden region to the right of the turning point x0 in Fig.
10. In mathematical terms,
+T
Z /2
S[x̄E ] =
1
(x̄˙ )2 + V (x̄E )
2 E
1
(x̄˙ )2 − V (x̄E ) + 2 V (x̄E )
2 E
dτ
−T /2
+T
Z /2
=
dτ
−T /2
+T
Z /2
=
dτ [E + 2 V (x̄E )]
−T /2
+T
Z /2
=T E + 2
dτ V (x̄E )
−T /2
Zx0
=T E + 4
0
Zx0
=T E + 4
0
dτ
dx̄E
V (x̄E ) + 4
dx̄E
ZxE
dx̄E
x0
V (x̄E )
+4
dx̄E p
2 [E + V (x̄E )]
dτ
V (x̄E )
dx̄E
ZxE
V (x̄E )
.
2 [E + V (x̄E )]
dx̄E p
x0
(8.158)
The first and second terms are both positive and non-vanishing. The
last term is negative and grows as the area underneath the (negative)
dτ
curve 4 dx̄
V (x̄E ) between x0 and xE . As limE→∞ xE = ∞, the last
E
contribution always dominates over the first two contributions to the
right hand side of Eq. (8.158). A single bounce is thus a saddle point
as it is a local maximum in the “direction” made of the submanifold
of classical path x̄E whereas it is a local minimum in the remaining
orthogonal “directions” in the space of all paths entering the path integral representation of the partition function.
A corollary to the unboundness of S[x̄E ] is the fact that −∂τ2 +
V 00 (x̄1 ) must have a negative eigenvalue. Indeed, one observes that the
velocity of a single bounce is an eigenfunction of [−∂τ2 + V 00 (x̄1 )] with
vanishing eigenvalue that obeys the hard-wall boundary conditions and
supports one node. Since the velocity of a single bounce is an eigenstate
of [−∂τ2 + V 00 (x̄1 )] with vanishing eigenvalue and supports one and only
one node, there must exist one and only one nodeless eigenfunction of
−∂τ2 +V 00 (x̄1 ) with negative eigenvalue. The counterpart to the constant
K in Eq. (8.116c) must then be imaginary for a single bounce.
8.5. INSTANTONS IN QUANTUM MECHANICS
447
V (x)
x0 xE x
V0
Figure 10. A classical path for an inverted potential
well −V (x) with a metastable maximum at x = 0 by
which the particle starts at the origin x = 0 with kinetic
energy E, reaches the turning point xE past the classical
turning point x0 , and “rolls” back to the origin x = 0
after time T .
It is now possible to salvage a physical interpretation of the semiclassical expansions (8.92), (8.95), and (8.106) around multi-bounce
trajectories. The fact that the counterpart to the constant K in Eq. (8.116c)
is imaginary for bounces means that the ground state in the expansion (8.106) is not the true ground state of the Hamiltonian but must
be interpreted as an unstable ground state with the complex energy
ε0 = Re ε0 + iIm ε0 and the inverse lifetime
2 Im ε0 = |K| e−S1 /~ ,
Γ0 := (8.159)
~ whereby S1 given by Eq. (8.155). The tunneling rate (8.159) should be
compared with Eq. (8.41). To reach this conclusion it is sufficient to
replace Eq. (8.117a) by
n
∞ X
i(K/2) e−S1 /~ T
1
−ĤT /~
hx = 0|e
|x = 0i
∝p
× [1 + O(~)]
n!
Det (−∂τ2 + ω 2 ) n=0
√
−S /~
∝ ω e−ωT /2 ei(K/2) e 1 T × [1 + O(~)] .
(8.160)
Upon the analytical continuation T = iT , Eq. (8.160) implies that the
amplitude for a spinless point particle with unit mass on the real line
to remain at the metastable minimum x = 0 of a cubic-like potential
decays exponentially fast with T .
There is a subtlety with the factor 1/2 appearing in (K/2) from
Eq. (8.160). The factor 1/2 arises from the full Gaussian path integration about the single-bounce saddle-point. The Gaussian path integral
is not convergent since it contains the divergent Riemann integral
Z∞
dE −(S1 /~)− 1 (S100 E 2 /~)+···
2
√
e
.
(8.161)
2π~
0
448
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
Here, E is a “deviation” about the single-bounce saddle-point E = 0
in the unstable direction corresponding to Fig. 10. The unstable direction is encoded by the fact that S100 is negative. Hence, the Gaussian
Riemann integral over E is divergent. To make sense of this divergent
integral one replaces the path of integration 0 ≤ E < ∞ along the half
line by the path of integration E = iz where 0 ≤ z < ∞ along half of
the imaginary axis,
Z∞
1
dz
1 1
00 2
√
(8.162)
e−(S1 /~)− 2 (|S1 | z /~)+··· = p 00 e−S1 /~ .
2 |S1 |
2π~
0
The replacement K → (K/2) originates from the factor 1/2 in Eq. (8.162).
8.6. The quantum-dissipative Josephson junction
In section 8.4.2.1, we introduced the Caldeira-Leggett (CL) model
[84] of dissipative quantum mechanics, which is defined by the partition
function ( ˙ denotes τ derivative, 0 denotes q derivative)
Z
0
Zβ = Nbath D[q] e−Sβ /~ ,
(8.163a)
with the additive decomposition of the action
Sβ0 := S0 + S1 + Sext
(8.163b)
into three contributions. There is the dissipative-free action
Zβ
dτ
S0 :=
M 2
q̇ + V (q)
2
(8.163c)
0
=
X M
$l
2
p
2
$l q(+$l ) q(−$l ) + β [V (q)]$ δ$l ,0 .
l
There is the dissipative action
2
Z+∞ Zβ
q(τ ) − q(τ 0 )
0 η
S1 :=
dτ
dτ
4π
|τ − τ 0 |
−∞
0
X η
=
|$ | q
q
.
2 l (+$l ) (−$l )
$
(8.163d)
l
Finally, there is the driving term
Zβ
Sext := −
dτ Fext (τ ) q(τ )
(8.163e)
0
=−
X
$l
Fext (+$l ) q(−$l ) .
8.6. THE QUANTUM-DISSIPATIVE JOSEPHSON JUNCTION
449
Here, all trajectories q(τ ) entering the path integral are periodic in
imaginary time,
1 X
2π
q(τ ) = q(τ + β) = √
l,
l ∈ Z.
q$l e−i$l τ ,
$l =
β
β $
l
(8.164)
The CL model is constructed so that, after analytical continuation
τ = it to real time t, the particle of mass M with coordinate q(t) along
the real line is subjected to a force −V 0 (q) arising from a potential V (q)
of the cubic type with a single metastable minimum, to a frictional
force −η q̇(τ ), and, finally, to an external force (or source term) Fext (t)
in the classical limit ~ → 0. The kernel of S1 , which is non-local in
imaginary (Matsubara) time, and the proportionality constant Nbath
are the remnants of the interaction between the particle of mass M
and a bath made of infinitely many independent harmonic oscillators.
A semi-classical estimate of the probability per unit time Γ0 for the
particle to tunnel out from the metastable minimum of V (q) is [see
Eq. (8.159) and (8.134)], in the absence of an external force,
B0
Γ0 = A0 e− ~ [1 + O(~)] ,
B 0 = S0 [q̄1 ] + S1 [q̄1 ],
r
1/2
B 0 Det D0 0
A =
,
2 π ~ Det0 D1 (8.165a)
and
D0 q(τ ) = −∂τ 2 + $0
2
η
q(τ ) +
πM
Z+∞
q(τ ) − q(τ 0 )
dτ 0
,
(τ − τ 0 )2
−∞
D1 q(τ ) =
Z+∞
1
η
q(τ ) − q(τ 0 )
−∂τ 2 +
V 00 (q̄1 ) q(τ ) +
dτ 0
.
M
πM
(τ − τ 0 )2
−∞
(8.165b)
This estimates relies on saddle-point approximations to the path integral about multi-bounce trajectories q̄n , n = 1, 2, · · · , that are assumed
to behave like n non-interacting single-bounce trajectories q̄1 . The
prime over the functional determinant of the non-local propagator D1
says that the zero eigenvalue of D1 (corresponding to a uniform translation of the bounce along Matsubara time) is to be omitted. The
absolute value in A0 is needed as bounces are not local minima of the
action but saddle-points, i.e., there exists one negative eigenvalue of
D1 . This is not so for the propagator D0 which is evaluated at the
single local minimum of V (q), ω02 ≡ V 00 (q = qmin ). Equation (8.165)
holds for all values of the damping coefficient η.
450
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
The quantum-dissipative Josephson junction is related to the CL
model by the identifications
q(τ ) −→ φ− (τ )
(8.166a)
between the particle position in the CL model and the Josephson phase,
2
~
M −→ C
[units are: (charge2 /energy) (energy2 ×time2 /charge2 )= energy×time2 ]
e∗
(8.166b)
between the particle mass in the CL model and the capacity,
V (q) −→ −
~
I cos φ− ,
e∗ 0
I0 =
2 e∗ UJ
,
~
(8.166c)
between the metastable potential in the CL model and the Josephson
coupling,
2
1
~
η −→
,
[units are: (energy×time/charge2 )−1 (energy2 ×time2 /charge2 )= energy×time]
Rs e∗
(8.166d)
between the friction in the CL model and the Ohmic resistance
Fext (τ ) −→ −
~
I(τ ),
e∗
(8.166e)
between the driving forces, and
Sβ0 −→ S0 kin + S0 int + S1 + Sext ,
(8.166f)
whereby we have split the dissipative-free action into the dissipativefree kinetic action
S0 kin
~
= ∗
e
Zβ
dτ
1 ~
C (φ̇ )2 ,
2 e∗ −
(8.166g)
0
and the dissipative-free interacting action
S0 int
~
= ∗
e
Zβ
dτ (−I0 ) cos φ− ,
(8.166h)
0
while the dissipative action is
~
S1 = ∗
e
Z+∞ Zβ
φ− (τ ) − φ− (τ 0 ) 2
0 1 1 ~
dτ dτ
4π Rs e∗
|τ − τ 0 |
−∞
0
~ X1 1 ~
= ∗
|$ | φ
φ
,
e $ 2 Rs e∗ l −(+$l ) −(−$l )
l
(8.166i)
8.6. THE QUANTUM-DISSIPATIVE JOSEPHSON JUNCTION
451
and the driving action has become
Sext
~
= ∗
e
Zβ
dτ I(τ ) ϕ− (τ ).
(8.166j)
0
There is one essential difference between the original CL and the dissipative Josephson junction, namely the fact that the Josephson angle
φ− is a compact degree of freedom (it is defined modulo 2 π, i.e., on
the unit circle). This difference invalidates the assumption that multibounces (for I 6= 0) or multi-instantons configurations (for I = 0) form
a dilute non-interacting gas. As a result, we will see that the dissipationless regime 1/Rs → 0 and strong dissipation regime 1/Rs → ∞ are
not smoothly connected, to the contrary of the CL model (8.165) for
which the limits η → 0 and η → ∞ are smoothly connected.
With the introduction of
• the quantum resistors for Cooper pairs
~
=: R~ ,
e∗2
2 π R~ =
2π~
h
= ∗2 =: Rh ,
∗2
e
e
(8.167a)
• the ratio of the quantum resistor to the Ohmic resistance
R
1 Rh
1
1 ~
= ~ =
=:
α,
∗2
Rs e
Rs
2 π Rs
2π
(8.167b)
• the Josephson potential
−
~
I cos φ− = −2 UJ cos φ− ,
e∗ 0
(8.167c)
the partition function for the dissipative Josephson junction becomes
Z
0
(8.168a)
Zβ = Nbath
D[φ− ] e−Sβ /~ ,
with the additive decomposition of the action
Sβ0 := S0 kin + S0 int + S1 + Sext
(8.168b)
in terms of four contributions: the dissipation-free kinetic action
Zβ
S0 kin =
dτ
2
~
C R h ∂ τ φ−
4π
0
X ~
=
C Rh $l2 φ−(+$l ) φ−(−$l ) ,
4π
$
l
(8.168c)
452
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
the dissipation-free interacting action
Zβ
S0 int =
dτ (−2 UJ ) cos φ−
(8.168d)
0
Xp
=
β (−2 UJ ) cos φ− $ δ$l ,0 ,
l
$l
the dissipative action
Z+∞ Zβ
φ− (τ ) − φ− (τ 0 ) 2
0 ~ α
S1 =
dτ dτ
4π 2 π
|τ − τ 0 |
−∞
(8.168e)
0
X ~
=
α |$l | φ−(+$l ) φ−(−$l ) ,
4π
$
l
and the driving action
Zβ
dτ
Sext =
~
I(τ ) φ− (τ )
e∗
0
(8.168f)
X ~
=
I
φ
.
e∗ (+$l ) −(−$l )
$
l
Whereas the Josephson potential −2 UJ cos(φ− ) is local in Matsubara
time, S1 only becomes local after performing a Fourier transformation
to Matsubara frequencies.
From now on, we only consider the case when the time-independent
external current (bias) vanishes,
I = 0.
(8.169)
Comparison of S0 kin and S1 suggests that there are two distinct
regimes of frequencies:
(1) The regime of weak dissipation
C Rh |$l | α ⇐⇒ |$l | 1
1/Rs
α=
.
C Rh
C
(8.170a)
In this regime, the propagator defined by
2 π/~
C Rh $l2 + α |$l |
2 π/~
α
=
+O
C Rh $l2
C Rh |$l |
D$l :=
(8.170b)
decays quadratically fast with large frequencies to a good approximation.
8.6. THE QUANTUM-DISSIPATIVE JOSEPHSON JUNCTION
453
(2) The regime of strong dissipation
C Rh |$l | α ⇐⇒ |$l | 1
1/Rs
.
α=
C Rh
C
(8.170c)
In this regime,
2 π/~
C Rh $l2 + α |$l |
2 π/~
C Rh |$l |
=
+O
α |$l |
α
D$l =
(8.170d)
is inversely proportional to small frequencies to a good approximation.
Comparison of S0 kin + S1 and S0 int suggests that there are two distinct
regimes of Josephson coupling:
(1) A regime
C Rh × UJ
1,
~
C Rh × UJ
α,
~
(8.171a)
in which perturbation theory in powers of the Josephson coupling might be sensible.
(2) A regime
C Rh × UJ
1,
~
C Rh × UJ
α,
~
(8.171b)
in which a good semi-classical approximation when α = 0
might remain sensible for finite α.
454
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
8.7. Duality in a dissipative Josephson junction
To simplify notation, we rewrite Eq. (8.168) as
Z
Zβ = Nbath
0
D[φ] e−Sβ /~ ,
Sβ0 = S0 kin + S0 int + S1 + Sext ,
Zβ
S0 kin =
0
X 1
1
m $l2 φ(+$l ) φ(−$l ) ,
dτ m (∂τ φ)2 =
2
2
$
l
Zβ
S0 int =
dτ (−y) cos φ =
Xp
β (−y) (cos φ)$ δ$l ,0 ,
l
$l
0
2 X
Z+∞ Zβ
φ(τ ) − φ(τ 0 )
η
0 η
dτ dτ
=
|$l | φ(+$l ) φ(−$l ) ,
S1 =
4π
|τ − τ 0 |
2
$
−∞
0
l
Zβ
Sext =
dτ J(τ ) φ(τ ) =
X
J(+$l ) φ(−$l ) ,
$l
0
(8.172a)
whereby 0 ≤ φ ≤ 2 π is the angular coordinate of a “particle” on a
circle of unit radius and
~
C Rh ,
2π
~
I(τ ).
e∗
(8.172b)
In the absence of an external force (or source), J(τ ) = 0, correlation
(Green) functions are obtained from
m≡
*
n
Y
j=1
+
y ≡ 2 UJ ,
Zβ
~
α,
2π
J(τ ) ≡
∂ n Zβ
1
:= (−~)
.
Zβ ∂J(τ1 ) · · · ∂J(τn ) J(τ )=0
n
φ(τj )
η≡
(8.172c)
From now on, ~ = 1. Our goal is to approximate the partition function
in regimes (8.171a) and (8.171b), respectively.
8.7.1. Regime m y 1 and m y η. When the characteristic
energy scale y is the smallest in the problem (aside from the temperature β −1 ), it appears natural to expand formally the partition function
in powers of y. In this context we are going to reinterpret y/2 as the
8.7. DUALITY IN A DISSIPATIVE JOSEPHSON JUNCTION
455
fugacity of a neutral plasma of classical point-like particles with coordinates 0 ≤ τ ≤ β interacting through the propagator
1 X
D(τ ) :=
D$l e−i$l τ
β $
l
See Eqs. (8.170b) and (8.170d)
=
1
1 X
e−i$l τ
2
β $ m $l + η |$l |
l
Z+∞
In the limit β −1 y
≈
d$
1
τ
e−i$
(8.173a)
,
2
2 π m $ + η |$|
−∞
and Zβ as the grand-canonical partition function of this classical plasma.
Here, the divergence due to the pole at the origin of the integral on the
right-hand side needs to be regulated. This is achieved by taking the
difference between D(τ ) and D(τ 0 ) to extract

η
1

if r := m
|τ | 1,
− 2η r,
Dreg (τ ) ≈
(8.173b)

− 1 ln r, if r := η |τ | 1.
πη
m
To justify this reinterpretation, we first write
Z
Zβ = Nbath D[φ] e−(S0 kin +S1 )−(S0 int +Sext ) .
(8.174)
Observe that

e−S0 int = exp +y
Zβ

dτ cos φ(τ )
0
β
∞
n Z
X
1 nY
y
dτj cos φ(τj )
=
n!
n=0
j=1
0
β
∞
n Z
X
1 y n Y
=
dτj e+iφ(τj ) + e−iφ(τj )
n! 2
n=0
j=1 0
 β

β
R
Rβ
n Z
∞
Y
X
+i dτ δ(τ −τj )φ(τ )
−i dτ δ(τ −τj )φ(τ )
1 y n

=
dτj e 0
+e 0
n!
2
n=0
j=1
0
Rβ
Zβ
Zβ
n
∞
X
X
− dτ Jn,m (τ ) φ(τ )
n!
1 y n
dτ1 · · · dτn
e 0
=
,
n!
2
(n
−
m)!
m!
m=0
n=0
0
0
(8.175)
whereby the fact that the integrand on the penultimate line is independent of the ordering of “charges” of the same sign has been used,
456
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
and
Jn,m (τ ) := −i
n−m
X
≡ −i
j=1
n
X
δ(τ − τj ) −
m
X
!
δ(τ − τl )
l=1
(8.176)
ej δ(τ − τj ),
j=1
1 =: e1 =: · · · =: en−m =: −en−m+1 · · · =: −en .
Insertion of Eq. (8.175) in the partition function gives
Zβ
Zβ
∞
n
X
X
1 y n
n!
Zβ =
dτ1 · · · dτn
n! 2
(n − m)! m!
n=0
m=0
0
*
×
−
e
Rβ
0
dτ (Jn,m +J)(τ ) φ(τ )
(8.177a)
+
,
0
S0 kin +S1
where
Z
h· · · iS
0 kin +S1
D[φ] (· · · ) e−S0 kin −S1 .
:= Nbath
(8.177b)
Since the argument of the exponential in · · · is linear in the integration
variable φ and since the argument −S0 kin − S1 entering the Boltzmann
weight over which averaging h· · · iS0 kin +S1 is to be performed with is
quadratic in φ, we are dealing with a (bosonic) Gaussian path integral
for given n and given m,
* Rβ
+
−
e
dτ (Jn,m +J)(τ ) φ(τ )
=
0
S0 kin +S1
+ 12
N
√ bath0 e
Det D
Rβ
0
dτ
Rβ
dτ 0 (Jn,m +J)(τ ) D(τ −τ 0 ) (Jn,m +J)(τ 0 )
0
δn,2 m .
(8.178)
As we have explained above Eq. (8.173b), D(τ − τ 0 ) contains the singular term D(τ − τ 0 ) − Dreg (τ − τ 0 ). The reason for which the neutrality
condition
n = 2m
(8.179)
must hold is that
1
= 0,
$→0 D$
(8.180a)
lim D$
(8.180b)
lim
i.e.,
$→0
8.7. DUALITY IN A DISSIPATIVE JOSEPHSON JUNCTION
457
diverges. Thus, integration over φ$=0 is only well defined if
Zβ
0=
Zβ
dτ (Jn,m +J)(τ ) = −i(n−2 m)+ dτ J(τ ),
0
m, n = 0, 1, 2, · · · ,
0
(8.181)
Rβ
i.e., n = 2 m and
dτ J(τ ) = 0. Hence, the vanishing eigenvalue (zero
0
mode) of D−1 must be omitted in (Det D)−1/2 as is implied by the
use of (Det0 D)−1/2 . As a corollary, we can also replace D(τ − τ 0 ) by
Dreg (τ − τ 0 ) on the right-hand side of Eq. (8.178) as the τ -independent
divergent contribution D(τ − τ 0 ) − Dreg (τ − τ 0 ) drops out from the
argument of the exponential in the integrand as a consequence of the
neutrality condition.
In summary, when the external source J(τ ) is set to zero,
2n
P
2 Zβ
Zβ
∞ − 21
ej Djk ek
Nbath X 1
y 2n
j,k=1
dτ
·
·
·
dτ
e
,
Zβ = √
1
2n
2
Det0 D n=0 n!
0
0

η
1

if rjk := m
|τj − τk | 1,
− 2η rjk ,
Djk ≡ D(τj − τk ) ∼

− 1 ln r , if r := η |τ − τ | 1,
jk
jk
k
πη
m j
(8.182a)
defines the grand-canonical partition function at the temperature β −1 y and fugacity y/2 of a neutral plasma made of classical point-like particles with coordinates 0 ≤ τ ≤ β interacting through the two-body
potential
1 X
D$l e−i$l |τj −τk | ,
β $
D(τj − τk ) :=
D$l :=
l
m $l2
1
.
+ η |$l |
(8.182b)
In the presence of the source term J(τ ), the term
Zβ
Zβ
Sext =
dτ
0
1
+
2
Zβ
Zβ
dτ
X
$l
−i
!
ej δ(τ − τj ) D(τ − τ 0 ) J(τ 0 )
j=1
0
0
=
dτ 0
2n
X
dτ 0 J(τ ) D(τ − τ 0 ) J(τ 0 )
0
1
−iρ2n (−$l ) + J(−$l )
2
D$l J(+$l )
(8.183)
458
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
(a)
y(1
(b)
¯ (⌧ )
1
+4⇡
+4⇡
+2⇡
+2⇡
cos )
0
⌧ =0
⌧=
1/!0
¯ (⌧ )
6
⌧3 ⌧4
0
2⇡
2⇡
4⇡
4⇡
⌧1
⌧5
⌧2
⌧6
Figure 11. Instanton configurations. (a)pA singleinstanton configuration φ1 with ω0 × β := y/m × β
finite and located at β/2. (b) A 6-instantons configuration φ6 in the limit (1/ω0 )/β → 0. Anti-instantons are
located at τ1 , τ2 , and τ3 , instantons are located at τ4 , τ5 ,
and τ6 .
must be added to the action
ρ(τ ) is defined to be
ρ2n (τ ) :=
2n
X
ej δ(τ −τj ),
1
2
P2n
ρ2n $l
j=1
j,k=1 ej
1
:= √
β
Djk ek . The plasma density
Zβ
dτ ρ2n (τ ) e
+i$l τ
0
2n
1 X
=√
e e+i$l τj ,
β j=1 j
(8.184)
in the sector with 2n particles.
8.7.2. Regime m y 1 and m y η. When the geometrical
√
mean m y of the characteristic scales in S0 kin and S0 int becomes arbitrarily large relative to 1 or η, the partition function Zβ reduces to a
summation over all the periodic trajectories φ(τ ) that are local minima
of the action S0 kin + S0 int , i.e., satisfy
¨ − y sin φ,
0 = mφ
φ(τ ) = φ(τ + β).
(8.185)
Approximate solutions to Eq. (8.185) can be constructed from the instanton φ1 and anti-instanton φ−1 solutions
r
y
φ1 (τ ) = 4 arctan exp (ω0 τ ) ,
ω0 :=
,
(8.186a)
m
and
φ−1 (τ ) = −φ1 (τ )
(8.186b)
¨ − y sin φ
0 = mφ
(8.186c)
respectively, to
8.7. DUALITY IN A DISSIPATIVE JOSEPHSON JUNCTION
459
by writing
φ(τ ) ≈
2n
X
2n
X
ej φ1 (τ −τj ) ≡ φ2n (τ ),
j=1
ej = 0,
ej = ±1,
j = 1, · · · , 2n.
j=1
(8.186d)
p
The characteristic time 1/ω0 = m/y is the time needed for φ1 (τ )
to change by ±2 π. Such trajectories made of a sequence of kinks
when ej = +1 and anti-kinks when ej = −1 become exact solutions
of Eq. (8.185) in the limit when the width of the kink 1/ω0 is much
smaller than the separation between the kinks,
1/ω0
→ 0,
|τj − τk |
j, k = 1, · · · , 2n.
(8.187)
An example with 6 instantons is depicted in Fig. 11. We have seen in
section 8.5.4 that the mean density of kinks is of the order
K[φ1 ] × e−(S0 kin [φ1 ]+β y+S0 int [φ1 ]) ,
s
s
S0 kin [φ1 ] + β y + S0 int [φ1 ]
Det (−∂τ2 + ω02 )
,
K[φ1 ] =
2π
Det0 −∂τ2 + y cos(φ1 )
S0 kin [φ1 ] + β y + S0 int [φ1 ] = +4 (m y)1/2 + O(e−2 ω0 β ),
(8.188)
√
which is indeed negligible for very large m y (β fixed). 12 Integrating
fluctuating trajectories about each local minimum φ2n up to Gaussian
order yields the partition function
2
Zβ
Zβ
∞ X
1
2n
Zβ = p
z
dτ1 · · · dτ2n e−S1 [φ2n ] ,
2
2
n!
Det (−∂τ + ω0 ) n=0
Nbath × e+yβ
0
0
(8.189a)
where the fugacity z is
z = K[φ1 ] e−(S0 kin [φ1 ]+β y+S0 int [φ1 ])
= K[φ1 ] e−4 (m y)
1/2 +O(e−2 ω0 β ).
(8.189b)
12
The contribution β y comes about by rewriting the interaction as −y cos φ =
y(1−cos φ)−y and working with the potential y(1−cos φ) as opposed to the
potential
−y cos φ as is indicated in Fig. 11. An overall multiplicative factor exp (−1)2 β y
must then be accounted for in the partition function. The 0 in Det0 means removal
of the zero mode. Finally, we make use of the constant of motion m (φ˙ 1 )2 − 2 × (1 −
cos φ) y = 0 to express, with the explicit form of φ1 , S0 int [φ1 ] in terms of S0 kin [φ1 ].
Rβ
There follows S0 kin [φ1 ] + β y + S0 int [φ1 ] = 4 m ω02 cosh2dτ(ω τ ) . Integration over τ
0
tanh(ω0 β)
.
ω0
−2 ω0 β
gives S0 kin [φ1 ] + β y + S0 int [φ1 ] = 4 m ω02
right-hand side becomes 4 m ω0 1 + O(e
) .
0
In the limit, ω0 β 1, the
460
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
The combinatorial factor 1/(n!)2 follows from the property that the
centers of instantons of a given charge are time ordered. Time ordering
can then be removed at the price of the combinatorial factor 1/(n!)2
since the interaction S1 [φ2n ] is a two-body interaction that is translation
invariant as we shall verify explicitly shortly. The new feature brought
by the dissipation compared to the dissipationless periodic Hamiltonian
of section 8.5.5 is that instanton now interact through S1 [φ2n ]. Hence, it
is not possible to integrate freely over the instanton centers τ1 , · · · , τ2n .
We now turn to the evaluation of the interaction between instantons
induced by dissipation, i.e., by the coupling to the bath. Let h$l be
the Fourier transform with respect to Matsubara time of (∂τ φ1 )(τ ),
∂τ φ1
1 X
(τ ) = √
h$l e−i$l τ ,
β $
h$l
1
=√
β
Zβ
dτ ∂τ φ1 (τ ) e+i$l τ ,
0
l
(8.190)
where $l = 2βπ l and l ∈ Z. As we shall see shortly, the most important
property of h$l is
2π
h$l =0 = √ .
(8.191)
β
Making use of the neutrality condition, we can express the Fourier
transform
Zβ
1
φ2n $l = √
(8.192)
dτ φ2n (τ ) e+i$l τ
β
0
of φ2n (τ ) in terms of h$l by taking the τ derivative of Eq. (8.186d) and
Fourier transforming it,
−i$l φ2n $l =
2n
X
ej h$l e
+i$l τj
j=1
,
2n
X
j=1
ej = 0 =⇒ φ2n $l = +i
2n
h$l X
$l
ej e+i$l τj ,
j=1
(8.193)
for all l ∈ Z. Here, the neutrality condition can be used together with
Eq. (8.191) to avoid an inconsistency when l = 0. Evaluation of S1 [φ2n ]
is now straightforward,
X η
S1 [φ2n ] =
|$l | φ2n (+$l ) φ2n (−$l )
2
$
l
2n X
2n
X η
h(+$ ) h(−$ ) X
l
l
=
|$l |
ej ek e+i$l (τj −τk )
2
2
$
l
$l
j=1 k=1


2n X
2n
X
X
η
1

=
h
h
e+i$l (τj −τk ) (8.194a)
ej ek .
2 j=1 k=1 $ |$l | (+$l ) (−$l )
l
8.7. DUALITY IN A DISSIPATIVE JOSEPHSON JUNCTION
461
The underlined term defines the two-body interaction potential
η X β
0
h(+$l ) h(−$l ) e+i$l (τ −τ )
β $ |$l |
l


X
X
η 
 β h(+$ ) h(−$ ) e+i$l (τ −τ 0 )
=
+
l
l
β
|$l |
|$l |<ω0
|$l |≥ω0

2
0

−constant × η ρ , if ρ := ω0 |τ − τ | 1,
∼
(8.194b)

−4 π η ln ρ,
0
if ρ := ω0 |τ − τ | 1,
∆(τ − τ 0 ) :=
(constant is a numerical constant of order unity) in terms of which
S1 [φ2n ] =
2n 2n
1 XX
e ∆(τj − τk ) ek .
2 j=1 k=1 j
(8.194c)
The limiting form of the two-body interaction (8.194b)
lim ∆(τ ) ∼ −constant × η ρ2
τ →0
(8.195)
follows from expanding exp + i$l τ in powers of $l τR up to second
ω0
order owing to the limit ω0 |τ | 1. The integration
d$ is then
independent (divergent) of τ to zero-th order, vanishes to first order in
τ , and gives to second order in τ Eq. (8.195). As before, the condition
of charge neutrality allows us to ignore the diverging constant. The
limiting form of the two-body interaction (8.194b)
lim ∆(τ ) ∼ −4 π η ln (ω0 |τ |) ,
τ →∞
(8.196)
follows from observing that, in the limit ω0 |τ | 1, the sum over $l is
dominated by the contribution
R from $l near zero. One may then take
advantage of the fact that ω d$ |$|−1 is invariant under rescaling of
0
$ after insertion of Eq. (8.191) into Eq. (8.194b). Equation (8.196)
also follows from
Z∞
−
x
cos t
dt
= γ+ln x+
t
Zx
dt
cos t − 1
,
t
γ denoting Euler’s constant.
0
(8.197)
(See Eq. 8.230.2 from Ref. [57]). Correspondingly, S1 [φ1 ] is scale invariant below a frequency cutoff. Charge neutrality can then be understood
as following from the fact that it costs an infinite action to create a net
charge in the “thermodynamic limit” β → ∞.
462
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
In summary, when the external source J(τ ) is set to zero,
2
Zβ
Zβ
∞ X
− 12
1
2n
Zβ = p
z
dτ
·
·
·
dτ
e
1
2n
Det (−∂τ2 + ω02 ) n=0 n!
Nbath × e+yβ
0
z = K[φ1 ] e
−4 (m y)1/2 +O(e−2 ω0 β )
2n
P
j,k=1
ej ∆jk ek
0
,
∆jk ≡ ∆(τj − τk )
1 X
≡
∆$l e+i$l (τj −τk )
β $
 l
2

−constant × η ρjk , if ρjk := ω0 |τj − τk | 1,
∼

−4 π η ln ρ ,
if ρjk := ω0 |τj − τk | 1,
jk
(8.198a)
with
∆$l :=
ηβ
h
h
,
|$l | (+$l ) (−$l )
(8.198b)
defines the grand-canonical partition function at the temperature β −1 y and fugacity z of a neutral plasma made of classical point-like particles with coordinates 0 ≤ τ ≤ β interacting through the two-body
potential ∆(τj − τk ). The instanton expansion converges best with
√
small fugacity z, i.e., with large m y. In the presence of the source
term J(τ ), the term
Sext [φ2n ] = −i
X
$l
J(+$l )
2n
h(−$ ) X
l
$l
ej e−i$l τj
(8.199)
j=1
P
must be added to the action S1 [φ2n ] = 12 2n
j,k=1 ej ∆jk ek . Here, the
source couples linearly as opposed to Eq. (8.178) where the source
couples quadratically.
8.7.3. Duality. Duality is the observation [87] that Eqs. (8.182a)
and (8.198a) are related by the substitutions
y
↔ z,
2
η
↔ ω0 ,
m
1
↔ 4π η,
πη
(8.200)
in the absence of sources and if one is allowed to neglect the difference in
the core regions of the interaction potentials Djk and ∆jk . Duality implies that there is a one to one correspondence between the asymptotic
behavior at very low Matsubara frequencies or, equivalently, at very
large separations of Matsubara times, of correlation functions in the
regimes (8.171a) and (8.171b), respectively. If one knows the asymptotic behavior of one correlation function, say in regime (8.171a), one
can use the duality relations (8.200) to reconstruct the corresponding
,
8.8. RENORMALIZATION-GROUP METHODS
463
correlation function in the regime (8.171b). To do so, one must carefully account for the source term that allows to derive the correlation
function. For example, if one is after the two-point correlation function
∂ 2 Zβ
1
hφ(+$l ) φ(−$l ) iZβ =
,
(8.201a)
Zβ ∂J(−$ ) ∂J(+$ ) l
l
J=0
one finds that

h
i √
√

m y 1, m y η,
D
−1
−
D
hρ
ρ
i

$l
(+$ ) (−$ ) Zβ ,
 $l
l
hφ(+$l ) φ(−$l ) iZβ =


1
1
η |$l |
l
√
√
m y 1, m y η,
(8.201b)
∆$l hρ(+$ ) ρ(−$ ) iZβ ,
l
l
where the plasma density ρ(τ ) is defined to be
ρ(τ ) :=
2n
X
j=1
ej δ(τ −τj ),
ρ$ l
1
:= √
β
Zβ
+i$l τ
dτ ρ(τ ) e
0
2n
1 X
=√
e e+i$l τj ,
β j=1 j
(8.201c)
in the sector with 2n particles. Here, we also made use of Eqs. (8.183)
and (8.199), respectively, as well as the fact that D+$l = D−$l . Establishing duality between two regimes is useful insofar computations
can be carried out in either of one of the regimes. In the next section,
a renormalization-group calculation for the flow of the potential height
y in the regime (8.171a) is performed. As a by product of duality, the
corresponding flow can be obtained in the regime (8.171b).
8.8. Renormalization-group methods
In this section we are, following Ref. [88], going to focus our efforts
on a renormalization-group (RG) approach whenever the Josephson
coupling is the smallest energy scale in the problem (aside from the
temperature β −1 ) and frequencies are sufficiently small for the motion
of the quantum particle to be diffusive (i.e., dissipation S1 dominates
over the kinetic energy S0 kin ). We will then apply duality to investigate
the regime when the Josephson coupling is the largest energy scale in
the problem and frequencies are sufficiently small for the motion of the
quantum particle to be diffusive. The RG approach will allow us to
decide whether the diffusive limit is stable or unstable upon elimination
(integration) of high frequency modes. We shall argue that the answer
to this question depends on how large the Josephson coupling is.
8.8.1. Diffusive regime when m y 1 and m y η. In section 8.7.1 we performed a formal expansion of the partition function
in powers of y as y is the smallest energy scale aside from the temperature. Although this expansion is essential in establishing duality
with the regime m y 1 and m y η, it is of limited advantage to
464
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
the naive evaluation of correlation functions in the static limit. Indeed,
perturbation theory in powers of y breaks down in the diffusive regime
due to the fact that the free propagator is then proportional to 1/|$l |
in Matsubara frequency space to a very high accuracy as is indicated
by Eq. (8.170d). To tame the divergences plaguing perturbation theory
in powers of y, one relies on a scaling approach. Having identified the
upper frequency cutoff
Λ∼
α
η
1/Rs
∼
≡
C Rh
C
m
(8.202)
that defines the diffusive regime from Eq. (8.170c), the idea behind the
scaling approach is to distinguish between “slow” and “fast” Fourier
components φ$l in the Fourier expansion of the Josephson phase difference φ(τ ) and to integrate over the “fast” components in the partition
function (8.172a). One thus writes
 (1)
 φ$l , if |$l | < Λ − dΛ,
(8.203a)
φ$ l =
 (2)
φ$l , if Λ − dΛ ≤ |$l | ≤ Λ,
and
1
φ (τ ) := √
β
|$l |<Λ−dΛ
X
(1)
1
φ(2) (τ ) := √
β
−i$l τ
φ(1)
,
$l e
$l
(8.203b)
Λ−dΛ≤|$l |≤Λ
X
−i$l τ
φ(2)
,
$l e
$l
for the decomposition into slow and fast Fourier modes which, in turn,
is inserted into the partition function (8.172a):
Z
00
Zβ = Nbath
D[φ(1) , φ(2) ] e−Sβ +O(S0 kin ) ,
Sβ00 = S1 + S0 int + Sext ,
|$l |<Λ−dΛ
X
S1 =
$l
Zβ
S0 int =
η
(1)
(1)
|$ | φ
φ
+
2 l (+$l ) (−$l )
Λ−dΛ≤|$l |≤Λ
η
(2)
(2)
|$ | φ
φ
,
2 l (+$l ) (−$l )
X
$l
dτ (−y) cos φ(1) (τ ) + φ(2) (τ ) ,
0
|$l |<Λ−dΛ
Sext =
X
$l
Λ−dΛ≤|$l |≤Λ
(1)
J(+$ )
l
(1)
φ(−$ )
l
+
X
(2)
(2)
J(+$ ) φ(−$ ) .
l
l
$l
(8.203c)
8.8. RENORMALIZATION-GROUP METHODS
465
Here, we are neglecting the contribution
|$l |<Λ−dΛ
X
S0 kin =
$l
1
(1)
(1)
m $l2 φ(+$ ) φ(−$ ) +
l
l
2
Λ−dΛ≤|$l |≤Λ
X
$l
1
(2)
(2)
m $l2 φ(+$ ) φ(−$ )
l
l
2
(8.203d)
to the exact action in Eq. (8.172a).
(2)
Integration over the fast modes φ$l is free from divergences in view
of the lower frequency cut-off Λ − dΛ and yields an effective or renor(1) 00
malized action Sβ given by
Z
(1) 00
D[φ(1) ] e−Sβ
Zβ ≈ Nbath × N
(1) 00
Sβ
(1)
,
(1)
:= S1 + S0 int ,
|$l |<Λ−dΛ
(1)
S1
η
(1)
(1)
|$l | φ(+$ ) φ(−$ ) ,
l
l
2
$l

+
*
Zβ
:= − ln exp (−1)2 dτ y cos φ(1) (τ ) + φ(2) (τ )  ,
:=
(1)
S0 int
X
0
2
(8.204a)
in the absence of a source term. The notation h(· · · )i2 refers to
R
D[φ(2) ] exp −
Λ−dΛ≤|$l |≤Λ
P
$l
h(· · · )i2 :=
R
Λ−dΛ≤|$l |≤Λ
D[φ(2) ] exp −
P
$l
!
(2)
η
|$l | φ(+$ )
2
l
(2)
φ(−$ )
l
(2)
η
|$l | φ(+$ )
2
l
(2)
φ(−$ )
l
(· · · )
.
!
(8.204b)
(1)
In practice, the computation of S0 int cannot be performed exactly but
relies on a perturbative expansion in powers of y:
Zβ
*
(1)
S0 int
= − ln 1 + y
dτ cos φ(1) (τ ) + φ(2) (τ ) + O(y 2 )
0
*Zβ
= −y
0
+
dτ cos φ(1) (τ ) + φ(2) (τ )
2
+
+ O(y 2 ).
2
(8.205)
466
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
To calculate the expectation value of the cosine on the right-hand
side of Eq. (8.205), use the identity
R
2
dx e−a x /2 cos(x0 + x)
R
dx e−a x2 /2
R
0
0
2
dx e−a x /2 12 e+i(x +x) + e−i(x +x)
R
dx e−a x2 /2
1 +ix0 − 1 hx2 ix 1 −ix0 − 1 hx2 ix
2
2
+ e
e
2
2
1
2
e− 2 hx ix cos x0 ,
hx2 ix = 1/a,
a (8.206a)
> 0,
0
hcos(x + x)ix :=
=
=
=
with the identifications
|$l |<Λ−dΛ
1
x → φ (τ ) := √
β
0
X
(1)
1
x → φ (τ ) := √
β
−i$l τ
φ(1)
,
$l e
$l
Λ−dΛ≤|$l |≤Λ
X
(2)
2
1
x → φ(2) (τ ) =
β
−i$l τ
φ(2)
,
$l e
$l
Λ−dΛ≤|$l |≤Λ
X
2
1
+
β
(2)
(2)
φ(+$ ) φ(−$ )
l
l
$l
Λ−dΛ≤|$l |,|Ωl |≤Λ
X
(2)
(2)
φ(+$ ) φ(+Ω ) e−i$l τ −iΩl τ ,
l
l
$l 6=−Ωl
D
2 E
1
hx ix → φ(2) (τ )
=
β
2
Λ−dΛ≤|$l |≤Λ
2
X
(η |$l |)−1
$l
1 1 dΛ
β η Λ 2 π/β
1 dΛ
=
.
πη Λ
≈2×
In the limit β → ∞
(8.206b)
Thus,
(1)
S0 int = −y
1 dΛ
1−
+O
2πη Λ
"
dΛ
Λ
2 #! Zβ
dτ cos φ(1) (τ ) + O(y 2 )
0
Zβ
≡
0
dτ (−y ) cos φ (τ ) + O(y 2 ),
(1)
(1)
(8.207a)
8.8. RENORMALIZATION-GROUP METHODS
467
where
1 dΛ
1−
+O
2πη Λ
y (1) := y
"
dΛ
Λ
2 #!
.
(8.207b)
The action thus transforms covariantly upon integration over the fast
degrees of freedom, i.e., the changes induced to the action by integration over the fast degrees of freedom can be absorbed by assigning a
scale dependence to the coupling constants η and y according to the
transformation laws
η(Λ − dΛ) := η(Λ) + O(y 2 ),
(8.208a)
and
(
1 dΛ
y(Λ − dΛ) := y(Λ) 1 −
+O
2πη Λ
"
dΛ
Λ
2 #)
+ O(y 2 ). (8.208b)
In units in which ~ = 1, the dissipation strength η is dimensionless
whereas the fugacity y has dimensions of inverse time. It is desirable to
distinguish in the transformation law (8.208b) the components induced
by the dimension carried by y from an “intrinsic” component. To this
end, define the dimensionless dissipation strength
ye(1) :=
y(Λ)
Λ
(8.209)
in terms of which the RG equations (8.208) become
η(Λ − dΛ) = η(Λ) + O(e
y 2 ),
(8.210a)
and
ye(1 − d ln Λ) =
y(Λ − dΛ)
Λ −(dΛ
y(Λ)
=
Λ
1 dΛ
1−
+O
2πη Λ
+ O(e
y2)
(
= ye(1) 1 + 1 −
1
2πη
"
dΛ
Λ
" #)
2 #) (
2
dΛ
dΛ
1+
+O
Λ
Λ
dΛ
+O
Λ
"
dΛ
Λ
2 #)
+ O(e
y 2 ).
(8.210b)
In the limit dΛ → 0, we obtain the pair of differential equations
dη
= 0 + O(e
y 2 ),
d ln(Λ−1 )
d ye
1/(2 π)
= 1−
ye + O(e
y 2 ).
d ln(Λ−1 )
η
(8.211)
468
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
We are ready to answer the question: How does ye change as the
frequency cut-off Λ is decreased (or, equivalently, as Λ−1 is increased )?
We must distinguish three cases:
(1) For sufficiently small dissipation
1
,
(8.212)
2π
ye decreases with decreasing Λ (or, equivalently, increasing Λ−1 ).
The initial assumption that y is the smallest energy scale in
the problem aside from the temperature is consistent and we
can trust the RG approach. It is said that the Josephson interaction is irrelevant.
(2) For the critical value
η<
1
,
(8.213)
2π
ye does not change with decreasing Λ (or, equivalently, increasing Λ−1 ). It is said that the Josephson interaction is marginal.
(3) For sufficiently large dissipation
η=
1
,
(8.214)
2π
ye increases with decreasing Λ (or, equivalently, increasing Λ−1 ).
The initial assumption that y is the smallest energy scale in the
problem aside from the temperature is not consistent and we
can not trust the RG approach. It is said that the Josephson
interaction is relevant.
To access the regime in which the RG perturbative approach breaks
down we take advantage of duality.
η>
8.8.2. Diffusive regime when m y 1 and m y η. The
same RG approach can
be used on Eq. (8.198a) when the fugacity
z ∝ exp − 4 (m y)1/2 is small. The counterparts to Eq. (8.211) can
be obtained with the help of the duality relation (8.200). They are
given by
d η −1
= 0 + O(e
z 2 ),
d ln(Λ−1 )
d ze
η
= 1−
ze + O(e
z 2 ).
−1
d ln(Λ )
1/(2 π)
(8.215)
The answer to the question how does ze change as the frequency cut-off
Λ is decreased (or, equivalently, as Λ−1 is increased ) is:
(1) For sufficiently small dissipation
η<
1
,
2π
(8.216)
8.9. CONJECTURED PHASE DIAGRAM FOR A DISSIPATIVE JOSEPHSON JUNCTION
469
ze increases with decreasing Λ (or, equivalently, increasing
Λ−1 ).
The initial assumption that z ∝ exp −4 (m y)1/2 is the smallest energy scale in the problem aside from the temperature is
not consistent and we can not trust the RG approach. It is
said that the instanton fugacity is relevant.
(2) For the critical value
1
η=
,
(8.217)
2π
ze does not change with decreasing Λ (or, equivalently, increasing Λ−1 ). It is said that the instanton fugacity is marginal.
(3) For sufficiently large dissipation
1
η>
,
(8.218)
2π
ze decreases with decreasing Λ (or, equivalently, increasing
Λ−1 ).
1/2
The initial assumption that z ∝ exp −4 (m y)
is the smallest energy scale in the problem aside from the temperature is
consistent and we can trust the RG approach. It is said that
the instanton fugacity is irrelevant.
8.9. Conjectured phase diagram for a dissipative Josephson
junction
We have modeled the dissipative Josephson junction pictured in
Fig. 4 by the quantum theory defined in Eq. (8.168) or, equivalently,
Eq. (8.172a). In this quantum model, the phase difference φ− between
two superconductors can be used to decide if the dissipative Josephson junction is in its coherent or decoherent state. An oscillatory and
periodic time dependence of φ− in the asymptotic limit t → ∞ characterizes the coherent state of the dissipative Josephson junction. A
(time independent) sharp value of φ− in the asymptotic limit t → ∞
characterizes the decoherent state of the dissipative Josephson junction.
In the classical limit, the time dependence of φ− is governed by the
second order differential equation
~
1 ~
0 = C ∗ φ̈− +
φ̇ + I0 sin φ− + I.
(8.219)
e
Rs e∗ −
Without the flow of a resistive current between the two superconductors, i.e., when the dissipation vanishes
R
η=~ ~ =0
(8.220)
Rs
due to an infinite Ohmic resistance Rs = ∞, the relative phase φ− oscillates periodically in time for any non-vanishing capacitance C > 0.
The difference in the number of electrons between the two supercondutors is then sharp, this is the coherent state of the dissipative Josephson
470
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
(a)
1
ye
0
localized
1
delocalized
2⇡⌘
(b)
1
2UJ
⇤
0
decoherence
1
coherence
Rh /Rs
Figure 12. (a) Conjectured phase diagram for a dissipative Josephson junction based on the perturbative RG
equations obeyed by the dimensionless dissipation and
Josephson coupling constants η and ye, respectively. To
first order in ye, the flow is vertical. Points on the segment 0 ≤ 2 π η < 1 at ye = 0 are attractive fixed points
and realize a regime of delocalization of the Josephson
phase difference. Points on the segment 1 < 2 π η < ∞
at ye = ∞ are attractive fixed points and realize a regime
of localization of the Josephson phase difference. The
thick line at 2 π η = 1 represents a line of critical points.
(b) Same figure as before but with axis corresponding to
a dissipative Josephson junction.
junction that is selected by the capacitance. The effect introduced by
a finite Ohmic resistance 0 < Rs < ∞ is to damp the oscillatory time
dependence of the phase difference φ− between the two superconductors. In this classical limit, increasing the strength of the dissipation
has no other effect than to smoothly change the time dependence of
the phase difference φ− from the under-damped regime, in which an
oscillatory time dependence survives, to the over-damped regime, in
which the time dependence is purely exponentially damped. For any
non-vanishing dissipation, the phase difference φ− becomes asymptotically sharp in the limit t C × Rs , i.e., the dissipative Josephson is
in its decoherent state.
Quantum fluctuations change this classical picture dramatically in
that they can overcome the effect of weak dissipation. The condition
for quantum fluctuations to favor localization over delocalization of φ−
is that the Ohmic resistance Rs is sufficiently small (smaller than the
quantum resistance Rh ).
Indeed, the RG equations (8.211) and (8.215) suggest a two-dimensional
phase diagram with the scale dependent coupling constants η and ye as
horizontal and vertical axis, respectively, made of vertical flows and separated by a vertical line of fixed points as depicted in Fig. 12(a). When
8.10. PROBLEMS
471
the bare dissipation strength η is smaller than 1/(2 π), the Josephson
junction coupling constant y is irrelevant and the ground state is delocalized as the kinetic energy dominates over the potential energy. When
the bare dissipation strength η is larger than 1/(2 π), the Josephson
junction coupling constant ye is relevant and the ground state is localized as the potential energy dominates over the kinetic energy. The line
of critical points η = 2 π separates the delocalized from the localized
regime.
In the regime 0 ≤ η < 1/(2 π) [0 ≤ (Rh /Rs ) < 1], the asymptotic
value of the Josephson phase difference φ− as t → ∞ is not sharp.
Correspondingly, the value of the electron number difference N− is
sharp. In the regime 1/(2 π) < η [1 < (Rh /Rs )], the asymptotic value of
the Josephson phase difference φ− as t → ∞ is sharp. Correspondingly,
the value of the electron number difference N− is not sharp. Thus, the
conditions for quantum coherence and decoherence to hold are:
• When 0 ≤ η < 1/(2 π) [0 ≤ (Rh /Rs ) < 1], quantum coherence
is robust to weak dissipation.
• When 1/(2 π) < η (1 < (Rh /Rs )], quantum coherence is destroyed by strong dissipation, quantum decoherence rules.
This interpretation of the phase diagram in Fig. 12(a) is given in Fig.
12(b). The interplay of dissipation, through an Ohmic current flow between two superconductors coupled capacitively and through a Josephson (periodic) interaction, with quantum fluctuations arising from the
uncertainty relation between the relative phase and the difference in the
electron number of electrons on superconductors 1 and 2 making up a
dissipative Josephson junction has brought about a sharp distinction
between the weak and strong dissipative regimes.
8.10. Problems
8.10.1. The Kondo effect: a perturbative approach.
Introduction. Dissipation is ubiquitous in physics, for the notion of
an isolated system for which conservation laws hold is an idealization.
Electrons in condensed matter physics originate from atoms with which
they exchange quantum numbers such as energy, momentum, angular
momentum, etc. In a metal, the couplings between the electrons close
to the Fermi energy and the atoms change the ability of the electrons
to carry an electrical current as compared to the idealized limit by
which the electrons define a closed system. The diagonal contributions to the conductivity tensor are a direct measure of the dissipative
effects of these couplings. Electrical resistance is an example of dissipation and its character increasingly becomes quantum mechanical
as temperature is lowered. Anomalies in the electrical resistance at
low temperature might thus reveal subtle manifestation of quantum
472
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
mechanics through the presence or absence of dissipation. Superconductivity and the fractional quantum Hall effect have perhaps been
historically the most dramatic examples of anomalous dissipation (by
its absence), whose explanations have brought paradigm changes in the
understanding of many-body quantum mechanics. Another example of
a many-body quantum-mechanical effect revealed by an anomalous dissipation is provided by the Kondo effect. [89]
In a semi-classical treatment of electrical transport, we may approximate the resistivity of a metal at zero temperature by [recall Eq. (5.71)]
ρ≈
m
1
× .
2
ne e
τel
(8.221)
The lifetime τel accounts for the elastic scattering of a single-particle
Bloch state to any other Bloch state that is brought about by a static
perturbation that breaks the space group symmetry of the lattice (the
periodic potential of the ions in their crystalline ground state). 13 This
lifetime is calculated to the first non-vanishing order in perturbation
theory, whereby the perturbation is a deviation from perfect crystalline
order, for every wave vector on the Fermi surface and then averaged
over the Fermi surface. In the limit of perfect crystalline order, τel → ∞
and the resistivity vanishes. Otherwise, the resistivity is non-vanishing
but finite and non-universal.
At any non-vanishing temperature, we may approximate the resistivity of a metal by [recall Eq. (5.71)]
!
m
1
1
1
ρ≈
+
+ ··· .
(8.222)
×
+
ne e 2
τel τe-e (T ) τe-pho (T )
The inverse lifetime of the Fermi-liquid quasiparticles in a window of energy of order kB T centered about the Fermi energy acquires additional
additive channels. A Fermi-liquid quasiparticle can decay through (inelastic) electron-electron interactions, in which case it can be shown
that
1
∼ T 2.
(8.223)
τe-e (T )
A Fermi-liquid quasiparticle can also decay through electron-phonon
interactions (the dynamical counterpart to the static deviation from
perfect crystalline symmetry captured by 1/τel ), in which case it can
be shown that
1
∼ T5
(8.224)
τe-pho (T )
13
This perturbative expansion might break down. The theory of Anderson
localization aims at solving the cases (low dimensionality or strong perturbations)
when this perturbation expansion breaks down.
8.10. PROBLEMS
473
if only forward scattering is accounted for. There follows the temperature dependence
2
5
kB T
kB T
kB T
ρ(T ) ≈ ρ0 + ρ2
+ ρ5
+· · · ,
1. (8.225)
εF
~ ωD
εF
The non-universal numbers ρ0 , ρ2 , and ρ5 are positive and carry the
dimension of the resistivity. The second characteristic energy scale
besides the Fermi energy εF > 0 is here the Debye energy ~ ωD > 0. In
this approximation, the resistivity is a monotonous increasing function
of temperature that is dominated by the contribution from phonon
scattering at sufficiently large temperatures.
However, it has been known since 1934 that the resistance of gold
shows a minimum as a function of temperature when measured between
1 and 21 Kelvins. [90] Evidently, the right-hand side of Eq. (8.225) must
be augmented by a scattering channel that increases the resistivity with
decreasing temperature. Such a term was found by Kondo in 1964,
see Ref. [91], who realized that an effective antiferromagnetic coupling
between the spin of the conduction electrons and localized magnetic
impurities would produce the temperature dependence
5
D − εF
k T
kB TK
kB T
0
ρ(T ) ≈ ρ0 +ρK ln
. B 1,
+ρ5
+· · · ,
kB T
~ ωD
εF
εF
(8.226a)
with a resistivity minimum at the temperature
1/5
ρK
min
k B TK =
~ ωD ,
(8.227)
5 ρ5
if the electron-electron interaction is neglected. Here, this magnetic
scattering channel changes ρ0 to ρ00 > 0, while it produces the multiplicative constant ρK which is again positive, non-universal, and carries
the dimension of the resistivity. The band width D is an upper bound
to the Fermi energy 0 < εF < D.
The logarithmic growth
D − εF
ρK ln
(8.228)
kB T
with decreasing temperature cannot continue all the way to zero temperature. Upon lowering temperature, perturbation theory must break
down when
D − εF
ρK ln
(8.229)
kB T
is of the same order as ρ00 , i.e., when kB T is of the order
0
kB TK := (D − εF ) e−ρ0 /ρK .
(8.230)
474
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
Kondo conjectured a crossover to the contribution
D − εF
ρK ln
,
T TK ,
kB TK
(8.231)
at temperatures well below the Kondo temperature.
The failure of perturbation theory could signal two scenarios. On
the one hand, magnetic impurities are gradually screened by the conduction electrons on the way down to zero temperature. This screening
process is non-perturbative in the bare coupling between the electrons
and the magnetic impurities very much in the same way as screening of
a test point charge is non-perturbative in the electron charge for the jellium model. Nevertheless, the low temperature phase remains a Fermi
liquid one, as was conjectured by Kondo, i.e., with the low-temperature
dependence of all susceptibilities found for the non-interacting jellium
model of section 5.3. On the other hand, screening of the magnetic
impurities is not achieved, a non-Fermi liquid phase is selected at low
temperatures with deviations from the power laws found for the noninteracting jellium model of section 5.3.
Theoretical methods are thus needed to overcome the singular nature of perturbation theory for temperatures lower than the Kondo
temperature, a challenge known as the Kondo problem, in order to establish under what conditions the conjecture (8.231) and its implication
that a Fermi liquid phase is recovered for T TK holds.
As we have seen in sections 3.6, 4.6, and 8.8, and alluded to when
discussing the ingredients entering a rigorous proof of Luttinger theorem in section 6.10.2, one option to deal with systemic logarithmic divergences that invalidate perturbation theory is to do a resummation of
perturbation theory as dictated by a renormalization group calculation.
This program was first carried out by Anderson, Yuval, and Hamann
in the spirit of a one-loop renormalization group calculation, [92] and
then by Wilson who used a non-perturbative renormalization group
scheme requiring a numerical implementation. [93] The conclusion is
the same by either methods. A single band of non-interacting electrons
coupled antiferromagnetically to a dilute concentration of spin-1/2 localized impurities has a Fermi liquid ground state, thereby confirming
the conjecture of Kondo.
Perturbative estimates of transition probabilities. Exercise 1.1: We
are going to substantiate Eqs. (8.223) and (8.228).
(a) Justify Eq. (8.223) with the help of the zero-temperature estimate for the decay rate of a Fermi liquid quasiparticle derived
in footnote 2 from section F.2.
(b) Assume the additive decomposition
Ĥ(t) = Ĥ0 + eη t V̂
(8.232)
8.10. PROBLEMS
475
into two conserved Hermitean operators Ĥ0 and V̂ . The infinitesimal number η > 0 with the dimension of frequency
implements adiabatic switching on of the perturbation V̂ at
t = −∞. Fermi’s golden rule states that the transition prob(0)
ability per unit time from the initial energy eigenstate |Ea i
(0)
of Ĥ0 to the final energy eigenstate |Eb i of Ĥ0 is
*
+2
i Rt 0 0 d (0) − ~ 0 dt Ĥ(t ) (0) Wb←a := lim
Eb e
Ea t→∞ d t (8.233a)
2π
(0)
=
δ Ea(0) − Eb
Va b Vb a
~
to lowest order in an expansion in powers of the matrix elements
D
E
(0) Vdc := Ed V̂ Ec(0)
(8.233b)
in the basis made of the eigenstates of Ĥ0 . Apply Fermi’s
golden rule to the Fermi-liquid Hamiltonian (F.15) to derive
Eq. (8.223).
(c) Show that
2π (0)
(0)
δ Ea − Eb |Mb←a |2
Wb←a ≡
~
!
X
2π (0)
V
V
V
+
c.
c.
(0)
ab bc ca
=
δ Ea − Eb
Va b Vb a +
(0)
(0)
~
Ea − Ec
c6=a
(8.234a)
with the amplitude
Mb←a := Vb a +
X
Vb c V c a
(0)
c6=a
(0)
Ea − Ec
,
(8.234b)
up to second order in an expansion in powers of V̂ .
(d) Kondo chooses 14
X X
Ĥ0 :=
εk ĉ†k,σ ĉk,σ
(8.235a)
k∈BZ σ=↑,↓
14
Our conventions are
{ĉk,σ , ĉ†k0 ,σ0 } = δk,k0 δσ,σ0 ,
{ĉk,σ , ĉk0 ,σ0 } = {ĉ†k,σ , ĉ†k0 ,σ0 } = 0,
{ĉr,σ , ĉ†r0 ,σ0 } = δr,r0 δσ,σ0 ,
{ĉr,σ , ĉr0 ,σ0 } = {ĉ†r,σ , ĉ†r0 ,σ0 } = 0,
and
whereby
1 X †
ĉ†r,σ = √
ĉk,σ e−ik·r ,
N k∈BZ
1 X
ĉr,σ = √
ĉk,σ e+ik·r .
N k∈BZ
476
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
to encode a single-band of conduction electrons dispersing with
the single-particle energies εk and the perturbation
Ns
2J 1 X
V̂ :=
~ N i=1
X
k,k0 ∈BZ
0
e+i(k−k )·ri
X
ĉ†k0 ,α σ αβ ĉk,β · Ŝ i
(8.235b)
α,β=↑,↓
by which the conduction electrons couple through an antiferromagnetic Heisenberg exchange coupling J > 0 to a dilute
concentration Ns /N 1 of localized spin-1/2 degrees of freedom Ŝ i at the site r i from the lattice Λ obeying the algebra
h
i
X
2
3
Ŝiµ , Ŝjν = δij i~
µνγ Ŝiγ ,
µ, ν = 1, 2, 3 ≡ x, y, z,
Ŝ i = ~2 1.
4
γ=1,2,3
(8.235c)
The lattice Λ is here made of N sites.
We introduce the notation
~ X †
ŝk0 ,k :=
ĉ 0 σ αβ ĉk,β
2 α,β=↑,↓ k ,α
and define
2
Ns
X
1 X
2
0
e+i(k−k )·ri
V̂J⊥ ,J =
k
~ N i=1 0
k,k ∈BZ
J⊥
+
−
−
+
z
z
×
ŝk0 ,k Ŝi + ŝk0 ,k Ŝi + Jk ŝk0 ,k Ŝi .
2
(8.236)
(8.237)
Consider the direct product |k, αi⊗|i, βi of a single-particle
electronic state with momentum k and projection α of its spin
along the quantization axis with a localized spin at site i with
the projection β of its spin along the quantization axis. We
shall also denote with σ̄ the reversal of the projection σ along
the quantization axis. Compute the sixteen matrix elements
D
E E
D
00
00 00
00 k , α ⊗ i , β V̂J⊥ ,J k, α ⊗ i, β , α, β, α00 , β 00 =↑, ↓≡ +, −,
k
(8.238)
00 00
for given initial k, i and final k , i .
(e) Draw the Feynman diagram describing the amplitude for the
following process. An initial state |k, σi ⊗ |i, σ̄i with k just
above the Fermi sea overlaps with the virtual state |k00 , σ̄i ⊗
|i, σi with k00 outside the Fermi sea owing to the perturbation
V̂ . This virtual state overlaps with the final state |k0 , σi ⊗
|i, σ̄i with k0 chosen just above the Fermi sea owing to the
perturbation V̂ . Show that the amplitude of this Feynman
diagram is, at zero temperature and when the density of state
8.10. PROBLEMS
477
at the Fermi energy νF is non-vanishing, proportional to
J
2
ZD
X 1 − fFD (ε 00 )
ν(ε)
2
k
=J
dε
εk − εk00
εk − ε
00
k ∈BZ
εF
2
≈ J νF ln
εk − εF
D − εk
,
0 < εF . εk < D.
(8.239)
For comparison, estimate the order of magnitude of the integral
ZD
ν(ε)
.
(8.240)
dε
εk − ε
0
Hint: Assume that εk is just above the Fermi energy. Do the
change of variable ξ := ε − εF and decompose the density of
state ν̃(ξ) into the sum of a function of ξ that is even about
the Fermi energy ξ = 0 and one that is odd.
(f) Usually, there is no need to extend perturbation theory beyond
the first non-vanishing order if the expansion is convergent and
if the goal of perturbation theory is not merely to increase the
precision of the expansion. However, perturbation theory in
many-body physics is often not convergent as the function to
be expanded is singular in the expansion parameter. The dependence on the squared electric charge of the Thomas-Fermi
screening length (6.92) is a case at hand. Keeping this in mind,
we are now ready to finish the steps that lead to Eq. (8.228).
We need to evaluate Eq. (8.234a). To this end, the initial
state |ai, the final state |bi, and the virtual states |ci must be
supplied. We denote with
Y †
| · · · ; k, σk ; · · · i :=
ĉk,σ |0i
(8.241)
k
{k,σk }
a Slater determinant for the N conduction electrons. The
Fermi sea, a special case of such a Slater determinant, is denoted |FSi. We work in the approximation with one impurity, i.e., Ns = 1, and multiply intensive quantities calculated
for one impurity by the impurity concentration. Our initial
state contains one quasiparticle above the Fermi surface. The
quasiparticle has the momentum k, the single-particle energy
εk > εF , and the spin-1/2 quantum number α. The spin-1/2
impurity at the lattice site r i has the spin-1/2 quantum number σ. Thus,
|ai := ĉ†k,α |FSi ⊗ |r i , σi.
(8.242)
478
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
The final state has one quasiparticle above the Fermi surface
with the momentum k0 , the single-particle energy εk0 = εk >
εF , and the spin-1/2 quantum number α. The spin-1/2 impurity has the spin-1/2 quantum number σ. Thus,
†
|bi := ĉk0 ,α |FSi ⊗ |r i , σi.
(8.243)
There are four possible families of virtual states that we arrange pairwise. The first pair of virtual states is constructed
by applying V̂J⊥ =0,J=J on either the initial or final state. The
k
virtual state is then either labeled by the momentum p> above
the Fermi surface with the spin-1/2 quantum number α, in
which case
|ci := ĉ†p> ,α ĉk,α |ai,
εp> > εF ,
(8.244a)
|bi = ĉ†k0 ,α ĉp> ,α |ci,
εp> > εF ,
or it is labeled by the momentum p< of a quasihole below the
Fermi surface with the spin-1/2 quantum number α, in which
case
|ci := ĉ†k0 ,α ĉp< ,α |ai,
|bi = ĉ†p< ,α ĉk,α |ci,
εp< < εF ,
εp< < εF .
(8.244b)
The second pair of virtual states is constructed by applying
V̂J=J⊥ ,J =0 on either the initial or final state. The virtual state
k
is then either
|ci := ĉ†p> ,ᾱ ĉk,α Ŝ α |ai,
εp> > εF ,
|bi = ĉ†k0 ,α ĉp> ,ᾱ Ŝ ᾱ |ci,
εp> > εF ,
|ci := ĉ†k0 ,α ĉp< ,ᾱ Ŝ ᾱ |ai,
εp< < εF ,
(8.245a)
or
|bi = ĉ†p< ,ᾱ ĉk,α Ŝ α |ci,
εp< < εF .
(8.245b)
Show that the third-order term on the right-hand side of Eq. (8.234a)
is proportional to the integral (8.240) for the first pair (8.244)
of virtual states. Show that the third-order term on the righthand side of Eq. (8.234a) delivers the sum
X fFD (εp )
εk
≈ νF ln
(8.246)
ε − εp
εk − εF
p∈BZ k
for the second pair (8.245) of virtual states. Which one of
the four families of virtual states gives the single-particle result (8.239)? Which ones of the steps of this computation
are intrinsically many-body ones, i.e., would not be possible
within single-particle physics?
8.10. PROBLEMS
479
8.10.2. The Kondo effect: a non-perturbative approach.
Introduction. The model (8.235) studied by Kondo has a very small
concentration Ns /N 1 of spin-1/2 impurities, where Ns is the number of spin-1/2 impurities and N is the number of sites from the lattice
Λ on which the non-interacting electrons are hopping with the singleparticle dispersion εk . In this limit, each spin-1/2 impurity can be
treated independently of the others. This is why the model (8.235)
with Ns = 1 is known as the Kondo model.
Although Kondo solved the mystery of the resistivity minimum in
dilute magnetic alloys such as the metals Cu, Ag, Au, Mg, or Zn
with Cr, Mn, Fe, Mo, Re, or Os as impurities, he left opened the
(Kondo) problem of how to prove his conjecture that a spin-1/2 impurity is screened by one band of conduction electrons. This task can be
achieved in three different ways.
In the spirit of Anderson, Yuval, and Hamann, [92] an effective
partition function for the spin-1/2 impurity is obtained after approximately integrating out the conduction electrons. The effective action
is defined in imaginary time where it is non-local, a reflection of the
fact that the spectrum of the conduction electrons is gapless. It can be
studied with the help of the renormalization group. This approach is
very similar to that employed for a dissipative Josephson junction in
section 8.8.
In the spirit of Wilson, [93] a single-particle basis for the conduction electrons is chosen such that their hybridizations with the magnetic impurity is maximized and their energies are as close as possible
to the Fermi energy. In other words, electronic states that are far away
from the impurity in position space and have a single-particle energy
close to the bottom or top of the conduction band can be neglected.
This basis selection is similar to the one made in section 6.7.1.4 to
derive the Friedel oscillations of a classical point scatterer. One may
then integrate those single-particle basis states that reside in a small
window of large energies relative to the Fermi energy, as we did to
derive the one-loop flow of a repulsive interaction in a superconductor
in section 7.2.1 for example. In doing so, the Kondo coupling changes
slightly. By iterating this procedure numerically, the flow of the Kondo
coupling can be traced in a non-perturbative way all the way from the
regime of a free magnetic impurity (high temperature) down to vanishing temperature. [94] For a bare ferromagnetic Kondo coupling, the
Kondo coupling flows to zero in magnitude, i.e., the magnetic impurity
is essentially free as the temperature approaches zero. For a bare antiferromagnetic Kondo coupling, the Kondo coupling flows to infinity
in magnitude. In this limit, the Kondo hybridization term V̂ is minimized by forming a singlet between the electronic spin density and the
spin-1/2 impurity with an infinitely large gap to the triplet excitations.
In effect, the spin-1/2 impurity has been screened.
480
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
The conclusions of both approaches have been confirmed by nonperturbative analytical means with the help of the seminal observation
made by Andrei in Ref. [95] and by Wiegmann in Ref. [96] that a
mild approximation to the Kondo model brings it to a class of exactly
soluble models in (1 + 1)-dimensional position space and time.
We are going to construct a mean-field approximation of the Kondo
model that solves the Kondo problem. This mean-field approximation
is uncontrolled. However, it reproduces (by design) qualitatively the
crossover to the strong Kondo coupling regime with decreasing temperature uncovered by the aforementioned approaches. Before we do so
however, we are going to list some lattice models related to the Kondo
model.
Definitions. The Kondo model (8.235) has one magnetic spin-1/2
impurity (Ns = 1). In the opposite limit, the Kondo lattice model
has one spin-1/2 impurity on each site of the lattice Λ. Conduction
electrons from a single band hop between the sites of Λ. Hence, the
number of lattice sites N equals the number of magnetic impurities Ns
and we may write
Ĥ := Ĥ0 + V̂KL ,
(8.247a)
where (footnote 14 defines our normalization convention for Fourier
transformations)
X X †
1 X −ik·(ri −rj )
Ĥ0 :=
ĉi σ tij ĉj σ ,
tij :=
e
εk = t∗ji ,
N
i,j∈Λ σ=↑,↓
k∈BZ
(8.247b)
is the tight-binding representation of Hamiltonian (8.235a), while
4J X
~ X †
(8.247c)
V̂KL := 2
ŝi · Ŝ i ,
ŝi =
ĉi,α σ αβ ĉi β ,
~ i∈Λ
2 α,β=↑,↓
is the tight-binding representation of Hamiltonian (8.235b) for Ns =
N . The Kondo lattice model is more complicated than the Kondo
model. [97] The hopping of the electrons is strongly affected by constructive or destructive quantum interferences arising from the localized
spins. Conversely, if one imagine integrating out the fermions, there
follows a Heisenberg model with long-range oscillatory exchange couplings. The Kondo lattice model is often taken as a starting point to
understand a class of materials called heavy fermions. [98]
The mean-field method that we shall apply to the Kondo model
takes advantage of a representation of the localized spin-1/2 magnetic
impurities in terms of auxiliary bosons and fermions. This is a mere
trick devised to circumvent the lack of Wick theorem for operators
satisfying the SU (2) spin algebra. We already used this trick in section
6.10.3. Alternatively, we may ask the following question. Is there a
tight-binding model for more than one band of fermions that is related
to the Kondo model? The answer is affirmative as we now show.
8.10. PROBLEMS
481
The model in question is due to Anderson in 1961. [99] This model
and its variants are now called the Anderson model with various qualifiers, such as single impurity, periodic, SU (4) symmetric, etc. Its
relationship to the Kondo model was established by Schrieffer and
Wolff. [100]
Anderson wanted to construct a simple model such that it captures
the competition between the tendency for localized fermions to develop
a local magnetic moment by way of a generalization of Hund’s coupling
from atomic physics and the preference for these localized fermions to
lower their energy by hybridization with a conducting band of electrons. The localized electrons often originate from 3d or 4f atomic
orbitals. The letter c is conventionally reserved for the creation and
annihilation operators of conduction electrons. The letter f is conventionally reserved for the creation and annihilation operators of the
non-dispersing electrons. The two-band model known as the Anderson
model is then 15
Ĥ := Ĥc + Ĥf + Ĥc−f
(8.248a)
with
X X
Ĥc :=
εk ĉ†k,σ ĉk,σ
(8.248b)
k∈BZ σ=↑,↓
the non-interacting Hamiltonian for the conduction (c) electrons in the
Bloch representation,
N
N
Ĥf := εf
f
X
X
fˆr†i σ
fˆri σ + U
f X
fˆr† ↑ fˆri ↑
i
†
ˆ
ˆ
fr ↓ fri ↓ , (8.248c)
i
i=1
i=1 σ=↑,↓
the interacting Hamiltonian for the localized (f ) electrons in the position representation, and
N
Ĥc−f :=
f X XX
Vk e−ik·ri ĉ†k,σ fˆri σ + Vk∗ e+ik·ri fˆr†i σ ĉk,σ
k∈BZ σ=↑,↓ i=1
(8.248d)
the coupling between the conduction band of c electrons and the localized f electrons. The f electrons are not dispersing on their own, they
are localized on a subset made of Nf sites from the lattice Λ. Occupation of any one of these sites with two f electrons of opposite spins
cost the on-site energy U > 0. The coupling between ĉ†k,σ and fˆri σ
is mediated by a single-body term with the matrix elements Vk ∈ C
that we assume independent of the spin quantum number σ and of the
impurity site i (in magnitude for the latter). The choice for the normalization of Vk is motivated by requesting that Ĥc−f is proportional
15
All creation and annihilation operators obey the usual anticommutation relations appropriate to fermions. Creation and annihilation operators with different
symbols or labels have a vanishing anticommutator. The anticommutator of an
annihilation operator with its adjoint is unity.
482
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
√
to N when Nf = 1. We could have made the choice Vk → Vk / N .
With this choice, the coupling between the f electrons and the c electrons becomes weaker as the number of conduction electrons increases.
This is the spirit of the Caldeira-Leggett model that takes advantage
of the macroscopic size of the bath to couple each normal mode of
the bath to the finite number (Ns ) of degrees of freedom constituting
the “impurity”. In fact, this convention is required if we demand that
Hamiltonian (8.271b) scales like N in the thermodynamic limit [see
Eq. (8.328) at which stage we will
p need to use Eq. (8.271b)]. Another
possibility is to choose Vk → Ns /N Vk and to take the thermodynamic limit N → ∞ keeping the ratio Ns /N fixed, say Ns /N 1 in
the dilute limit. In thermodynamic equilibrium the total number of
electrons, the sum of c electrons and of f electrons, is fixed by the
chemical potential.
The phase diagram of the Anderson model at vanishing temperature
depends on the dispersion of the c electrons, on the cost εf to fill an
impurity site by a single 4f electron, on the on-site repulsive interaction
U of f electrons, on the hybridization matrix elements Vk , and on the
chemical potential µ that determines the total number Ne of electrons.
Exercise 1.1:
(a) When the number of impurity sites Nf is smaller than the
number N of lattice sites, the translation symmetry group of
the lattice Λ is broken in the Anderson model. The symmetry
group of the lattice Λ is recovered when Nf = N , in which
case the periodic Anderson model follows. Write down the
periodic Anderson model with the f creation and annihilation
operators expressed in terms of the crystal momentum k ∈ BZ
(the on-site interaction term should be written in terms of the
occupation number operator of f electrons in the first Brillouin
zone).
(b) Write down the Anderson model (8.248) with the c electron
creation and annihilation operators and their hybridization
matrix elements to the f electron creation and annihilation
operators expressed in terms of the site index i of the lattice
Λ.
When the number of impurity sites Nf = 1, the single-impurity
Anderson model is obtained. Without loss of generality, we declare that
the impurity site is at the origin of the coordinate system in position
space. This is the model we want to relate to the Kondo model. If
we use a path-integral representation, it is defined by the partition
function
Z
Z
∗
Z := D[c , c] D[f ∗ , f ] e−S
(8.249a)
with the action
S := Sc + Sf + Sc−f
(8.249b)
8.10. PROBLEMS
483
decomposing into the Euclidean action for the c electrons
Zβ
Sc :=
dτ
0
X X
c∗k,σ (∂τ + ξk ) ck,σ ,
(8.249c)
k∈BZ σ=↑,↓
the Euclidean action for the f electrons
"
#
Zβ
X
Sf := dτ
fσ∗ ∂τ + ξf fσ + U f↑∗ f↑ f↓∗ f↓ ,
(8.249d)
σ=↑,↓
0
and the Euclidean action that couples the c electrons and the f electrons
Zβ
X X
(8.249e)
Sc−f := dτ
Vk c∗k,σ fσ + Vk∗ fσ∗ ck,σ .
0
k∈BZ σ=↑,↓
As usual, all Grassmann-valued fields obey antiperiodic boundary conditions in imaginary time. If we opt to work with the Hamiltonian
formalism, we deduce from S the Hamiltonian
Ĥ := Ĥc + Ĥf + Ĥc−f ,
from Sc the Hamiltonian
X X
Ĥc :=
ξk n̂ck,σ ,
n̂ck,σ := ĉ†k,σ ĉk,σ ,
(8.250a)
(8.250b)
k∈BZ σ=↑,↓
from Sf the Hamiltonian
Ĥf := ξf n̂f↑ + n̂f↓ + U n̂f↑ n̂f↓ ,
n̂fσ := fˆσ† fˆσ ,
and from Sc−f the Hamiltonian
X X
Vk ĉ†k,σ fˆσ + Vk∗ fˆσ† ĉk,σ .
Ĥc−f :=
(8.250c)
(8.250d)
k∈BZ σ=↑,↓
All single-particle energies are now measured relative to the chemical
potential µ. This is why we changed the symbol from ε· to ξ· to denote
single-particle energies.
The single-impurity Anderson Hamiltonian (8.250) is constructed
to be exactly soluble in the limit of no hybridization, i.e., for Vk = 0
for all wave vectors from the first Brillouin zone. The spectrum of
Hamiltonian (8.250a) is the “addition” of the spectrum of Hamiltonian (8.250b) and of the spectrum of Hamiltonian (8.250c).
The spectrum of Hamiltonian (8.250b) is that of a non-interacting
gas of electrons, i.e., the Fermi sea ground state |FSic with all possible particle-hole excitations as excited states. The level spacing of
Hamiltonian (8.250b) above the ground state is of order ~ vF 2π/L. 16
16
Here, vF is the Fermi velocity and L the linear size of the lattice.
484
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
It becomes a continuum in the thermodynamic limit L → ∞ holding
the filling fraction Ne /N of the lattice fixed.
The spectrum of Hamiltonian (8.250c) consists of four orthonormal
energy eigenstates. There is the vacuum with the eigenenergy 0, i.e.,
the state |0if annihilated by fˆσ for both projections σ =↑, ↓ of the
spin along the quantization axis. There are two degenerate states with
eigenenergy ξf , the states
| ↑if := fˆ↑† |0if ,
| ↓if := fˆ↓† |0if .
(8.251)
There is one state with the eigenenergy 2ξf + U ,
| ↑↓if := fˆ↑† fˆ↓† |0if .
(8.252)
Since we are measuring energies relative to the Fermi energy, at the
Fermi energy the f electrons are in their vacuum state |0if . If we
choose ξf to be positive, the many-body eigenstates with one or two f
electrons present have an excitation energy bounded from below by
ξf > 0
(8.253)
2ξf + U > 0,
(8.254)
and
respectively. Thus, they are inoperative to stabilize a localized magnetic moment at temperatures below the threshold ξf /kB .
The regime of parameter space for Hamiltonian Ĥc + Ĥf that is the
most favorable to the formation of a localized magnetic moment in the
ground state of Hamiltonian Ĥc + Ĥf is when the chain of inequalities
ξf < 0 < 2ξf + U
(8.255)
holds, for the many-body ground state manifold of Ĥc + Ĥf is then
two-fold degenerate, given by
n
o
span |FSic ⊗ | ↑if , |FSic ⊗ | ↓if ,
(8.256)
and separated from all excited many-body energy eigenstates by the
gap of order ~ vF 2π/L in the sector of the Hilbert space for the conduction electrons and the gap |ξf | in the sector of the Hilbert space
for the f electrons. The chain of inequalities (8.255) is always met for
sufficiently large U .
Exercise 1.2:
(a) Calculate the magnetic susceptibility of the ground state manifold (8.256) and show that it obeys the Curie law for temperatures satisfying kB T |ξf |.
(b) Assume the inequalities
ξf < 2ξf + U < 0
(8.257)
8.10. PROBLEMS
485
and compute the magnetic susceptibility of Ĥc + Ĥf for temperatures satisfying kB T |2ξf + U |.
(c) Assume the inequalities
2ξf + U < ξf < 0
(8.258)
and compute the magnetic susceptibility of Ĥc + Ĥf for temperatures satisfying kB T |ξf |.
The question to be addressed is what happens to Curie’s law obeyed by
the ground state manifold (8.256) upon approaching zero temperature
once the average value
P
|Vk |2
k∈Fs
|Vk |2 Fs := P
(8.259)
1
k∈Fs
of the hybridization over the unperturbed Fermi surface (Fs as opposed
to FS for the Fermi sea) of the c electrons is switched on adiabatically.
Perturbation theory in the hybridization cannot address this issue for
the same reason as perturbation theory in the Kondo coupling fails.
This can be seen by brute force calculation when reaching the fourth
order of perturbation theory. Alternatively, we can map the singleimpurity Anderson model to the Kondo model when conditions (8.274)
are met, as first shown by Schrieffer and Wolff in Ref. [100].
The Schrieffer-Wolff transformation. We rewrite Hamiltonian (8.250)
as
Ĥ(λ) = Ĥ0 + λ Ĥ1 ,
(8.260)
where the real-valued dimensionless number λ is introduced for bookkeeping. Hamiltonian
Ĥ0 := Ĥc + Ĥf
(8.261)
is the unperturbed Hamiltonian of which we know the eigenstates and
eigenvalues. In particular, we can compute the magnetic susceptibility
χ0 (T ) of Ĥ0 . In fact we know that χ0 (T ) is singular (Curie singularity)
at T = 0. Hamiltonian
Ĥ1 := Ĥc−f
(8.262)
is the perturbation, which can be shown to be singular when computing
the corrections it brings about to the magnetic susceptibility χ0 (T ) of
Ĥ0 at temperature T , i.e., the sequence of functions χn defined for any
positive integer n by computing the correction to χ0 up to n-th order
in the perturbation Ĥ1 does not converge uniformly with decreasing
temperature to χ0 . This breakdown of perturbation theory signals
that the dependence on temperature of χ(T ) could still be singular or
it could be regular in the limit T → 0, but it cannot be captured by
doing perturbation theory in powers of Ĥ1 . If the manipulation
χ(T ) = χ0 (T ) + [χ(T ) − χ0 (T )]
(8.263)
486
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
seems innocuous at any non-vanishing temperature, it is not at T = 0
where χ0 (T ) is singular, and it is not to be expected that perturbation
theory in the operator that encodes the difference χ(T ) − χ0 (T ) is a
wise approach to capture the dependence of χ(T ) in the limit T → 0.
It would be wiser to seek a function χ?0 that shares the same leading
dependence on T for T → 0 as χ and then do the decomposition
χ(T ) = χ?0 (T ) + [χ(T ) − χ?0 (T )] .
(8.264)
This function χ?0 should be a property of a fixed point that is perturbatively close to the fixed point governing the dependence of χ(T ) on
T as T → 0. With the privilege of hindsight, this fixed point should be
governed by some strong coupling limit, but it had not been identified
prior to the work of Schrieffer and Wolff.
With no such a priori knowledge, as was the case for Schrieffer
and Wolff, we may keep Ĥ0 in the additive decomposition of Ĥ but
opt using a perhaps simpler term than Ĥ1 , one which would still be
singular if treated as a perturbation to Ĥ0 , but one that would offer new
insights on how to identify a more reasonable fixed point Hamiltonian
Ĥ0? compared to Ĥ0 .
Hence, we seek a similarity transformation generated by Ŝ = −Ŝ †
that acts on the space of creation and annihilation operators and such
that the transformed Hamiltonian
Ĥ ? (λ) := e+λ Ŝ Ĥ(λ) e−λ Ŝ
(8.265)
obeys
?
dĤ
dλ
!
(λ = 0) = 0,
(8.266)
in the hope that the perturbation Ĥ ? (λ) − Ĥ0 leads to useful insights
or simplifications. To this end, we may use the following expansion of
the similarity transformation (8.265),
i λ3 h h
ii
λ2 h
λ
Ŝ, [Ŝ, Ĥ(λ)] +
Ŝ, Ŝ, [Ŝ, Ĥ(λ)] +· · ·
Ĥ ? (λ) = Ĥ(λ)+ [Ŝ, Ĥ(λ)]+
1!
2!
3!
(8.267a)
and demand that
[Ŝ, Ĥ0 ] := −Ĥ1 .
(8.267b)
Insertion of the condition (8.267b) that defines the similarity transformation Ŝ into the expansion (8.267a) gives
(2 − 1) λ2
[Ŝ, Ĥ1 (λ)]
2!
i (4 − 1) λ4 h h
ii
(3 − 1) λ3 h
Ŝ, [Ŝ, Ĥ1 (λ)] +
Ŝ, Ŝ, [Ŝ, Ĥ1 (λ)] + · · · ,
+
3!
4!
(8.268)
Ĥ ? (λ) = + Ĥ0 +
8.10. PROBLEMS
487
as desired. Computing Ĥ ? to all orders is equivalent to diagonalizing
Ĥ, which we do not know how to do in the first place. However, the
expansion (8.268) becomes useful if there is a range of parameters for
which it may be truncated, say by dropping all terms of order λ3 or
higher.
Exercise 2.1:
(a) Verify Eqs. (8.267a) and (8.268). Hint: Equation (8.267a) is
closely related to the Baker-Campbell-Hausdorff formula on
the one hand, and to the equations of motion obeyed by operators in the Heisenberg picture in imaginary time on the other
hand [recall Eq. (5.144)].
(b) Verify that
"
Ŝ =
X X
k∈BZ σ=↑,↓
#
f
V
1
−
n̂
Vk n̂fσ̄
σ̄
ĉ† fˆ + k
ĉ†k,σ fˆσ − H.c.
ξk − ξf − U k,σ σ
ξk − ξf
(8.269)
solves Eq. (8.267b). Use this result to justify identifying the
two dimensionless ratios
rJ :=
ν̃˜F × h|Vk |2 iFs
,
|ξf + U |
rW :=
ν̃˜F × h|Vk |2 iFs
,
|ξf |
(8.270)
as the ones that control how good the expansion (8.268) is.
Here, ν̃˜F is the density of states per unit energy of the unperturbed c electrons at the Fermi energy, and we made use of
the definition (8.259) for the average over their unperturbed
Fermi surface.
(c) We use a more compact notation by which ĉk is a column
vector with the two components ĉk ↑ and ĉk ↓ and similarly for
fˆ. We also use the three-component vector σ to present the
three Pauli matrices. Verify that
i
1h
(1)
(2)
(3)
(4)
Ŝ, Ĥ1 = H1 + H1 + H1 + H1 ,
2
(8.271a)
where
(1)
H1 := −
1
4
Jk0 ,k ĉ†k0 σ ĉk · fˆ† σ fˆ ,
X
k,k0 ∈BZ
Jk0 ,k := Vk0 Vk∗
!
1
1
1
1
+
−
−
,
ξk − ξf − U
ξk0 − ξf − U
ξk − ξf
ξk0 − ξf
(8.271b)
488
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
would be a local Heisenberg exchange interaction in the first
Brillouin zone if Jk0 ,k was proportional to δk0 ,k ,
X 1
(2)
†
†
ˆ
ˆ
ĉk0 ĉk ,
H1 :=
Wk0 ,k + Jk0 ,k f f
4
k,k0 ∈BZ
!
(8.271c)
Vk0 Vk∗
1
1
Wk0 ,k :=
,
+
2
ξk − ξf
ξk0 − ξf
would be the sum of a local density of c electrons in the first
Brillouin zone with a local density-density interaction between
c and f electrons in the first Brillouin zone if Wk0 ,k and Jk0 ,k
were both proportional to δk0 ,k ,
X X 1
(3)
f
H1 := −
Wk,k + Jk,k n̂σ̄ n̂fσ ,
(8.271d)
2
k∈BZ σ=↑,↓
shifts the chemical potential of the f electrons, and
1 X X (4)
H1 :=
Jk0 ,k ĉ†k0 ,σ̄ ĉ†k,σ fˆσ fˆσ̄ + H.c.
(8.271e)
4 0
σ=↑,↓
k,k ∈BZ
would hybridize a pair of f electrons with a local pair (in the
first Brillouin zone) of c electrons if Jk0 ,k was proportional to
δk0 ,k .
(d) Verify that if k0 = k is on the unperturbed Fermi surface of
the c electrons, i.e., if ξk0 = ξk = 0, then
Jk0 ,k = 2 |Vk |2
U
,
ξf (ξf + U )
Wk0 ,k = −|Vk |2
1
.
ξf
(8.272)
(e) Equation (8.272) indicates that Jk0 ,k enters as an antiferromagnetic coupling for k0 = k on the unperturbed Fermi surface if
and only if
ξf (ξf + U ) < 0,
(8.273)
in which case the unperturbed ground state of Ĥf has one f
electron present. The condition (8.273) is met for any negative value of ξf provided ξf + U > 0. In the subspace of the
(2)
Hilbert space with one f electron present, Ĥ1 simplifies to
a one-body Hamiltonian that changes the dispersion of the c
P
(3)
electrons, Ĥ1 shifts the value of ξf by − k∈BZ Wk,k , and
(4)
Ĥ1 is inoperative, for it only has non-vanishing matrix elements between the subspaces with no and two f electrons
present. Explain why the two independent conditions
rJ 1
(8.274a)
8.10. PROBLEMS
489
and
rW 1
(8.274b)
ν̃˜F × Jk,k Fs 1
(8.274c)
imply that
and explain why, at sufficiently low temperatures, the approximation (the bookkeeping parameter λ has been set to the
number 1)
(1)
Ĥ ? ≈ Ĥ0 + Ĥ1
(8.275)
is a good one.
Having rewritten the single-impurity Anderson model as a Kondo
model shows that the Kondo problem is also present in the singleimpurity Anderson model. It also shows that the
that drives
coupling
the crossover from high to low temperature is Jk,k Fs in the singleimpurity Anderson model. Finally, the solution for the crossover from
high to low temperatures in either one of the two models can be applied
to the other model.
Mean-field approximation for the single-impurity Anderson model.
We are going to work with the representation (8.249) of the singleimpurity Anderson model. We are going to reproduce a mean-field
approximation done by Anderson in Ref. [99] that will give us a complementary perspective to the Schrieffer-Wolff one.
Exercise 3.1: Using a Hubbard-Stratonovich transformation verify
that the Euclidean action (8.249d) for the f electrons can be written
as
"
#
Zβ
X
U
U
fσ∗ ∂τ + ξf fσ +
Sf = dτ
f↑∗ f↑ − f↓∗ f↓ ϕ + ϕ2 .
2
4
σ=↑,↓
0
(8.276)
2
fˆ↑ − fˆ↓† fˆ↓
Hint: Make use of
fˆ↓ =
fˆ↑ + fˆ↓ −
and explain why and how the first term on the right-hand side of this
equality can be ignored. Observe that the Hubbard-Stratonovich field
z
ϕ couples linearly to the magnetic moment fˆ† σ2 fˆ of the f electron.
Exercise 3.2: The sum of the Euclidean actions (8.249c) and (8.249e)
can be written as
"
Zβ
X X
c∗k,σ − fσ∗ Vk∗ Gck (−Gck )−1
Sc + Sc−f = dτ
4 fˆ↑†
0
fˆ↑ fˆ↓†
fˆ↑†
fˆ↓†
2
fˆ↑†
k∈BZ σ=↑,↓
(8.277a)
#
× ck,σ − Gck Vk fσ + fσ∗ |Vk |2 Gck fσ ,
where we have introduced the single-particle Green function
(Gck )−1 := − (∂τ + ξk )
(8.277b)
490
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
for the unperturbed c electrons. After going to fermionic Matsubara
frequency space, we can independently shift the c∗ ’s and c’s Grassmann
integration variables and do their Grassmann integrations. Verify that
the integration over the c Grassmann variables delivers
Z
Z
0
0
Z ∝ D[ϕ]
D[f ∗ , f ] e−S0 −S1
(8.278a)
with the effective action
Zβ
X
0
∗
S0 = dτ
fσ ∂τ + ξf + Σ̂ fσ
0
(8.278b)
σ=↑,↓
and
S10
Zβ
=
dτ
U 2
U
∗
∗
f↑ f↑ − f↓ f↓ ϕ + ϕ .
2
4
(8.278c)
0
The f electrons have acquired the self-energy Σ̂, a non-diagonal operator in the imaginary-time representation that becomes diagonal in the
Matsubara frequency representation with the C-valued matrix elements
X |V |2
π
k
Σiωn =
,
ωn = (2n + 1),
n ∈ Z,
(8.278d)
iω
−
ξ
β
n
k
k∈BZ
from their hybridization with the c electrons. 17 The Fourier conventions of Eq. (6.22) have been used here. The action S00 +S10 is quadratic
in the f electrons. Verify that their integration delivers the partition
function
Z
00
Z ∝ D[ϕ] e−S [ϕ]
(8.279a)
with the effective action
"
#
Zβ
X
U
U
S 00 [ϕ] = dτ
ϕ2 −
Tr log ∂τ + ξf + Σ̂ + σ ϕ .
4
2
σ=+,−
0
(8.279b)
The trace must be done with functions of τ that obey antiperiodic
boundary conditions over the interval [0, β]. Show that
S 00 [+ϕ] = S 00 [−ϕ].
(8.280)
So far no approximations were made and we have access to the projection on the quantization axis of the magnetization
z
~
σ
1
δ
ln
Z
†
fˆ
fˆ :=
[B = 0],
(8.281)
2
β
δB
β
17
With the convention we chose for the normalization of Vk in Eq. (8.248d) or
in Eq. (8.250d) the self-energy is extensive, i.e., grows like N in the thermodynamic
limit.
8.10. PROBLEMS
and the dynamical susceptibility
1
δ 2 ln Z
0
χ(τ, τ ) :=
[B = 0],
β δB(τ ) δB(τ 0 )
491
(8.282)
on the impurity site.
B(τ
) is a source field that enters as the
Here,
† ~ σz ˆ
ˆ
additive term B(τ ) f 2 f (τ ) in the logarithm of S 00 .
Exercise 3.3: The first approximation that we are going to do is
on the self-energy (8.278d). Show that
X |V |2 ξ
k
k
Re Σiωn = −
(8.283a)
2
ωn + ξk2
k∈BZ
and
Im Σiωn = −
X |V |2 ω
n
k
.
2 + ξ2
ω
n
k
k∈BZ
(8.283b)
Justify the approximations
Re Σiωn
X |V |2 ξ
k
k
≡ δξf
≈−
2
ξk
k∈BZ
(8.284a)
and
Im Σiωn ≈ −∆ sgn ωn
(8.284b)
0 ≤ ∆ := π ν̃˜F |Vk |2 Fs ,
(8.284c)
X
(8.284d)
with [recall Eq. (8.259)]
where
ν̃˜F :=
δ(ξk )
k∈BZ
is the density of states per unit energy at the unperturbed Fermi energy.
Exercise 3.4: The second approximation that we make consists
in evaluating the functional derivative of the action (8.279) with the
approximation (8.284) for the self-energy and assuming that ϕ is independent of imaginary time. Neglecting fluctuations in imaginary time
ignores quantum fluctuations. Show that, at β = ∞,
1 dS 00
U
U
(ϕ) =
1 − ¯ ϕ + O ϕ2 ,
(8.285a)
β
dϕ
2
∆
where
2
∆2 + ξf + δξf
¯ := π
∆
.
∆
Hint: You will need the integral
Z
1
1
x+a
dx
= arctan
2
2
(x + a) + b
b
b
and the expansion
π 1
arctan x = − + · · ·
2 x
(8.285b)
(8.286)
(8.287)
492
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
for x 1.
Exercise 3.5: Show that by combining Eq. (8.280) with Eq. (8.285),
the condition
U
(8.288)
¯ >1
∆
guarantees that ϕ = 0 is a local maximum of S 00 (·) and that there exists
at least two minima of S 00 (·) at ±ϕ0 . Each minimum corresponds to one
out of the two possible orientations along the quantization axis of the
local magnetic moment. Comment on the similarities and differences
between the criterion (8.288) which is known as the Anderson criterion
for the emergence of a local magnetic moment in the single-impurity
Anderson model, and the Schrieffer-Wolff criterion (8.274) relating the
single-impurity Anderson model to the Kondo model.
Assume that the function S 00 (·) has two absolute minima at ±ϕ0
¯ and that ϕ = 0 is the absolute minimum of S 00 (·) otherwhen U > ∆
¯ drives
wise. Increasing the dimensionless and positive parameter U/∆
¯ = 1 from the
a continuous mean-field transition at the value U/∆
¯
regime 0 ≤ U/∆ ≤ 1 with one absolute minimum at ϕ = 0 of S 00 (·) to
¯ < ∞ with two absolute minima at ±ϕ of S 00 (·).
the regime 1 ≤ U/∆
0
Can we trust this mean-field prediction? The short answer is negative.
Short of a calculation, we can list the following arguments supporting
this conclusion.
We know that quantum fluctuations wipe out the semi-classical
quantum phase transition of a (0+1)-dimensional quantum field theory
that is local in time. This is the case of a quantum particle with
the kinetic energy (∂τ ϕ)2 and with the double-well potential energy
V (+ϕ) = V (−ϕ), whose ground state wave function is an even function
of ϕ with equal probability to have the quantum particle at any one of
the two minima of the potential well.
However, the difficulty with the functional S 00 [·] in Eq. (8.279) to
be integrated over in the path integral (8.279a) is that it is not local in
imaginary time, for the argument of the logarithm on the right-hand
side of Eq. (8.279b) encodes the coupling to a dissipative bath. Nonlocality in time opens the possibility that elimination of high-energy
quantum fluctuations (large values of ∂τ ϕ) modify the quantum dynamics of the remaining quantum fluctuations, i.e., the separation into
a kinetic and potential contribution to the action. In the case of a local
action in imaginary time with a double-well potential, integration of
high-energy quantum fluctuations preserves the double-well shape at
all energies. Anderson, Yuval, and Hamann on the one hand and Wilson on the other hand tell us that this is not the case for the quantum
problem (8.279).
We may imagine the following scenario against spontaneous symmetry breaking of the symmetry under ϕ → −ϕ of the quantum problem (8.279). At high energies set by the energy scale U the potential
8.10. PROBLEMS
493
has two absolute minima representing the two possible orientations of
localized spin-1/2 degree of freedom on the impurity site. At low energies this potential has one absolute minimum representing a localized
spin-1/2 degree of freedom that has been screened by the conduction
electrons through the formation of a singlet bound state. The flow of
this potential from high to low energies corresponds to the crossover
from Curie’s law χ(T ) ∝ (1/T ) at high temperatures to the Pauli
susceptibility χ(T ) ∝ (1/TK ) at temperatures well below the Kondo
temperature TK .
Large Nc expansion for the single-impurity Anderson model. The
mean-field approximation
00 Z
dS
−S 00 [ϕ]
−S 00 (ϕ̄)
Z ∝ D[ϕ] e
≈e
,
[ϕ̄] = 0,
(8.289)
dϕ
dictates that Curie’s law can only be obeyed if condition (8.288) holds.
However, this mean-field approximation is an uncontrolled approximation. Following Coleman in Ref. [101], we are going to introduce a
model for which (i) there exists an exact mean-field theory, and such
that (ii) the qualitative behavior expected from exact solutions of the
Kondo and related models is captured.
The quantum impurity problem is defined by the partition function
Z
Z
∗
Z := lim
D[c , c] D[f ∗ , f ] e−S
(8.290a)
U →∞
with the action
S := Sc + Sf + Sc−f
decomposing into the Euclidean action for the c electrons
Zβ
dτ
Sc :=
0
Nc
X X
c∗k,σ (∂τ + ξk ) ck,σ ,
(8.290b)
(8.290c)
k∈BZ σ=1
the Euclidean action for the f electrons
"N
! N
!#
Zβ
Nc
c
c
X
X
X
Sf := dτ
fσ∗ ∂τ + ξf fσ + U
fσ∗ fσ − 1
fσ∗ fσ
,
σ=1
0
σ=1
σ=1
(8.290d)
and the Euclidean action that couples the c electrons to the f electrons
Zβ
Sc−f :=
dτ
0
Nc
X X
k∈BZ σ=1
1
p
Vk c∗k,σ fσ + Vk∗ fσ∗ ck,σ .
Nc
(8.290e)
As usual, all Grassmann-valued fields obey antiperiodic boundary conditions in imaginary time.
This model differs from the single-site impurity Anderson model (8.249)
in two ways. First, the spin index is now a color index σ that runs from
494
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
1 to Nc . The limit of infinitely many colors, Nc → ∞, is the one for
which the mean-field approximation to come is exact. Second, we have
replaced the Hubbard on-site repulsive interaction by an on-site repulsive interaction that penalizes with the positive energy (n − 1) n U the
occupation of the impurity site by 1 < n ≤ Nc electrons. The limit
U → ∞ is taken from the outset, i.e., a counterpart to Anderson criterion (8.288) would be met if it exists. In the limit U → ∞, the Fock
space of the f electrons
Fphys
f
†
ˆ
:= span |0if , fσ |0if fˆσ |0if = 0,
n
{fˆσ , fˆσ†0 } = δσ,σ0 ,
{fˆσ† , fˆσ†0 } = {fˆσ , fˆσ0 } = 0,
σ, σ 0 = 1, · · · , Nc
(8.291a)
o
is (Nc + 1)-dimensional. In contrast, the Fock space for the conduction
electrons
(
Nc nk,σ
Y Y
Fc = span
ĉ†k,σ
{ĉk,σ , ĉ†k0 ,σ0 } = δk,k0 δσ,σ0 ,
|0ic ĉk,σ |0ic = 0,
σ=1
k∈BZ
)
{ĉ†k,σ , ĉ†k0 ,σ0 }
= {ĉk,σ , ĉk0 ,σ0 } = 0,
nk,σ = 0, 1,
0
k, k ∈ BZ,
0
σ, σ = 1, · · · , Nc
(8.291b)
is 2N ×Nc -dimensional. The Fock space over which the partition function
Z is to be traced is
F := Fc ⊗ Fphys
.
f
(8.291c)
Implementing constraints is inherently a strong coupling problem
as the limit U → ∞ makes explicit. The constraint
Nc
X
fˆσ† fˆσ ≤ 1
(8.292)
σ=1
is difficult to implement analytically because it is an inequality. The
slave boson method was devised to turn this constraint into an operator
equality on a different Hilbert space. [69, 70, 101] We introduce the
infinite-dimensional bosonic Fock space
(
Fb = span
∞
Y
(b̂† )n
√ |0ib b̂ |0ib = 0,
n!
n=0
)
[b̂, b̂† ] = 1,
[b̂† , b̂† ] = [b̂, b̂] = 0
(8.293a)
8.10. PROBLEMS
495
and the 2Nc -dimensional fermionic Hilbert space
( N
c
Y
n
Fs = span
ŝ†σ σ |0is ŝσ |0is = 0,
{ŝσ , ŝ†σ0 } = δσ,σ0 ,
σ=1
)
{ŝ†σ , ŝ†σ0 } = {ŝσ , ŝσ0 } = 0,
nσ = 0, 1,
σ, σ 0 = 1, · · · , Nc .
(8.293b)
The (Nc + 1)-dimensional physical Fock space is
!
)
(
Nc
X
ŝ†σ ŝσ |bi⊗|si = |bi⊗|si .
Fphys
|bi⊗|si|bi ∈ Fb , |si ∈ Fs ,
b̂† b̂+
b×s :=
σ=1
(8.293c)
phys
Exercise 4.1: Show that an isomorphism between Ff and Fphys
b×s
is established by the maps
|0if 7→ b̂† |0ib ,
fˆσ† |0if 7→ ŝ†σ |0is ,
σ = 1, · · · , Nc ,
fˆ† 7→ ŝ† b̂,
σ = 1, · · · , N ,
σ
σ
fˆσ 7→ b̂† ŝσ ,
(8.294)
c
σ = 1, · · · , Nc .
Hint: Verify first that the fermionic algebra of the fˆ’s is preserved
under this mapping. Verify then that the matrix elements of the fˆ’s in
Fphys
are in one-to-one correspondence with those of the b̂† ŝ’s in Fphys
f
b×s .
Exercise 4.2: Convince yourself that there are uncountably many
distinct ways of representing the (Nc + 1)-dimensional physical Fock
space Fphys
with auxiliary bosonic and fermionic operators. Hint: Do
f
this by way of three examples. Define a transformation by which the
phases of the ŝ and b̂ operators are changed without changing their
algebra and leaving the fˆ operators unchanged. Second, allow the b
bosons to acquire the same index σ as the s fermions. Third, choose the
b̂ operators to be spinless fermions and the ŝ operators to be bosons.
This representation is called the slave-fermion representation.
Exercise 4.3: Verify that the partition function (8.290) can be
presented as the partition function
Z
Z
Z
Z
∗
∗
Z := D[c , c] D[λ] D[s , s] D[b∗ , b] e−S ,
(8.295a)
where the action
Zβ
S := Sc + Ss + Sb + Sc−b×s − i
dτ λ
0
(8.295b)
496
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
is here decomposed into the bilinear Euclidean action for the c electrons
Zβ
Sc :=
dτ
Nc
X X
c∗k,σ (∂τ + ξk ) ck,σ ,
(8.295c)
k∈BZ σ=1
0
the bilinear Euclidean action for the s electrons
Zβ
Ss :=
Nc
X
dτ
s∗σ ∂τ + ξf + iλ sσ ,
(8.295d)
σ=1
0
the bilinear Euclidean action for the b bosons
Zβ
Sb :=
dτ b∗ (∂τ + iλ) b,
(8.295e)
0
and the Euclidean action that couples the c electrons to the s electrons
and b bosons
Zβ
Nc
X X
dτ
Sc−b×s :=
k∈BZ σ=1
0
1
p
Vk c∗k,σ sσ b∗ + Vk∗ s∗σ b ck,σ . (8.295f)
Nc
All the Grassmann integration variables c∗ , c , s∗ , and s are independent and obey antiperiodic boundary conditions in imaginary time.
The complex-valued field b∗ is the complex conjugate of b, the latter obeying periodic boundary conditions in imaginary time. The
real-valued field λ enforces the projection onto the physical Hilbert
space (8.293c). It obeys periodic boundary conditions in imaginary
time.
Exercise 4.4:
(a) Verify that the partition function (8.295) is invariant under
the U (1) local gauge transformation
s∗σ → s∗σ e+iφ ,
∗
sσ → e−iφ sσ ,
∗ +iφ
b →b e ,
λ → λ + ∂τ φ,
b→e
−iφ
σ = 1, · · · , Nc ,
(8.296)
b,
for any real-valued and smooth function φ : [0, β] → [0, 2π[, τ 7→
φ(τ ) that obeys periodic boundary conditions in imaginary
time.
(b) Use the notation
†
n̂b×s := b̂ b̂ +
Nc
X
σ=1
ŝ†σ
ŝσ ,
∗
nb×s := b b +
Nc
X
σ=1
s∗σ sσ .
(8.297)
8.10. PROBLEMS
497
Show that
0 = n̂b×s (τ1 ) − 1 · · · n̂b×s (τn ) − 1 Z
n
o
Tr e−β Ĥ n̂b×s (τ1 ) − 1 · · · n̂b×s (τn ) − 1
n
o.
:=
−β
Ĥ
Tr e
(8.298)
Hint: Add a source field to the action S that couples to nb×s −
1. Express the correlation function (8.298) as a functional
derivative of the partition with source field. Use the U (1) local
gauge symmetry (8.296) to show that the partition function
with source field equals the partition function without source
field.
Exercise 4.5: Choose the parametrization
p
p
b∗ (τ ) = Nc ρ(τ ) e−iθ(τ ) ,
b (τ ) = Nc ρ(τ ) e+iθ(τ ) ,
(8.299)
in terms of the real-valued amplitude field ρ : [0, β] → [0, ∞[, τ 7→ ρ(τ )
and the real-valued phase field θ : [0, β] → [0, 2π[, τ 7→ θ(τ ). The
transformation law of the measure is
N
(8.300)
D[b∗ , b] = D[ρ, θ],
db∗ (τ ) db(τ ) = c d[ρ2 (τ )] dθ(τ ).
2
Verify that the partition function (8.295a) does not depend on the
phase field θ and is given by
Z
Z
Z
Z
∗
∗
Z ∝ D[c , c] D[λ] D[s , s] D[ρ] e−S ,
(8.301a)
where the action
Zβ
S := Sc + Ss + Sρ + Sc−ρ×s − iNc
dτ
0
λ
Nc
(8.301b)
is here decomposed into the bilinear Euclidean action for the c electrons
Zβ
Sc :=
dτ
0
Nc
X X
c∗k,σ (∂τ + ξk ) ck,σ ,
(8.301c)
k∈BZ σ=1
the bilinear Euclidean action for the s electrons
Zβ
Nc
X
s∗σ ∂τ + ξf + iλ sσ ,
Ss := dτ
0
(8.301d)
σ=1
the bilinear Euclidean action for the amplitude of the b bosons
Zβ
Sρ := Nc
dτ ρ (∂τ + iλ) ρ,
0
(8.301e)
498
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
and the Euclidean action that couples the c electrons to the s electrons
and b bosons
Zβ
Nc
X X
Sc−ρ×s := dτ
ρ Vk c∗k,σ sσ + Vk∗ s∗σ ck,σ .
(8.301f)
0
k∈BZ σ=1
Observe that the replacements
Vk →
q
Zβ
h|Vk
|2 i
Fs ,
Sρ → Nc
dτ
0
h|Vk |2 iFs 2 ξf
ρ −
J0
Nc
,
J0 > 0,
(8.302)
define the large Nc limit of the Coqblin-Schrieffer model solved by Read
and Newns in Ref. [69]. 18
Exercise 4.6: Verify that integration over the c Grassmann variables followed by integration over the s Grassmann variables gives the
partition function
Z
Z
Z = D[λ]
D[ρ] e−Nc β F [λ,ρ] ,
(8.303a)
1 Zβ
1
iλ
F [λ, ρ] := − Tr log ∂τ + ξf + iλ + ρ Σ̂ ρ +
dτ ρ (∂τ + iλ) ρ −
.
β
β
Nc
0
(8.303b)
So far no approximations were made. Hint: All the preparatory work
has been done when deriving Eqs. (8.278) and (8.279) from which we
borrow the definition of the self-energy operator Σ̂.
Exercise 4.7: Convince yourself that, in the limit Nc → ∞, the
partition function (8.303a) simplifies to
X
Z∝
lim e−Nc β F (λ̄,ρ̄) ,
(8.304a)
{λ̄,ρ̄}
Nc →∞
where a pair {λ̄, ρ̄} is a saddle-point solution to the equations
δF
0=
[λ, ρ]
(8.304b)
δρ(τ )
and
δF
0=
[λ, ρ],
(8.304c)
δλ(τ )
18
In the Coqblin-Schrieffer model, [102] the field b is not needed to enforce a
constraint, it is a Hubbard-Stratonovich field introduced to decouple an interaction
between the conduction and impurity electrons. The field λ retains its role as a
U (1) temporal gauge field enforcing the constraint that there be no less and no
more than one electron at the impurity site. For this reason, the partition function
of the Coqblin-Schrieffer model has no separate dependences on ξf and iλ. It only
depends on the linear combination ξf + iλ.
8.10. PROBLEMS
499
that is an absolute minimum (possibly degenerate) of the functional (8.303b).
We seek all pairs of solutions {λ̄, ρ̄} that are static in imaginary time,
as is expected in thermodynamic equilibrium given a conserved Hamiltonian.
Exercise 4.8: From now on, we are going to make the analytical
continuation λ → −iλ. For ρ and λ independent of imaginary time,
show that the functional (8.303b) becomes
1
1 X
2
2
F (−iλ, ρ) = −
log −iωn + ξf + λ + ρ Σiωn + λ ρ −
β ω
Nc
n
I
dz ˜
1
2
2
=
f (z) log −z + ξf + λ + ρ Σz + λ ρ −
.
2πi FD
Nc
Γ
(8.305)
The closed path Γ in the z-complex plane is oriented counterclockwise
and runs parallel and infinitesimally close to the left and right sides
of the imaginary axis. The self-energy Σiωn acquired by the s electrons as a result of their hybridization to the c electrons was defined in
Eq. (8.278d) and approximated by
Z
X
˜
ν̃(ξ)
2
˜
δ(ξ − ξk ). (8.306)
Σiωn ≈ |Vk | Fs
,
ν̃(ξ)
:=
dξ
iωn − ξ
k∈BZ
Hint: Use Eq. (6.59).
Exercise 4.9: It is time to make some assumptions about the
nature of the conduction band. We need to distinguish two cases. For
˜ = 0) is non-vanishing. We then make the
the standard case, ν̃˜F ≡ ν̃(ξ
simplifying assumption
ξ
ξ
˜
˜
ν̃(ξ) = ν̃F Θ
+1 −Θ
−1 ,
(8.307)
D
D
where Θ is the Heaviside function and the positive energy D is half
the bandwidth. This is a mild assumption with no qualitative, only
quantitative, consequences. The less common case, is that of the powerlaw dependence
ξ
Cr
ξ
r
˜
ν̃(ξ) = 1+r × |ξ| Θ
+1 −Θ
−1
(8.308)
D
D
D
with r > −1 a real number and Cr a positive dimensionless number
(for r = 0, C0 = D ν̃˜F ) was first explored by Withoff and Fradkin in
Ref. [103]. (On a square lattice with nearest-neighbor hopping and
at half-filling, the density of states is logarithmically divergent. This
would be another instance deviating from the standard Fermi liquid
scenario.) The choices r = 1 and r = 2 apply to graphene and bilayer
graphene, respectively. Verify that the approximation (8.306) has the
500
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
two analytical continuations
Z
PV
2
˜
Σω∓i0+ ≈ |Vk | Fs
dξ ν̃(ξ)
± iπ δ(ω − ξ)
ω−ξ
(8.309a)
Z+D r h|Vk |2 iFs
ξ
PV
dξ
± iπ δ(ω − ξ)
= Cr
D
D
ω−ξ
−D
where
PV
x
denotes the principal value of 1/x. Hence,

+D
R

(ξ/D)r


dξ
,
if |ω| > D,

ω−ξ


−D
h|Vk |2 iFs 
Re Σω∓i0+ ≈ Cr
!

D
+

ω−0
+D
R
R

r


+
dξ (ξ/D)
, otherwise,

ω−ξ

+
−D
ω+0
(8.309b)
while
ω
i
h|Vk |2 iFs ω r h ω
Θ
+1 −Θ
−1 .
D
D
D
D
(8.309c)
as it will be needed when solving the saddle-point
Im Σω∓i0+ ≈ ±π Cr
Compute Re Σω∓i0+
Eqs. (8.340).
Exercise 4.10: We are going to solve the single-impurity Anderson
model in the limit Nc → ∞ when the density of states per unit energy
of the conduction electrons is non-vanishing at the Fermi energy and
smooth across the band.
(a) Assumption (8.307) allows to define [recall Eq. (8.284c)]
(8.310)
∆ := π ν̃˜F |Vk |2 .
Fs
Show that Eq. (8.305) can be written
Z+D
F (−iλ, ρ) =
dω ˜
f (ω) arctan
π FD
−D
19
∆ ρ2
ω − ξf − λ
!
+λ
1
ρ −
Nc
2
.
(8.311)
Hint: Explain why you may deform the integration path Γ
according to the rule
Z+∞
Z−∞
I
dz →
d(ω + i0+ ) +
d(ω − i0+ ).
(8.312)
Γ
−∞
+∞
You may then use
log w − log w∗ = ln |w| + iarg w − ln |w∗ | − iarg w∗ = 2iarg w (8.313)
19
If non-vanishing, the real part of the self-energy renormalizes ξf . We use the
same symbol ξf for the renormalized value.
8.10. PROBLEMS
501
for any complex valued w, and in particular for the case when
w = ω − ξf − λ + i∆ ρ2 .
At zero temperature, the Fermi-Dirac distribution becomes the
Heaviside function Θ(−ω). The integral over the bandwidth on the
right-hand side of Eq. (8.311) can be done in closed form with the help
of
Z
a
b−x
a
a
2
2
dx arctan
= ln a +(b−x) −b arctan
+x arctan
x−b
2
a
x−b
(8.314)
and
π
arctan(x) + arctan(1/x) = .
(8.315)
2
(b) Verify that, to leading order in ξf /D, λ/D, or ∆ ρ2 /D,
∆ ρ 2 ∆ ρ 2 λ2 + ∆ 2 ρ 4
∆ ρ2 λ
+ arctan
+
ln
F − i(λ − ξf ), ρ = −
π
π
λ
2π
D2
1
+ (λ − ξf ) ρ2 −
.
Nc
(8.316)
The large Nc limit of the Coqblin-Schrieffer model that we
defined by the replacement (8.302) is, to leading order in ξf /D,
λ/D, or ∆ ρ2 /D,
∆ ρ 2 ∆ ρ 2 λ2 + ∆ 2 ρ 4
∆ ρ2 λ
+ arctan
+
ln
F − i(λ − ξf ), ρ = −
π
π
λ
2π
D2
λ
∆ ρ2
−
+
.
π ν̃˜F J0 Nc
(8.317)
Observe that the free energy of the single-impurity Anderson model
in the large Nc limit depends on three microscopic energy scales within
the approximation that we made. There is the energy scale ξf to occupy
the impurity site with one electron. There is the hybridization energy
scale ∆, the product of the squared coupling between the conduction
and impurity electrons with the density of states per unit energy of the
conduction electrons averaged over the Fermi surface of the conduction
electrons. There is the band width 2 D of the conduction electrons. The
argument of the logarithm on the right-hand side of both Eqs. (8.316)
and (8.317) defines the new energy scale
q
kB TK (λ, ρ) := (ξf + λ)2 + ∆2 ρ4 ,
(8.318)
the Kondo energy scale in the limit Nc → ∞. The Kondo energy scale
is the natural unit in which the bandwidth of the conduction electrons
is to be measured on a logarithmic scale.
502
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
(c) Show that the saddle-point equations
dF
(−iλ, ρ) = 0
dρ
and
dF
dλ
(−iλ, ρ) = 0
have the solutions
q
2
λ̄ + ξf + ∆2 ρ̄4 = D e−π λ̄/∆
and
(8.319a)
1
∆ ρ̄2
1
arctan
=
− ρ̄2 ,
π
Nc
λ̄ + ξf
(8.319b)
(8.320a)
(8.320b)
respectively. The term 1/Nc on the right-hand side of Eq. (8.320b)
is the only term present in the large Nc saddle-point solutions
q
2
˜
λ̄ + ξf + ∆2 ρ̄4 = D e−1/(ν̃F J0 )
(8.321a)
and
1
1
∆ ρ̄2
=
arctan
π
Nc
λ̄ + ξf
(8.321b)
to the Coqblin-Schrieffer model. Here, it can be safely dropped
in the limit Nc → ∞ of Eq. (8.320). The solution
ρ̄2 = 0
(8.322)
to the saddle-point equation (8.320b) in the limit Nc → ∞
decouples the conduction electrons from the impurity. Show
that a non-vanishing solution
1
0 < ρ̄2 ≤
(8.323a)
2
to Eq. (8.320b) is then possible if and only if
∆
< −1
π(λ̄ + ξf )
(8.323b)
in the limit Nc → ∞.
Equations (8.320) and (8.321) supply the dependence of the Kondo
energy scale (8.318) on the microscopic parameters of the single-impurity
Anderson model and the Coqblin-Schrieffer model in the limit Nc → ∞,
respectively. The limit Nc → ∞ of the Coqblin-Schrieffer model is simpler than that of the single-impurity Anderson model in the following
sense. Both Eqs. (8.316) and (8.317) depend on the bandwidth of the
conduction electrons through the term
k T (λ, ρ)
∆ ρ2
k T (λ, ρ) ∆ ρ2
D0
∆ ρ2
ln B K
=
ln B K 0
+
ln ,
π
D
π
D
π
D
(8.324)
8.10. PROBLEMS
503
where 2 D0 > 0 is the new band width. If the free energies (8.316) and
(8.317) are independent on the choice of the band width, i.e., invariant
under the infinitesimal transformation D → D0 , there must follow that
the microscopic parameters of the theory obey RG equations, as we are
going to verify for the Coqblin-Schrieffer model.
(d) Show that the infinitesimal transformation D → D0 on the
free energy (8.317) can be absorbed by an infinitesimal change
of the microscopic coupling J0 . Derive the first-order differential equation obeyed by J0 that guarantees form invariance
of the free energy (8.317) under the infinitesimal transformation D → D0 . If µ := ln D0 /D, show that J0 (µ) flows to 0 as
µ → ∞. This is an example of asymptotic freedom by which
the running coupling constant vanishes at high energies. Show
that J0 (µ) flows to ∞ as D0 → kB TK . Explain why asymptotic
freedom implies that the impurity spin susceptibility at high
temperatures obeys Curie’s law, if we assume that the analysis at Nc → ∞ remains qualitatively correct at Nc = 2. Explain why asymptotic freedom implies that the impurity spin
susceptibility at temperatures below the Kondo temperature
obey the Pauli law, if we assume that the analysis at Nc → ∞
remains qualitatively correct at Nc = 2. The specific heat
coefficient [recall Eq. (5.67)], the zero-temperature spin susceptibility [recall Eq. (5.100b)], and the charge susceptibility
at the impurity site and at zero temperature were computed
for the Nc → ∞ limit of the Coqblin-Schrieffer model by Read
and Newns in Ref. [69]. They behave as would be expected in
a Fermi liquid.
(e) Verify that if we assume a non-vanishing solution to Eq. (8.320b)
2
in the limit Nc → ∞ with λ̄ + ξf ∆2 ρ̄4 then Eq. (8.320a)
gives
kB TK (λ̄, ρ̄) ≈ D e−π|ξf |/∆ .
(8.325)
The parametric regime for which Eq. (8.325) holds is the Kondo
limit of the single-impurity Anderson model. In this Kondo
limit, we may borrow the results from exercise 4.10(d) on the
Nc → ∞ limit of the Coqblin-Schrieffer model provided we do
the identification
π |ξf |
1
→
.
(8.326)
˜
∆
ν̃F J0
(f) Verify that, provided ρ̄ 6= 0,
Gf (iωn ) :=
ρ̄2
iωn − ξf − λ̄ + iρ̄2 ∆ sgn ωn
(8.327)
is the effective Green function at the impurity site for both
the single-impurity Anderson model and the Coqblin-Schrieffer
504
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
model in the limit Nc → ∞. Verify that, upon analytical
continuation iωn → ωn ± i0+ , the imaginary part of the singleparticle Green function displays a peak at the energy ω =
ξf + λ̄ with a width ρ̄2 ∆ and weight ρ̄2 . Verify that ρ̄2 <
1/2 for the single-impurity Anderson model is a consequence
of the saddle-point equation. Is there any constraint on ρ̄2
for the Coqblin-Schrieffer model arising from the saddle-point
equation or on physical grounds? Explain why the weight
1 − ρ̄2 of the single-particle Green function (8.327) should be
distributed on physical grounds over two broad peaks at ξf
and ξf + U for the single-impurity Anderson model and only
one broad peak at ξf for the Coqblin-Schrieffer model if we
assume that we may apply the results of the limit Nc → ∞ to
the case of Nc = 2 and a large but finite U .
Exercise 4.11: We have argued that any model that couples one
Bloch band of non-interacting conduction electrons to a single pointlike impurity with internal degrees of freedom is characterized by no
less and no more than two fixed points, provided the following two
conditions hold. First, the density of states per unit energy of the
unperturbed conduction electrons is constant. Second, the coupling is
effectively that of an antiferromagnetic exchange interaction with an
effective spin-1/2 in the Kondo regime.
There is a fixed point at high energy (high temperature) at which
the conduction electrons decouple from the impurity. Perturbation
theory in the coupling to the impurity is valid in the vicinity of this fixed
point. There is a fixed point at low energy (low temperature) at which
the conduction electrons are strongly coupled locally to the impurity
so as to screen it. The Kondo energy scale (temperature) signals the
crossover between the two fixed points. Although the impurity breaks
translation symmetry, the strong-coupling fixed point is nevertheless
thought of as a local Fermi liquid. This interpretation is justified by
the fact that the two-point Green function at the impurity site has the
analytical structure expected from a Fermi liquid.
A scenario with a flow between a decoupled fixed point and a strong
coupling fixed point without any intervening quantum critical point is
not always true. The zero-temperature phase diagram becomes much
richer if we relax two assumptions that we made so far. The multichannel Kondo effect allows for a mismatch between twice the number of
conducting bands and the number of internal degrees of freedom at the
impurity site. [104] The condition of a non-vanishing density of states
per unit energy at the Fermi energy can also be relaxed. [103]
We shall now study the situation of a density of states per unit
energy that vanishes in a power-law fashion at the Fermi energy that
was first investigated by Withoff and Fradkin in Ref. [103].
8.10. PROBLEMS
505
Equations (8.275) and (8.271b) play a central role in what follows.
We consider the Coqblin-Schrieffer model on a lattice made of N lattice
˜
sites. The density of states per unit energy ν̃(ξ)
for the conduction
electrons is given by Eq. (8.308). Hence, it is supported on the interval
[−D, +D] and singular at the Fermi energy ξ = 0 if r 6= 0. Moreover,
it is extensive in the number N of lattice sites with the convention we
made for the normalization of the Hubbard-Stratonovich field of the
Coqblin-Schrieffer model. The coupling of the conduction electrons to
the f electrons on the single impurity site subject to the constraint it
be occupied by exactly one of them is denoted by J(D). We opt for
the convention that J(D) carries units of energy and scales like 1/N
with the number of lattice sites. This choice corresponds to having
Hamiltonian (8.271b) scales like N . [Recall the discussion that followed
the definition of Hamiltonian (8.248d).] We also choose the convention
for which J(D) > 0 corresponds to an antiferromagnetic Heisenberg
exchange coupling. This convention for the sign of J(D) is opposite to
the convention leading to Eq. (8.271b). Correspondingly,
˜
g(D) := ν̃(D)
J(D)
(8.328)
is a dimensionless coupling constant that is intensive with respect to
its scaling relative to the number of lattice sites N .
Instead of doing perturbation theory as Schrieffer and Wolff did to
reach Eq. (8.275), we imagine that we integrate out all electrons in a
thin shell below the bandwidth D under the assumption that J(D) is
small. The resulting bandwidth is D0 where D − D0 > 0 is very small,
i.e.,
D − D0
D0
= d` ⇐⇒
= e−d`
(8.329)
D
D
with d` infinitesimal. We still use Eq. (8.271b) with the assumption
that all the changes resulting from the integration over the electrons
with energies between D0 and D are a mere small additive shift that
can be absorbed covariantly by changing J(D) to J 0 (D0 ). In other
words, the action with the smaller bandwidth D0 and the dimensionful
coupling constant J 0 (D0 ) takes the same form as the action with the
original bandwidth D and the dimensionful coupling J(D). Accordingly,
˜
J 0 (D0 ) := J(D) + ν̃(D)
J 2 (D) d`.
(8.330)
We seek to express the left-hand side in terms of the original bandwidth
D. To this end, we express Eq. (8.330) in terms of the dimensionless
couplings g 0 (D0 ) and g(D) and the density of states per unit energy
˜ 0 ) and ν̃(D),
˜
ν̃(D
respectively,
g 0 (D0 )
g(D) g 2 (D)
=
+
d`.
˜ 0)
˜
˜
ν̃(D
ν̃(D)
ν̃(D)
(8.331)
506
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
If we assume that the homogeneity relation
0 +r
D
0
˜ ) = ν̃(D)
˜
˜
ν̃(D
= ν̃(D)
e−r d`
D
(8.332)
holds for D0 /D ≤ 1 and if we do the expansion
g 0 (D0 ) e+r d` = g(D) + dg(D) + r g(D) d`
(8.333)
on the left-hand side of Eq. (8.331), there follows the one-loop beta
function
dg
(`) = −r g(`) + g 2 (`).
(8.334)
d`
(a) Draw the mean-field phase diagram implied by Eq. (8.334).
Convince yourself of the following. For negative −1 < r < 0
and positive initial g, the flow is to strong coupling away from
the fixed point g = 0. For r = 0, the linear term on the righthand side drops out, while the positive sign of the quadratic
term indicates that a positive g is marginally relevant, it also
flows to strong coupling, though much more slowly than when
−1 < r < 0. The case of r = 0 is of course that of a constant
density of states per unit energy. For r > 0, the flow is to
the stable fixed point g = 0 as long as the initial condition is
compatible with r g(`) > g 2 (`). The one-loop beta function
vanishes if the condition g(`) = r is met. This critical point
is unstable, for the flow is to strong coupling as soon as the
initial condition satisfies the condition 0 < r g(`) < g 2 (`).
The crude scaling analysis that we made relies on the homogeneity
relation (8.332).
˜
(b) Verify that the density of states per unit energy ν̃(ξ)
given by
Eq. (8.308) is homogeneous of degree r, i.e.,
ν(κ ξ) = κr ν(ξ)
(8.335)
when 0 < |κ| ≤ 1.
(c) Show that the number of lattice sites
Z+D
˜
dξ ν̃(ξ)
N (D) :=
(8.336)
−D
is homogeneous of order r + 1, i.e.,
N (κ D) = κr+1 N (D)
(8.337)
when 0 < κ ≤ 1.
Exercise 4.12: Equation (8.334) is the poor man’s one-loop beta
function for any impurity problem in the Kondo regime. This is a perturbative calculation. The new feature brought about by a density of
states that vanishes in a power-law fashion at the Fermi energy is the
8.10. PROBLEMS
507
possibility of a new fixed point. It is desirable to confirm this perturbative RG result in a non-perturbative fashion and to decide if this
critical point represents a non-Fermi liquid critical point. We are going
to establish the existence of an unstable and non-Fermi-liquid critical
point that intervenes between the stable fixed points at vanishing and
infinite coupling in the limit Nc → ∞ of the Coqblin-Schrieffer model
for the case 0 < r < 1/2. We refer the reader to Ref. [103] for a more
detailed analysis.
We use the approximation (8.309) to the self-energy of the f electrons acquired after integrating out the conduction electrons. Instead
of the single energy scale (8.310) that applies when r = 0, we need to
introduce the two energy scales,
h|Vk |2 iFs
1 0
∆r := π
Cr
(8.338)
π
D
for the imaginary and real parts of the self-energy, respectively.
∆00r :=
(a) Show that the free energy at zero temperature for the CoqblinSchrieffer model is given by


Z+D


ω r
∆00r ρ2 D
dω ˜


Fr (−iλ, ρ) =
fFD (ω) arctan 

+D
R
π


ε r PV
0
2
ω − ξf − λ − ∆r ρ
dε D ω−ε
−D
−D
+
∆00r
2
ξf + λ
ρ
−
π Cr J0 /D
Nc
(8.339)
in the limit Nc → ∞ and to leading order in ξf /D, λ/D,
∆0 ρ2 /D, or ∆00 ρ2 /D. Hint: Modify the derivation of Eq. (8.311)
as demanded.
(b) Write the explicit form of the saddle-point equations
dFr
(−iλ, ρ) = 0
(8.340a)
dρ
and
dFr
dλ
(−iλ, ρ) = 0
(8.340b)
at zero temperature, i.e., when f˜FD (ω) = Θ(−ω).
(c) Verify that ρ̄ = 0 is a solution to the saddle-point equation (8.340a).
(d) Show that the saddle-point equation (8.340a) also admits a
solution to
1 dFr
(−iλ, ρ) = 0
(8.341)
ρ
dρ
508
8. A SINGLE DISSIPATIVE JOSEPHSON JUNCTION
for ρ̄ = λ̄ = ξf = 0 that defines a critical value
(c)
J0 (r) ∝ r
(8.342)
for any r > 0. This critical value is not predicated on being
small, as consistency demands when using poor man’s scaling.
It thus applies to graphene (r = 1) or to some stackings of n
consecutive layers of graphene (r = n).
(e) Argue why this solution can be interpreted as an unstable critical point between two non-Fermi-liquid phases of the CoqblinSchrieffer model for any r > 0. Hint: Assume (or better
(c)
(c)
show) that J0 (r) is such that any J0 > J0 (r) admits a solution to the saddle-point equations with ρ̄ > 0 while any
(c)
0 ≤ J0 < J0 (r) admits only the trivial solution ρ̄ = 0.
Why should we trust the large Nc expansion? After all, the saddlepoint approximation (8.285) predicts a spurious phase transition. Kondolike problems can be thought of as realizations of boundary critical
phenomena in (1 + 1)-dimensional conformal field theories. [105] For
example, observables at the impurity site in the Kondo problem (with
r = 0) such as the spin susceptibility, the decay of the two-point Green
function in time, etc., obey scaling laws that can be computed approximately and then compared to the power laws of boundary critical
phenomena in (1 + 1)-dimensional conformal field theories. In particular, the critical exponents that can be extracted from a large Nc
expansion of the Kondo-like problem (with r = 0) can be compared to
the exact boundary critical exponents in (1 + 1)-dimensional conformal
field theories. Their agreement is a measure of the quality of the large
Nc expansion. Cox and Ruckenstein in Ref. [106] showed that the extension of the large Nc expansion to the multichannel Kondo problem
with a non-singular density of states at the Fermi level already reproduces to leading order the corresponding boundary critical exponents
in (1 + 1)-dimensional conformal field theories and that these exponents are not modified to each order in the 1/Nc expansion. We refer
the reader to the review by Vojta in Ref. [107] of the developments,
mostly numerical, beyond the large Nc limit of Withoff and Fradkin.
CHAPTER 9
Abelian bosonization in two-dimensional space
and time
Outline
It is shown that the two-dimensional massive Thirring model is
related to the two-dimensional Sine-Gordon model through Abelian
bosonization. The bosonized solution to the quantum-xxz spin-1/2
chain and to the single-impurity problem in a Luttinger liquid are given.
9.1. Introduction
Quantum field theory in (1 + 1)–dimensional (position) space and
time is more tractable than in higher-dimensional space and time.
It is also of relevance to both classical statistical mechanics in twodimensional (position) space and to condensed matter physics in onedimensional (position) space. The trademark of quantum field theory
in (1 + 1)–dimensional space and time is the “equivalence” between
Bose and Fermi fields. [108, 109, 110, 111, 112] This equivalence is
dubbed Abelian bosonization and has been extremely fruitful in applications to condensed matter physics. By way of a comparison between
the two-dimensional Sine-Gordon model and the two-dimensional massive Thirring model, we shall illustrate the physical concepts at the
root of Abelian bosonization. This will also allow us to connect a
model of classical statistical mechanics such as the two-dimensional
XY model to the quantum physics of interacting electrons constrained
to one-dimensional (position) space.
The two-dimensional Sine-Gordon model is defined by the partition
function
Z
ZSG;t,h := D[φ] e−SSG;t,h ,
(9.1a)
with the action
Z
SSG;t,h :=
d2 x LSG;t,h ,
(9.1b)
and the Lagrangian density
1
h
(∂µ φ)2 − cos φ,
(9.1c)
2t
t
for the real-valued scalar field φ. As usual, we are working in twodimensional Euclidean space, i.e., time is imaginary.
LSG;t,h :=
509
510 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
The two-dimensional massive Thirring model describes a massive
and interacting quantum field theory for a spinor in (1+1)–dimensional
Euclidean space and time (i.e., time is imaginary). It is defined by the
partition function
Z
ZTh;m,g := D[ψ̄, ψ] e−STh;m,g ,
(9.2a)
with the action
Z
STh;m,g :=
d2 x LTh;m,g ,
(9.2b)
and the Lagrangian density
2
g
LTh;m,g := ψ̄ iσµ ∂µ − m ψ −
ψ̄ σµ ψ .
(9.2c)
2
The spinor field ψ̄ and ψ are Grassmann valued. They each transform according to the spin-1/2 representation of the two-dimensional
Euclidean Poincaré group, i.e., they have two components on which
the two-vector of Pauli matrices σµ = (σ1 , σ2 ) act. We are using the
summation convention over repeated indices, i.e., σµ ∂µ ≡ σ1 ∂1 + σ2 ∂2 .
The dimensionless coupling constant g measures the strength of the
current-current interaction (jµ )2 ≡ j12 + j22 with the conserved current
jµ := ψ̄ σµ ψ,
µ = 1, 2,
(9.3)
that results from the invariance of the partition function under the
global U (1) gauge transformation
ψ̄ = ψ̄ 0 e+iα ,
ψ = e−iα ψ 0 ,
α ∈ R.
(9.4)
In the massless limit m = 0, LTh;m=0,g is also invariant under the global
U (1) axial gauge transformation
ψ̄ = ψ̄ 0 e−iα5 γ5 ,
ψ = e−iα5 γ5 ψ 0 ,
γ5 ≡ −iσ1 σ2 ,
α5 ∈ R.
(9.5)
For any non-vanishing g, the symmetry of LTh;m=0,g under the transformation (9.5) is broken by the measure of the partition function
ZTh;m=0,g as is shown in section 9.2.2. The mass term m ψ̄ ψ is symmetric under the transformation (9.4), but it breaks explicitly the
symmetry of LTh;m=0,g under the transformation (9.5), since ψ̄ ψ =
ψ̄ 0 e−2iα5 γ5 ψ 0 .
Abelian-bosonization rules encode the fact that the two-dimensional
Sine-Gordon and two-dimensional Thirring models are equivalent in the
sense that some local fields share the same correlation functions in both
theories. This correspondence is summarized in table 1.
9.2. Abelian bosonization of the Thirring model
9.2.1. Free-field fixed point in the massive Thirring model.
Before undertaking a justification of the Abelian bosonization rules, it
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
511
Table 1. Abelian bosonization rules in 2d Euclidean space.
SG model
1
(∂µ φ)2
8π
1
∂ φ
2π µν ν
TH model
ψ̄iσµ ∂µ ψ
ψ̄iσµ ψ
−im
π
h
t
is useful to gain some familiarity with the two-dimensional Thirring
model by studying the massless free-field fixed point m = g = 0,
Z
ZTh;0,0 := D[ψ̄, ψ] e−STh;0,0 ,
Z
(9.6)
STh;0;0 := d2 x LTh;0,0 ,
LTh;0,0 := ψ̄ iσµ ∂µ ψ.
At the massless free-field fixed point, engineering and scaling dimensions of ψ̄ and ψ are equal and given by 1/2 ([ψ̄] = [ψ] = length−1/2 ).
Scale invariance of STh;0,0 under simultaneous rescaling of the coordinates and fields imply 1
hψ(x)ψ̄(y)iZTh;0,0 ∼
1
,
|x − y|
(9.7)
if we are to neglect the spinor structure all together. To account for
the spinor structure, observe that (σ0 is the unit 2×2 matrix in spinor
space)
0
∂1 − i∂2
σ µ ∂µ =
,
∂1 + i∂2
0
2
(9.8)
σµ ∂µ = σ0 ∂12 + ∂22 ≡ σ0 ∆,
2
1
x
∆−1 (x) = + ln
,
4π
a2
where a is the short distance cutoff, i.e., the lattice spacing. Hence,
−1
hψ(x)ψ̄(0)iZTh;0,0 = iσµ ∂µ
(x)
= −iσµ ∂µ σ0 ∆−1 (x)
2 1
x
= −iσµ ∂µ
ln
.
4π
a2
(9.9)
1
Z
We are using the convention
Z
∗
dψ ∗ dψ e−ψ ψ = dψ ∗ dψ (−)ψ ∗ ψ = 1,
for the Grassmann integration.
Z
dψ ∗ dψ ψ ψ ∗ e−ψ
∗
ψ
Z
=
dψ ∗ dψ ψ ψ ∗ 1 = 1,
512 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
It is convenient at this stage to rotate the Cartesian coordinates of
two-dimensional Euclidean space by π/4. If
x± := x1 ± ix2 ,
then
∂∓ ≡ ∂1 ∓ i∂2 = 2
(9.10a)
∂
∂x±
(9.10b)
and
hψ(x)ψ̄(0)iZTh;0,0
1
i
0
x1 − ix2
=
−
0
2π x2 x1 + ix2
!
1
0
i
x1 +ix2
=
−
.
(9.10c)
1
0
2π
x1 −ix2
The representation in terms of the Pauli matrices chosen here defines the so-called Dirac representation of the Thirring model. In the
Dirac representation, the Euclidean Feynman propagator (9.10c) is offdiagonal. From now on, we choose units in which a = 1.
Whereas the Feynman propagator (9.10c) is not diagonal, the propagator hψ(x) ψ ∗ (y)iZTh;0,0 , where
ψ ∗ := ψ̄ σ1 ⇐⇒ ψ̄ =: ψ ∗ σ1 ,
(9.11)
is diagonal. To see this, introduce first the third Pauli matrix
γ5 ≡ −iσ1 σ2 = σ3 .
(9.12)
Observing that P± := (1/2)(σ0 ± γ5 ) form a complete set of projectors
onto the spinor subspace, the chiral representation,
ψ−
∗
∗
∗
ψ := ψ− ψ+ ,
ψ :=
,
(9.13a)
ψ+
is defined by
1
(σ ∓ γ5 ) ,
2 0
With this definition,
∗
ψ±
:= ψ ∗
ψ± :=
1
(σ ∓ γ5 ) ψ.
2 0
(9.13b)
ψ̄ iσµ ∂µ ψ = ψ ∗ σ1 iσµ ∂µ ψ
= ψ ∗ i (σ0 ∂1 + iγ5 ∂2 ) ψ
∗
∗
= ψ−
i (∂1 + i∂2 ) ψ− + ψ+
i (∂1 − i∂2 ) ψ+ ,
(9.14)
and
∗
hψ(x) ψ (0)iZTh;0,0
1 x− 0
i
=
−
2π x2 0 x+
!
1
0
i
x+
=
−
.
0 x1
2π
(9.15)
−
The chiral basis displays explicitly the fact that LTh;0,0 has more than
the global U (1) gauge invariance of Eq. (9.4). Indeed, Eq. (9.14) is
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
513
invariant under the local U (1) × U (1) gauge transformations defined
by 2
∗ 0 +iα(x+ )
∗
e
,
=: ψ−
ψ−
ψ− =: e−iα(x+ ) ψ− 0 ,
∗
∗ 0 +iβ(x− )
ψ+
=: ψ+
e
,
ψ+ =: e−iβ(x− ) ψ+ 0 .
(9.16)
Here, α(x+ ) is any holomorphic function of x+ = x1 + ix2 , whereas
β(x− ) is any antiholomorphic function of x− = x1 − ix2 . 3 The corresponding conserved (Noether) currents
∗
j− := j1 − ij2 = 2 ψ−
ψ− ,
∗
j+ = j1 + ij2 = 2 ψ+
ψ+ ,
(9.19a)
obey
0 = ∂+ j− = ∂− j+ .
(9.19b)
The U (1) × U (1) local gauge invariance (9.16) has dramatic consequences on correlation functions. For example, consider the two bilinears
∗
Ψ+− (x) := (ψ+
ψ− )(x),
∗
Ψ−+ (x) := (ψ−
ψ+ )(x).
(9.20)
These bilinears appear in the “standard”
ψ ∗ σ1 ψ = + Ψ−+ + Ψ+− ,
(9.21)
ψ̄ γ5 ψ = −iψ ∗ σ2 ψ = − Ψ−+ − Ψ+− ,
(9.22)
ψ̄
ψ=
and “axial” mass terms
respectively. Now, the 4n-point correlation function
* n
+
Y
Ψ−+ (xj ) Ψ+− (yj )
j=1
(9.23)
ZTh;0,0
2
This local gauge invariance is a manifestation of conformal invariance in two
dimensions.
3 Equation (9.14) is rewritten
∗
∗
ψ̄ iσµ ∂µ ψ = ψ−
i(2∂z̄ ) ψ− + ψ+
i(2∂z ) ψ+ ,
(9.17)
whereby we have introduced the notation
x+ := x1 + ix2 ≡ z,
x− := x1 − ix2 ≡ z̄,
1
(x + x− ),
2 +
i
x2 = − (x+ − x− ),
2
x1 =
∂+ := ∂1 + i∂2 ≡ (2∂z̄ ),
(9.18)
∂− := ∂1 − i∂2 ≡ (2∂z ).
514 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
factorizes into the product of holomorphic and antiholomorphic functions. For example,
Ψ−+ (x) Ψ+− (y) Z
=
Th;0,0
=
=
=
=
∗
∗
ψ− (x) ψ+ (x) ψ+
(y) ψ− (y) Z
Th;0,0
∗
∗
− ψ− (y) ψ− (x) ψ+ (x) ψ+ (y) Z
Th;0,0
∗
∗
− ψ− (y) ψ−
(x) Z
× ψ+ (x) ψ+
(y) Z
Th;0,0
Th;0,0
2
i
1
1
− −
×
2π
y+ − x+ x− − y−
2
i
1
1
+ −
×
.
(9.24)
2π
x+ − y + x− − y −
In general,
*
n
Y
+
*
= (−1)n
Ψ−+ (xj ) Ψ+− (yj )
j=1
n
Y
+
∗
ψ− (yj ) ψ−
(xj )
j=1
ZTh;0,0
*
×
n
Y
ZTh;0,0
+
∗
ψ+ (xk ) ψ+
(yk )
k=1
,
ZTh;0,0
(9.25a)
where (Sn is the permutation group of n elements)
*
n
Y
j=1
+
∗
ψ− (yj ) ψ−
(xj )
=
ZTh;0,0
X
σ∈Sn
sgn(σ)
n
Y
∗
ψ− (yσj ) ψ−
(xj )
j=1
ZTh;0,0
n X
n
Y
1
i
sgn(σ)
= −
2π
(yσj )+ − (xj )+
j=1
σ∈Sn
n X
n
Y
i
1
n
sgn(σ)
= (−1) −
2π
(xj )+ − (yσj )+
j=1
σ∈Sn
n
i
1
n
= (−1) −
det
2π
(xj )+ − (yk )+ j,k=1,··· ,n
n
i
= (−1)n −
(−1)n(n−1)/2
2π
Q (xj )+ − (xk )+ (yj )+ − (yk )+
1≤j<k≤n
×
n Q
(xj )+ − (yk )+
j,k=1
(9.25b)
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
515
and
*
n
Y
+
∗
ψ+ (xk ) ψ+
(yk )
k=1
n
i
−
(−1)n(n−1)/2
2π
Q (xj )− − (xk )− (yj )− − (yk )−
=
ZTh;0,0
×
1≤j<k≤n
n Q
.
(xj )− − (yk )−
j,k=1
(9.25c)
Thus,
* n
Y
Ψ−+ (xi ) Ψ+− (yi )
i=1
xj − x 2 y j − y 2
k
k
Q
+
=
ZTh;0,0
−
i
2π
2n
1≤j<k≤n
.
n Q
x − y 2
j
k
j,k=1
(9.26)
Up to an overall multiplicative factor, the same correlation function is
obtained in the two-dimensional Sine-Gordon model with h = 0 and
t = 4π provided one identifies
Ψ−+ →
1 +iφ
e ,
2πi
Ψ+− →
1 −iφ
e .
2πi
(9.27)
This result is consistent with the Abelian bosonization rule h/t ←→
−im/π.
Another consistency check of the Abelian bosonization rules amounts
to comparing the generating functional
Z
2
ZTh;0,0 [Jµ ] := D[ψ̄, ψ] exp − d x ψ̄ σµ i∂µ + Jµ ψ
= Det σµ i∂µ + Jµ
Z
(9.28)
for the vector current-current correlation function in the two-dimensional
Thirring model with the generating functional (∂˜µ ≡ µν ∂ν implies
516 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
∂˜µ ∂˜ν = δµν ∆ − ∂µ ∂ν )
Z
Dφ e−
R
1
d2 x[ 2t
(∂µ φ)2 +β(∂˜µ φ)Jµ ]
Z
Dφ e−
R
1
φ(−∆)φ−βφ(∂˜µ Jµ )]
d2 x[ 2t
ZSG;0,0 [Jµ ] :=
=
β2 t
2
R
d2 x
R
d2 y (∂˜µ Jµ )(x)(−∆)−1 (x−y)(∂˜ν Jν )(y)
= [Det (−)∆/t]−1/2 e+
β2 t
2
R
d2 x
R
d2 y(∆ξ)(x)(−∆)−1 (x−y)(∆ξ)(y)
= [Det (−)∆/t]−1/2 e+
β2 t
2
R
d2 x(−∆ξ)ξ
= [Det (−)∆/t]−1/2 e+
β2 t
2
R
d2 x(∂µ ξ)2
R
d2 x
= [Det (−)∆/t]−1/2 e+
2
β t
−1/2 + 2
≡ [Det (−)∆/t]
e
R
h
i
∂ ∂
d2 yJµ (x) δµν δ(x−y)− µ∆ ν Jν (y)
(9.29)
.
Here, we have assumed that the source Jµ is a smooth vector field, i.e.,
that the decomposition
Jµ = ∂µ χ + ∂˜µ ξ
(9.30)
in terms of the pure-gauge ∂µ χ and the transverse component ∂˜µ ξ is
valid everywhere in space. Moreover, we assume that we can safely
drop all boundary terms when performing partial integrations.
The fermionic determinant on the second line of Eq. (9.28) for an
arbitrary source Jµ needs to be computed. This is done below using
the method of Fujikawa with the result
R 2
Det σµ i∂µ + Jµ
1
2
= e− 2π d x (∂µ ξ)
Det σµ i∂µ
(9.31)
≡e
1
− 2π
R
d2 x
R
h
i
∂ ∂
d2 y Jµ (x) δµν δ(x−y)− µ∆ ν Jν (y)
.
The right-hand side on the second line is defined by the right-hand side
on the first line. Hence, we must have
1
β 2t = − ,
π
(9.32)
if the two generating functions are to produce the same correlation
functions. With the choice m = 0 and t → 4π in the two-dimensional
Sine-Gordon model, we find that
β=−
i
2π
(9.33)
agrees with the Abelian bosonization rule ψ̄ iσµ ψ ←→ (1/2π)∂˜µ φ.
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
517
9.2.2. The U (1) axial-gauge anomaly. We want to compute
the fermionic determinant
Z
Z
2
Det σµ i∂µ + Jµ = D[ψ̄, ψ] exp − d x ψ̄ σµ i∂µ + Jµ ψ .
(9.34)
By assumption (Jµ is taken sufficiently smooth), we can always decompose the source Jµ into rotation and divergence free contributions,
Jµ = ∂µ χ + ∂˜µ ξ.
(9.35)
We can then perform the change of Grassmann variables
ψ̄ =: ψ̄ 0 e−iχ+γ5 ξ ,
ψ =: e+iχ+γ5 ξ ψ 0 ,
(9.36)
under which
L := ψ̄ σµ i∂µ + Jµ ψ
0 −iχ+γ5 ξ
˜
= ψ̄ e
σµ i∂µ + ∂µ χ + ∂µ ξ e+iχ+γ5 ξ ψ 0
= ψ̄ 0 iσµ ∂µ ψ 0 .
(9.37)
To reach the last equality, we made use of σµ γ5 ∂µ = −iµν σν ∂µ =
+iµν σµ ∂ν = +iσµ ∂˜µ . The fact that the transformation (9.36) decouples
the spinor from the source field is unique to two dimensions. On general
grounds, a change of integration variables costs a Jacobian. What
is the Jacobian Jf of the transformation (9.36)? Fujikawa was the
first to propose a method to calculate the Jacobian Jf associated to
Eq. (9.36). [113]
The Grassmann measure is defined in terms of the Grassmannvalued expansion coefficients ām and an of the fields ψ̄ and ψ, respectively, in the basis of the Dirac operator σµ (i∂µ + Jµ ). In other words,
!
D[ψ̄, ψ] =
Y
!
Y
dām
m
ψ̄(x) =
X
ψ(x) =
X
n
,
n
hm|xi ām ≡
m
dan
X
ϕ†m (x) ām ,
(9.38a)
m
an hx|ni ≡
X
an ϕn (x),
n
where ϕn is the complete set of orthonormal eigenspinors with eigenvalues λn of σµ (i∂µ + Jµ ),
σµ i∂µ + Jµ ϕn (x) = λn ϕn (x),
Z
X
†
ϕn (x) ϕn (y) = σ0 δ(x − y),
d2 x ϕ†m (x) ϕn (x) = δm,n .
n
(9.38b)
518 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
Similarly, after performing the change of integration variables (9.36),
!
!
Y
Y
da0n0 ,
D[ψ̄ 0 , ψ 0 ] =
dā0m0
n0
m0
ψ̄ 0 (x) =
X
ϕ†m0 (x) ā0m0
=
m0
ψ 0 (x) =
X
ϕ†m (x) ām e+iχ(x)−γ5 ξ(x) ,
(9.39)
m
X
a0n0 ϕn0 (x) =
X
n0
an e−iχ(x)−γ5 ξ(x) ϕn (x).
n
The relationship between the expansion coefficients (ām , an ) and (ā0m0 , a0n0 )
is linear,
Z
X
0
ām0 =
Ūm0 ,m ām ,
Ūm0 ,m = d2 x ϕ†m (x) e+iχ(x)−γ5 ξ(x) ϕm0 (x),
m
a0n0
=
X
Z
Un0 ,n an ,
Un0 ,n =
d2 x ϕ†n0 (x)e−iχ(x)−γ5 ξ(x) ϕn (x).
n
(9.40)
Under the linear transformation (9.40), the transformation law of the
measure is
Y
−1 Y
dā0m0 = Det Ū
dām ,
m0
Y
n0
m
da0n0
−1
= (Det U )
Y
(9.41)
dan .
n
Notice that it is not the determinant of the linear transformation that
appears on the right-hand side, as would be the case for Riemann
4
integrals, but the inverse determinant.
−1
We first evaluate Det Ū
and (Det U )−1 by assuming that Jµ is
infinitesimal,
Jµ = ∂µ (δχ) + ∂˜µ (δξ),
(9.42)
where the rotation-free contribution from δχ and the divergence-free
contribution from δξ are infinitesimal. We then integrate the result.
On the one hand,
−1
Z
−1
2
†
Det Ū
=
Det δm,m0 + d x ϕm (x) [+i(δχ)(x) − γ5 (δξ)(x)] ϕm0 (x)
Z
2
†
=
Det δm,m0 − d x ϕm (x) [+i(δχ)(x) − γ5 (δξ)(x)] ϕm0 (x)
o
n
R
Tr ln δm,m0 − d2 x ϕ†m (x)[+i(δχ)(x)−γ5 (δξ)(x)]ϕm0 (x)
= e
−
= e
4
PR
m
d2 x ϕ†m (x)[+i(δχ)(x)−γ5 (δξ)(x)]ϕm (x)
.
This is so because the Grassmann
is constructed
√ ∗ √such that
R integral
∗
dψ
√
√
dψ ∗ dψ exp(−ψ ∗ A ψ) = A, i.e., A dψ
exp
−
(
Aψ )( Aψ)
=
A
A
R ∗
∗
A dζ dζ exp(−ζ ζ) = A.
R
(9.43)
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
519
2
To reach the last equality, we made use of ln(1 − x) = −x − x2 − · · · .
On the other hand,
−1
Z
−1
†
2
(Det U )
=
Det δn0 ,n + d x ϕn0 (x) [−i(δχ)(x) − γ5 (δξ)(x)] ϕn (x)
Z
†
2
=
Det δn0 ,n − d x ϕn0 (x) [−i(δχ)(x) − γ5 (δξ)(x)] ϕn (x)
n
o
R
Tr ln δn0 ,n − d2 x ϕ†n0 (x)[−i(δχ)(x)−γ5 (δξ)(x)]ϕn (x)
= e
−
= e
PR
n
d2 x ϕ†n (x)[−i(δχ)(x)−γ5 (δξ)(x)]ϕn (x)
.
(9.44)
2
To reach the last equality, we again made use of ln(1 − x) = −x − x2 −
· · · . We thus find that the infinitesimal Jacobian
Q
Q
dām
dan
m
n
= Det Ū (Det U )
δJf := (9.45)
Q 0
Q 0
dām0
dan0
m0
n0
only depends on the infinitesimal generator (δξ) of the divergence-free
contribution to Jµ ,
δJf = e−
R
d2 x (δξ)(x) A5 (x)
A5 (x) := 2 ×
X
,
(9.46)
ϕ†n (x) γ5 ϕn (x).
n
The function A5 is called the “axial anomaly” if it is non-vanishing,
for it implies that quantum fluctuations encoded by the measure in the
path integral break the axial symmetry of the Lagrangian density.
To make sense of the “axial anomaly” A5 (x) when
Jµ → Jµ + (δJ)µ ,
(δJ)µ = ∂˜µ (δξ),
(9.47)
we need to regularize the summation on the right-hand side of Eq. (9.46),
for a given background Jµ . This is done by choosing the following
gauge-invariant regularization,
X
2
2
A5 (x) := 2 × lim
ϕ†n (x) γ5 e−(λn ) /M ϕn (x)
M →∞
= 2 × lim
M →∞
n
X
ϕ†n (x) γ5 e−[iσµ ∂µ +σµ Jµ (x)]
2
/M 2
(9.48)
ϕn (x).
n
Since it is not possible to construct the eigenfunctions ϕn (x) explicitly for an arbitrary source Jµ , we trade the summation over n by a
summation over the momenta of the plane-wave basis. This is done by
insertion of the resolution of the identity twice. More precisely, for any
operator O acting on the spinor subspace with the representation Oab
520 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
with a, b = 1, 2, we may write
Tr |xi O hx| ≡
2
XX
n
=
2 XX
XX
n
=
hn, a|xi Oab hx|n, bi
a,b=1
hn, a|kihk|xi Oab hx|k 0 ihk 0 |n, bi
k0
k
2
XX X
k
!
X
hk 0 |
k0 a,b=1
=
|n, bihn, a| |kihk|xiOab hx|k 0 i
n
|
Z
(9.49)
a,b=1
{z
}
=δa,b
d2 k
hk|xi (tr O) hx|ki.
(2π)2
(9.50)
where the label n refers to the energy eigenstate, the indices a, b = 1, 2
refer to the spinor indices, and tr refers to the trace over the spinor
indices (hx|ki is a mere C number). Thus, after having traded the basis
that diagonalizes the Dirac operator σµ (i∂µ + Jµ ) for the plane-wave
basis, the axial anomaly in the background Jµ becomes
Z
A5 (x) = 2 × lim tr
M →∞
2
d2 k
−ikx −[iσµ ∂µ +σµ Jµ (x)] /M 2 +ikx
γ
e
e
e (9.51)
.
(2π)2 5
If we introduce the notation
Dµ := i∂µ + Jµ ,
µ = 1, 2,
(9.52a)
for the covariant derivative and
Fµν (x) := ∂µ Jν (x) − ∂ν Jµ (x),
µ, ν = 1, 2,
(9.52b)
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
521
for the field strength of the background vector potential, needed are
then
2
X
σ µ Dµ
2
=
µ=1
2
X
σµ σµ Dµ Dµ +
µ=1
=
2
X
=
σ0 Dµ Dµ +
2
X
X
σµ σν Dµ Dν − Dν Dµ
µ<ν
σ0 Dµ Dµ +
1X
σµ σν − σν σµ Dµ Dν − Dν Dµ
2 µ<ν
σ0 Dµ Dµ +
2
1 X
[σ , σ ][D , D ]
4 µ,ν=1 µ ν µ ν
µ=1
=
σµ σν Dµ Dν
µ6=ν
µ=1
2
X
X
µ=1
2
X
2
i X
σ0 Dµ Dµ +
[σµ , σν ] ∂µ Jν − ∂ν Jµ
=
4 µ,ν=1
µ=1
2
X
2
i X
σ0 Dµ Dµ +
[σ , σ ]F ,
≡
4 µ,ν=1 µ ν µν
µ=1
(9.52c)
and (summation convention over repeated indices reinstated)
2 +ikx
i
+ikx
σµ Dµ e
=e
σ0 −kµ + Dµ −kµ + Dµ + [σµ , σν ] Fµν (x) .
4
(9.52d)
The axial anomaly in the background Jµ is now
2
d2 k
−ikx −(σµ Dµ ) /M 2 +ikx
γ
e
e
e
M →∞
(2π)2 5
Z
i
d2 k
2
2
γ5 e−(k −2kµ Dµ +Dµ Dµ + 4 [σµ ,σν ] Fµν (x))/M
= 2 × lim tr
2
M →∞
(2π)
Z
d2 p
−p2 +2pµ Dµ /M −Dµ Dµ /M 2 − 4i [σµ ,σν ] Fµν (x)/M 2
2
γ
e
= 2 × lim M tr
5
M →∞
(2π)2
Z
d2 p −p2
≡ 2 × lim M 2
e gM (x, p),
(9.53a)
M →∞
(2π)2
Z
A5 (x) = 2 × lim tr
where we have introduced the auxiliary function
n
i
gM (x, p) := tr γ5 1 + 2pµ Dµ /M − Dµ Dµ /M 2 − [σµ , σν ]Fµν (x)/M 2
4
o
1
+ (2pµ Dµ )2 /M 2 + O(M −3 )
2
(9.53b)
522 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
to shorten the notation. After performing the trace over the 2 × 2
matrices, the axial anomaly in the background Jµ turns into
Z
A5 (x) = 2 ×
= 2×
d2 p −p2
i tr
γ
[σ
,
σ
]F
(x)
e
(−)
5
µ
ν
µν
(2π)2
4
1
i
(−) tr {(+i)σ2 σ1 [4 × σ1 σ2 F12 (x)]}
4π
4
1
F (x)
π 12
1
=
F (x).
2π µν µν
=
(9.54)
Having obtained with Eq. (9.46) the infinitesimal change δJf under
Eq. (9.47) of the Jacobian Jf in the background Jµ , we can reconstruct
the Jacobian Jf itself by the composition of infinitesimal δJf ,
!
Jf =
Y
δJf = exp ln
Y
ξ
δJf
!
= exp
ξ
X
ln δJf
.
(9.55)
ξ
To this end, we express δFµν in terms of δξ,
(δJµ ) = µν ∂ν (δξ) ⇐⇒ −(δF12 ) = ∂2 (δJ1 ) − ∂1 (δJ2 )
= µµ0 ∂µ0 (δJµ )
= µµ0 µν ∂µ0 ∂ν (δξ)
= −µ0 µ µν ∂µ0 ∂ν (δξ)
= (−)2 δµ0 ν ∂µ0 ∂ν (δξ)
= ∆(δξ).
(9.56)
In the same way,
− F12 = ∆ξ.
Since Jf =
Q
(9.57)
δJf is obtained by exponentiation of
ξ
R
δξ ln(δJf ), we
ξ
deduce that


Z
Jf = exp −
d2 x
Z
(δξ)(x)A5 (x)
ξ


Z
= exp −
d2 x
Z
ξ
1
(δξ)(x) F12 (x) .
π
(9.58)
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
523
With the help of the decomposition (9.30), we conclude that


Z
Z
1
Jf = exp + d2 x (δξ)(x) (∆ξ)(x)
π
ξ
Z
1
2
= exp +
d x ξ(x)(∆ξ)(x)
2π
Z
1
2
2
= exp −
d x (∂µ ξ) (x)
2π
Z
Z
∂µ ∂ν
1
2
2
Jν (y)
(9.59)
.
d x d y Jµ (x) δµν δ(x − y) −
≡ exp −
2π
∆
The first calculation of the axial anomaly goes back to Schwinger’s
solution of quantum electrodymanics in (1+1)–dimensional Minkowski
space and time (QED2 ). [114]
It is important to point out that, had we chosen to regularize the
axial anomaly in Eq. (9.46) without respecting gauge invariance, say
by writing
X
2
) /M 2 ϕ (x), (9.60)
A5 (x) := 2 × lim
ϕ†n (x)γ5 e−(iσµ ∂µ
n
M →∞
n
we would have then found that A5 = 0. We are thus faced with the
choice between preserving gauge invariance at the expense of the anomaly, or an anomaly free theory at the expense of gauge invariance.
Where does the anomaly come from? On the one hand, the anomaly
comes from the zero-mode sector, i.e., the eigenspace of σµ Dµ with
vanishing eigenvalues according to Eq. (9.48). Eigenvalues λn 6= 0 are
irrelevant since they do not contribute in the limit M → ∞. On the
other hand, the anomaly originates from all plane waves, including
ones with very large momenta, according to Eq. (9.54) in the planewave basis. The transformation (9.50), from the eigenbasis of σµ Dµ to
the plane wave basis, that we might naively believe to be unitary, is
thus far from benign for the operator O ≡ γ5 that anticommutes with
σ µ Dµ .
9.2.3. Abelian bosonization of the massless Thirring model.
The generating function for the current-current correlation functions in
the massless Thirring model is
Z
ZTh;g [Jµ ] := D[ψ̄, ψ] exp(−STh;g ),
Z
(9.61)
STh;g := d2 x LTh;g ,
2
g
LTh;g := ψ̄ σµ i∂µ + Jµ ψ −
ψ̄ σµ ψ .
2
524 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
With the help of a Hubbard-Stratonovich transformation,
Z
Z
Z
1
2
2
D[ψ̄, ψ] exp(−Scov ),
ZTh;g [Jµ ] ∝ D[Bµ ] exp −
d x Bµ
2g
Z
Scov := d2 x Lcov ,
Lcov := ψ̄ σµ i∂µ + Jµ + Bµ ψ.
(9.62)
By shifting the functional integration over the auxiliary field Bµ to
Bµ = Aµ − Jµ ,
(9.63)
the generating function becomes
Z
Z
Z
2
1
2
ZTh;g [Jµ ] ∝ D[Aµ ] exp −
d x Aµ − Jµ
D[ψ̄, ψ] exp(−Scov ),
2g
Z
Scov := d2 x Lcov ,
Lcov := ψ̄ σµ i∂µ + Aµ ψ.
(9.64)
Assuming that the local decomposition
Aµ = ∂µ χ + ∂˜µ ξ
holds,
5
(9.65)
the decoupling transformation
ψ̄ =: ψ̄ 0 e−iχ+γ5 ξ ,
ψ =: e+iχ+γ5 ξ ψ 0 ,
(9.67)
results in the following representation of the generating function,
Z
R 2
R 2
2
2
1
1
˜
ZTh;g [Jµ ] ∝ D[χ, ξ] e− 2g d x (∂µ χ+∂µ ξ−Jµ ) e− 2π d x (∂µ ξ)
Z
× D[ψ̄ 0 , ψ 0 ] e−Sfree ,
(9.68)
Z
Sfree := d2 x Lfree ,
Lfree := ψ̄ 0 iσµ ∂µ ψ 0 .
Here, we made use of the fact that the bosonic Jacobian for going from
Aµ to (χ, ξ) is independent of (χ, ξ). We also made use of Eq. (9.59).
5
The gauge field Aµ is topologically trivial, i.e., it has vanishing winding
number (magnetic flux)
Z
Z
Z
1
1
1
d2 x µν Fµν =
d2 x F12 = −
d2 x (∆ξ) = 0.
(9.66)
2π
π
π
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
Integration over the pure-gauge contribution ∂µ χ yields [as
0]
Z
1
− 2g
D[χ] e
R
d2 x [(∂µ χ)2 −2(∂µ χ)Jµ ]
By Eq. (9.30) Jµ = ∂µ χ0 + ∂˜µ ξ 0
Z
=
1
D[χ] e− 2g
1
R
2
1
R
2
1
R
d2 x
R
R
525
R
d2 x (∂µ χ)(∂˜µ ξ) =
d2 x [χ(−∆)χ+2χ(∂µ Jµ )]
−1
2
∝ e+ 2g R d x R d y (∂µ Jµ )(x)(−∆) (x−y)(∂ν Jν )(y)
1
2
2
0
−1
0
= e+ 2g d x d y (∆χ )(x)(−∆) (x−y)(∆χ )(y)
0
= e+ 2g R d x (−∆χ )χ
1
2
0 2
= e+ 2g d x (∂µ χ )
≡ e+ 2g
R
0
d2 y Jµ (x)
∂µ ∂ν
∆
Jν (y)
.
(9.69)
Note that the argument on the right-hand side of Eq. (9.69) has the opposite sign to the corresponding (longitudinal) term in Eq. (9.29). After
integration over the pure-gauge component ∂µ χ of Aµ , we conclude that
the generating function for current-current correlation functions reads
Z
h
i
o
R 2 n
R
∂ ∂
g
1
d x (1+ π
(∂µ ξ)2 −2(∂˜µ ξ)Jµ + d2 y Jµ (x) δµν δ(x−y)− µ∆ ν Jν (y)
− 2g
)
ZTh;g [Jµ ] ∝ D[ξ]e
Z
× D[ψ̄ 0 , ψ 0 ] e−Sfree ,
Z
Sfree := d2 x Lfree ,
Lfree := ψ̄ 0 iσµ ∂µ ψ 0 .
(9.70)
This is not quite yet the canonical form for a generating function
since the argument of the exponential is quadratic in the source Jµ for
the current. Preferred is a linear dependence on Jµ . To remedy this
deficiency, we make use of Eq. (9.29) with
r
α
1
β2
1
1
φ → θ,
t→ ,
β → −i
,
β 2t = − →
= − (9.71)
α
g
π
α
g
to dispose of the term quadratic in the source Jµ in the argument of
the Boltzmann weight on the right-hand side of Eq. (9.70) by the introduction of an auxiliary real-valued scalar field θ in the path integral.
Hence,
Z
R 2
g
1
2
2
˜
ZTh;g [Jµ ] ∝ D[ξ, θ] e− 2g d x {(1+ π )(∂µ ξ) +α g(∂µ θ) −2[∂µ (ξ−gβθ)]Jµ }
Z
× D[ψ̄ 0 , ψ 0 ] e−Sfree ,
Z
Sfree := d2 x Lfree ,
Lfree := ψ̄ 0 iσµ ∂µ ψ 0 .
(9.72)
526 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
The penultimate step consists in the linear transformation
g
θ1
1/g −β
ξ
ξ
b
β
θ1
=
⇐⇒
=
(9.73)
,
θ2
a
b
θ
θ
θ2
b + aβg −a 1/g
with the ratio of the two adjustable complex-valued parameters a and
b chosen such that the coefficient
2 g
1
g
αa
1+
bβ −
(9.74)
b + aβg
g
π
g
of the cross term (∂µ θ1 )(∂µ θ2 ) vanishes. Since β 2 /α = −1/g, we must
then have that
a g β
= 1+
,
b π α
g/b
π b
ξ
b
β
θ1
β
θ1
=
=−
.
β2
g
θ
θ2
θ2
b −a 1/g
1 + 1 + π α g −a 1/g
(9.75)
In terms of the scalar fields θ1 and θ2 , the generating function reads
Z
R 2
1
2
2
˜
ZTh;g [Jµ ] ∝ D[θ1 , θ2 ] e− 2 d x [a1 (∂µ θ1 ) +a2 (∂µ θ2 ) −2(∂µ θ1 )Jµ ]
Z
× D[ψ̄ 0 , ψ 0 ] e−Sfree ,
(9.76a)
Z
Sfree := d2 x Lfree ,
Lfree := ψ̄ 0 iσµ ∂µ ψ 0 ,
with the coefficients
2 g
1
g 2
2
1+
b + αa
a1 : =
b + aβg
g
π
g
g
a2
=
+α 2g
2 1 +
π
b
1 + ab βg
"
2 #
g
g
g 2 β
= h
g
2 i2 1 + π + α 1 + π
α
1 + 1 + πg βα g
g g
g 2
= − 1+
2 1 +
π
π
1 − 1 + πg
h
i
g
g
g
=
−
1+
g 2
π
π
−π
g
= −π 1 +
,
(9.76b)
π
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
527
and
2 1
g 2 α
1+
β + 2
g
π
g
g
α
1
g
2
β +
h
i2 1 +
b2
π
g
g β2
1+ 1+ π αg
h i
α
1
g
+1
− 1 +
b2 1 − 1 + g 2
π
π
α1
−π 2 .
b g
a2 :=
=
=
=
g
b + aβg
(9.76c)
If we choose
b = β,
(9.77a)
ξ = −π (θ1 + θ2 )
(9.77b)
we find the simple relation
as well as the positive coefficient
a2 = +π.
(9.77c)
The real-valued scalar field θ1 has a negative definite kinetic energy.
To bring this field to a canonical form, we perform the final change of
variables
θ1 =:
i
φ,
2π 1
θ2 =:
1
φ,
2π 2
(9.78)
under which we find that
Z
R 2
g
1
1
i
2
2
˜
ZTh;g [Jµ ] ∝ D[φ1 , φ2 ] e− d x [ 8π (1+ π )(∂µ φ1 ) + 8π (∂µ φ2 ) − 2π (∂µ φ1 )Jµ ]
Z
× D[ψ̄ 0 , ψ 0 ] e−Sfree ,
Z
Sfree := d2 x Lfree ,
Lfree := ψ̄ 0 iσµ ∂µ ψ 0 ,
ψ̄ψ = ψ̄ 0 e+2γ5 ξ ψ 0 ,
1
ξ = − (iφ1 + φ2 ) .
2
(9.79)
Observe that φ1 (φ2 ) enters with (without) an i in ψ̄ψ = ψ̄ 0 exp(+2γ5 ξ) ψ 0 =
ψ̄ 0 exp(−γ5 (iφ1 + φ2 ))ψ 0 . This fact is crucial to establish the correspondence between the two-dimensional Sine-Gordon model and the
two-dimensional Thirring model. In the mean time, we have established the Abelian bosonization rules of the massless two-dimensional
528 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
Thirring model
1
(∂ φ )2 ,
8π µ 1
i
ψ̄ σµ ψ ←→ − (∂˜µ φ1 ),
2π
ψ̄ iσµ ∂µ ψ ←→
(9.80)
that imply
2
1 g
g
ψ̄ σµ ψ ←→
1+
(∂µ φ1 )2 .
(9.81)
ψ̄ iσµ ∂µ ψ −
2
8π
π
Integration over φ2 , ψ̄ 0 , and ψ 0 only changes the proportionality factor
between the fermionic and bosonic generating functions and is thus of
no consequences when calculating current-current correlation functions.
Keeping the explicit dependence on ψ̄ 0 , ψ 0 , and φ2 as in Eq. (9.79)
is needed to establish the equivalence between the two-dimensional
massive Thirring model and the two-dimensional Sine-Gordon model.
9.2.4. Abelian bosonization of the massive Thirring model.
Abelian bosonization of the two-dimensional massive Thirring model
follows the steps of Abelian bosonization of the massless two-dimensional
Thirring model up to Eq. (9.79), which becomes
Z
R 2
g
1
1
i
2
2
˜
ZTh;m,g [Jµ ] ∝ D[φ1 , φ2 ] e− d x [ 8π (1+ π )(∂µ φ1 ) + 8π (∂µ φ2 ) − 2π (∂µ φ1 )Jµ ]
Z
× D[ψ̄ 0 , ψ 0 ] e−Sm ,
Z
Sm := d2 x Lm ,
Lm := ψ̄ 0 iσµ ∂µ − m e−γ5 (iφ1 +φ2 ) ψ 0 .
(9.82)
Equivalence between the two-dimensional massive Thirring model (9.82)
and the two-dimensional Sine-Gordon model
Z
R 2
i
˜
ZSG;t,h [Jµ ] := D[φ] e−SSG;t,h + 2π d x (∂µ φ)Jµ ,
Z
SSG;t,h := d2 x LSG;t,h ,
(9.83)
h
1
2
LSG;t,h := (∂µ φ) − cos φ,
2t
t
h
im
4π
t=
,
∝− ,
1 + πg
t
π
is established by comparing term by term the expansions of the twodimensional Thirring (ZTh;m,g [Jµ ]) and two-dimensional Sine-Gordon
(ZSG;t,h [Jµ ]) current-current generating functions in powers of m and
h/(2t), respectively.
9.2. ABELIAN BOSONIZATION OF THE THIRRING MODEL
529
A generic charge-neutral term in the two-dimensional Thirring expansion of order 2n is of the form (hardcore condition is assumed)
2n
Z
2
Z
2
|
2
Z
d xn d y 1 · · · d 2 y n
{z
}
no x’s or y’s equal
× f1 × f2 × f˜ (x1 , · · · , xn , y1 , · · · , yn ),
| {z } | {z }
d x1 · · ·
m
Z
≡xn
≡y n
(9.84a)
where the integrand is the product of three functions. The first function
is
Z
R 2
g
1
i ˜
2
f1 (xn , y n ) := D[φ1 ] e− d z [ 8π (1+ π )(∂µ φ1 ) − 2π (∂µ φ1 )Jµ ]
× e−i[φ1 (x1 )+···+φ1 (xn )]+i[φ1 (y1 )+···+φ1 (yn )]
Z
R 2
i ˜
1
2
=
D[φ] e− d z [ 2t (∂µ φ) − 2π (∂µ φ)Jµ ]
(9.84b)
× e−i[φ(x1 )+···+φ(xn )]+i[φ(y1 )+···+φ(yn )] .
The second function is
Z
R 2 1
2
f2 (xn , y n ) := D[φ2 ] e− d z 8π (∂µ φ2 )
×e
(9.84c)
−[φ2 (x1 )+···+φ2 (xn )]+[φ2 (y1 )+···+φ2 (yn )]
.
The third function is
Z
R 2
0∗
0
0∗
0
∗
0
f˜(xn , y n ) := D[ψ 0 ± , ψ±
] e− d z (ψ − i∂+ ψ− +ψ + i∂− ψ+ )
∗
∗
∗
∗
0
0
0
0
× (ψ 0 − ψ+
)(x1 ) · · · (ψ 0 − ψ+
)(xn )(ψ 0 + ψ−
)(y1 ) · · · (ψ 0 + ψ−
)(yn ).
(9.84d)
Because there is no i that multiplies φ2 in e±φ2 on the right-hand side
of f2 , it is found that
2n
i
f2 (xn , y n ) f˜(xn , y n ) = −
,
2π
no two points in arguments equal,
(9.85)
by Eqs. (9.26) and (9.27). Owing to this cancellation, φ1 can be identified with the real-valued scalar field φ in the two-dimensional SineGordon model as represented in Eq. (4.55a), thereby establishing the
Abelian bosonization rules for the two-dimensional massive Thirring
model in terms of the two-dimensional Sine-Gordon model.
530 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
One interesting consequence of the Abelian bosonization rules relating the two-dimensional Thirring model to the two-dimensional SineGordon model is that the Sine-Gordon model at reduced temperature
1
t = 4π ⇐⇒ g = 0
⇐⇒ K = in 2d–XY model
(9.86)
π
is equivalent to a free-fermion (relativistic) theory.
9.3. Applications
We present three applications of the Abelian bosonization rules to
quantum systems in one-dimensional position space. We limit ourselves to two cases of spinless fermions on a lattice, for simplicity, when
dealing with interacting fermions, as it can be shown that the case of
fermions carrying spin-1/2 reduces to two copies of bosonized spinless
fermions (this is the phenomenon of spin and charge separation in onedimensional position space). We also consider a quantum magnet and
show how it reduces to spinless fermions, before taking advantage of the
Abelian bosonization rules to turn this model into one for interacting
bosons.
9.3.1. Spinless fermions with effective Lorentz and global
U (1) gauge symmetries. We consider the Hamiltonian
Ĥkin := −t
N X
ĉ†j ĉj+1 + ĉ†j+1 ĉj ,
(9.87a)
j=1
where
{ĉi , ĉ†j } = δij ,
{ĉ†i , ĉ†j } = {ĉi , ĉj } = 0,
that acts on the Fock space
( N
Y † mj ĉj
F := span
|0i ĉj |0i = 0,
i, j = 1, · · · , N, (9.87b)
)
ĉj = ĉj+N ,
mj = 0, 1
j=1
(9.87c)
subject to the condition that the average number of spinless fermions
in the grand-canonical ensemble is Nf . The parameter t that sets the
energy scale is taken to be positive.
Periodic boundary conditions in a finite system of length L = N a
have been chosen, as we are interested in the thermodynamic limit,
which is defined by the total number of sites N → ∞ and the average
number of electrons Nf → ∞ while holding their ratio Nf /N ≤ 1 fixed.
In this limit, our choice of boundary conditions does not affect the
conclusions that we draw below. To cover all grounds (see below), we
9.3. APPLICATIONS
531
assume that the density of electrons is commensurate to the lattice.
For definitiveness, we choose
Nf =
N
,
2
(9.88a)
i.e., the filling fraction
Nf
(9.88b)
N
of the lattice is one-half. This choice requires that N is even and that
the Fermi wave vector is
kF = π/2
(9.88c)
ν :=
in the thermodynamic limit. The many-body ground state is then the
non-interacting Fermi sea
|km |<kF
Y
|FSi =
c†km |0i,
km
N
1 X ikm j †
c†km := √
e
cj ,
N j=1
(9.88d)
with km = Nπ m and m = − N2 , · · · , + N2 − 1. The Fermi sea obeys the
isotropy condition
E 1
D FS c†j cj FS = ,
j = 1, · · · , N.
(9.89)
2
In preparation for taking the continuum limit a → 0, we perform
the local gauge transformation
ĉ2j = (−i)2j fˆei = e−2ikF j fˆei ,
ĉ2j+1 = (−i)2j+1 fˆoi = e−2ikF (j+1/2) fˆoi ,
(9.90a)
with i ≡ 2j, that leaves the fermionic algebra (9.87b) unchanged and
under which
N/2 h
i
X
Ĥkin = it
fˆei† fˆoi − fˆo(i−1) + fˆoi† fˆe(i+1) − fˆei .
(9.90b)
i=1
The (naive) continuum limit of Eq. (9.90b) is the Dirac Hamiltonian
ZL
ĤD =
dx η̂1† ivF ∂x η̂2 + η̂2† ivF ∂x η̂1
0
ZL
=
(9.91a)
dx η̂1† η̂2†
0
ivF ∂x
0
whereby
vF := (2a) t,
fˆei −→
√
2a η̂1 (x),
ivF ∂x
η̂1
,
0
η̂2
fˆoi −→
√
2a η̂2 (x),
(9.91b)
in one-dimensional position space. All derivatives of higher order than
one have been dropped here. This approximation should be good in
532 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
the very close vicinity of the two Fermi points ±kF . In the chiral basis
η̂− :=
q
1
2
η̂+ :=
q
1
2
(η̂1 + η̂2 )
⇐⇒
(η̂1 − η̂2 )
η̂1 :=
q
1
2
η̂− + η̂+
η̂2 :=
q
1
2
η̂− − η̂+
(9.92a)
the Dirac Hamiltonian is diagonal
ZL
ĤD =
dx
†
η̂−
ivF ∂x η̂− −
†
η̂+
ivF ∂x η̂+
0
ZL
=
(9.92b)
†
†
dx η̂−
η̂+
+ivF ∂x
0
0
0
η̂−
.
−ivF ∂x
η̂+
The partition function at the inverse temperature β (the Boltzmann
constant is set to unity) can be represented by a path integral over
Grassmann coherent states obeying antiperiodic boundary conditions
in the imaginary-time direction,
Z
ZD = D[η ∗ , η] e−SD ,
Zβ
ZL
0
LD =
(9.93)
dx LD ,
dτ
SD =
0
∗
η−
∗
(∂τ + ivF ∂x ) η− + η+
(∂τ − ivF ∂x ) η+ .
∗
is independent of η± , the
By taking advantage of the fact that η∓
change of integration variable
∗
∗
η∓
=: iψ∓
,
η∓ =: ψ∓ ,
(9.94)
brings the partition function (9.93) to the desired form [see Eq. (9.14)],
namely
Z
ZD ∝ D[ψ ∗ , ψ] e−SD ,
Zβ
SD =
ZL
0
(9.95)
dx LD ,
dτ
0
∗
∗
LD = ψ−
i (∂τ + ivF ∂x ) ψ− + ψ+
i (∂τ − ivF ∂x ) ψ+ ,
provided one identifies τ with x1 , x with x2 , and sets the Fermi velocity
vF to one,
x1 := τ,
x2 := x,
vF ≡ 1.
(9.96)
9.3. APPLICATIONS
(i)
+
+
k4
k1
533
(ii)
k4
k3
k2
+
k1
(iii)
+
k3
k2
k4
+
k1
k3
k2
+
Figure 1. (i) Forward, (ii) backward, and (iii) Umklapp scattering processes in reciprocal space due to a
quartic contact interaction in position space.
Note that these identifications imply, upon Abelian bosonization, that
∗
∗
ψ−
ψ− + ψ+
ψ+ (τ, x)
1
→ +
(∂ φ) (τ, x),
2πi x
∗
∗
ψ̄ σ2 ψ (τ, x) = +i ψ−
ψ− − ψ+
ψ+ (τ, x)
1
→ −
(∂ φ) (τ, x).
2πi τ
ψ̄ σ1 ψ (τ, x) =
(9.97a)
(9.97b)
Equation (9.97a) tells us that the imaginary-time component ψ̄ σ1 ψ of
the (relativistic) two-current ψ̄ σµ ψ becomes the space (x) derivative
of a real-valued scalar field φ upon Abelian bosonization. Equation
(9.97b) tells us that the space component ψ̄ σ2 ψ of the (relativistic)
two-current ψ̄ σµ ψ becomes the imaginary time (τ ) derivative of a realvalued scalar field φ upon Abelian bosonization.
The partition function (9.95) defines a free-field fixed point. The
engineering dimension of the Dirac field equals its scaling dimension
and is given by 1/2 in units where ~ = vF = 1 as
[ψ∓ ] = (length)−1/2 .
(9.98)
The engineering dimension of the local bilinears
∗
∗
(ψ−
ψ− ± ψ+
ψ+ )(x, τ )
(9.99a)
and
∗
∗
ψ−
ψ+ ± ψ+
ψ− (x, τ )
(9.99b)
are 1. These are infrared relevant perturbations to the Dirac free-field
fixed point. The engineering dimension of the local (contact) quartic
interactions
∗
∗
ψ− ± ψ+
ψ+ )2 (x, τ )
(9.100a)
(ψ−
and
∗
∗
ψ−
ψ+ ± ψ+
ψ−
2
(x, τ )
(9.100b)
534 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
is 2. These are marginal perturbations to the Dirac free-field fixed
point that encode (see Fig. 1): 6
• (i) Forward scattering in reciprocal space,
∗
∗
δ(k4 + k3 − k2 − k1 ) ψ+
(k4 ) ψ−
(k3 ) ψ− (k2 ) ψ+ (k1 ).
(9.103)
• (ii) Backward scattering in reciprocal space,
∗
∗
δ(k4 + k3 − k2 − k1 ) ψ−
(k4 ) ψ+
(k3 ) ψ− (k2 ) ψ+ (k1 ).
(9.104)
• (iii) Umklapp scattering in reciprocal space,
∗
∗
δ(k4 + k3 − k2 − k1 + K) ψ−
(k4 ) ψ−
(k3 ) ψ+ (k2 ) ψ+ (k1 ),
(9.105a)
where K is any vector from the reciprocal lattice, i.e.,
2π
K=
n,
n ∈ Z.
(9.105b)
a
Umklapp scattering demands a filling fraction (9.88b) that is
commensurate with the lattice in order to satisfy momentum
conservation up to momenta from the reciprocal lattice.
Higher powers of Eqs. (9.100a) and (9.100b) are infrared irrelevant at
the free-field fixed point.
By demanding that any quartic interaction satisfies the effective
Lorentz symmetry and the global U (1) gauge symmetry (9.4) of the
free-field Dirac theory (9.95), we are lead to consider the generic quartic
interaction
2
2 i
λ1 h ∗
∗
∗
∗
Lint = −
ψ− ψ− + ψ+ ψ+ − ψ− ψ− − ψ+ ψ+
2
(9.106)
2 λ3 ∗
2
λ2 ∗
∗
∗
ψ− ψ+ + ψ+ ψ− +
ψ− ψ+ − ψ+ ψ− ,
+
2
2
which is parametrized by the three dimensionless coupling constants
λ1 , λ2 , λ3 ∈ R. The Abelian bosonization rules
2
2
1
∂φ
1
∂φ
ψ̄ iσµ ∂µ ψ →
+
,
(9.107a)
8π ∂x1
8π ∂x2
6
It might seem peculiar to worry about interactions induced by taking powers
of Eq. (9.99) in that they appear to vanish in the continuum limit due to the
fermionic algebra
∗
∗
0 = ψ−
ψ−
ψ+ ψ+ (x, τ ).
(9.101)
However, these fields are highly singular at short distances. To deal with such
ambiguities one must rely on the regularization procedure known as point-splitting
by which
∗
∗
∗
∗
ψ−
ψ−
ψ+ ψ+ (x, τ ) → ψ−
(x + 4 , τ ) ψ−
(x + 3 , τ ) ψ+ (x + 2 , τ ) ψ+ (x + 1 , τ )
(9.102)
and the limit 1,2,3,4 → 0 is carefully taken so as to extract any singular C-number
that appears in the expectation value of the right-hand side due to short distance
singularities of the free-field Green functions.
9.3. APPLICATIONS
for the Dirac kinetic energy,
1
∂φ
ψ̄ σ1 ψ → +
,
2πi ∂x2
535
1
ψ̄ σ2 ψ → −
2πi
∂φ
∂x1
,
(9.107b)
for the Dirac current,
1 cos φ
,
πi 2a
for the Dirac masses, and
ψ̄ ψ → +
ψ̄ γ5 ψ → −
1 sin φ
,
π 2a
(9.107c)
1 e+iφ
1 e−iφ
∗
,
ψ+
ψ− →
,
(9.107d)
2πi 2a
2πi 2a
for the chiral masses imply that the fermionic partition function
∗
ψ−
ψ+ →
Z
Z=
∗
−
D[ψ , ψ] e
+∞
R
−∞
d2 x (LD +Lint )
(9.108a)
can be represented by the bosonic partition function
Z
−
+∞
R
d2 x L
,
D[φ] e −∞
1
λ3 sin2 φ
λ1
λ cos2 φ
L=
+
.
1+
(∂µ φ)2 − 22
8π
π
2π (2a)2
2π 2 (2a)2
Z=
(9.108b)
When using the Abelian bosonization rules for the Dirac mass ψ̄ ψ
and the axial mass ψ̄ γ5 ψ, we have divided the cosine and the sine of
the bosonic field φ by the lattice spacing (2a) of the sublattice made
of even sites to insure that the bosonic interaction has the units of
(length)−2 and with a bias towards the microscopic model constructed
from Eq. (9.87a) by the addition of some short-range interaction between lattice fermions. The choice of the length scale (2a) is arbitrary,
as we could equally have chosen a length scale that differs from (2a)
by a numerical factor of order 1, say if the hopping took place only
between nearest-neighbor even sites. The length scale entering the
bosonic interaction cannot be fixed by field theory alone, as to do so
would require a detailed knowledge of the physics at short distances.
To put it differently, many different microscopic models could be described at long distances and low energies by the field theory (9.108b),
although they would differ at short distances and high energies. This
ambiguity is reflected by the fact that the ratio m/h of dimensionful
couplings cannot be fixed from the sole data provided by the fermionic
and bosonic field theories in the bosonization table 1.
The one-loop renormalization-group (RG) flows obeyed by the three
dimensionless coupling constants λ1 , λ2 , and λ3 can be deduced along
the same lines as we derived the Kosterlitz-Thouless-RG flows in section
4.6.
536 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
9.3.2. Quantum xxz spin-1/2 chain. The quantum Hamiltonian for the quantum xxz spin chain is defined by
Ĥ
xxz
:= J⊥
N X
Ŝjx
x
Ŝj+1
+
Ŝjy
y
Ŝj+1
+ Jz
j=1
N
X
z
Ŝjz Ŝj+1
j=1
N
N
X
J X + −
+
z
+ Jz
Ŝjz Ŝj+1
Ŝj Ŝj+1 + Ŝj− Ŝj+1
,
= ⊥
2 j=1
j=1
(9.109a)
whereby
Ŝj+ := Ŝjx + iŜjy ,
Ŝj− := Ŝjx − iŜjy .
(9.109b)
The local spin operators obey the SU (2) (spin) algebra
h
i
β
α
Ŝk , Ŝl = iαβγ Ŝkγ δkl ,
k, l = 1, · · · , N,
α, β, γ = x, y, z,
(9.109c)
and the periodic boundary conditions
j = 1, · · · , N.
Ŝ j+N = Ŝ j ,
(9.109d)
The dimensionful coupling constants J⊥ and Jz are real valued. Without loss of generality, we may choose J⊥ to be positive, for a rotation
about the z axis in spin space by π for every other spins renders J⊥
positive.
Before defining the Hilbert space on which Ĥ xxz is defined, we recall
that the spin algebra (9.109c) can be rewritten
[Ŝj+ , Ŝk− ] = 2Ŝjz δjk ,
(9.110a)
for j, k = 1, · · · , N . For any s such that 2s is a non-vanishing and
positive integer and for any site j = 1, · · · , N , the local Hilbert space
can be constructed from the highest-weight state |sij , here defined by
the condition
[Ŝjz , Ŝk+ ] = +Ŝj+ δjk ,
[Ŝjz , Ŝk− ] = −Ŝj− δjk ,
Ŝj+ |sij = 0,
by repeated application of the ladder operator Ŝj− ,
2s
−
−
Hj := span |sij , Ŝj |sij , · · · , Ŝj
|sij .
(9.110b)
(9.110c)
The Hilbert space for the quantum xxz spin-s chain is then
H
xxz
:=
N
O
j=1
Hj .
(9.110d)
9.3. APPLICATIONS
537
The representation s = 1/2 of the spin algebra (9.109c) is specified
by demanding that the Casimir operator
2
Ŝ j := (Ŝjx )2 + (Ŝjy )2 + (Ŝjz )2
1 + −
=
Ŝj Ŝj + Ŝj− Ŝj+ + (Ŝjz )2
2
1 1
+1
=
2 2
(9.111)
holds locally, i.e., for any site j = 1, · · · , N . In the spin-1/2 representation,
2
Ŝj+ Ŝj− = Ŝ j − Ŝjz Ŝjz − 1
3 1
− + Ŝjz
4 4
1
j = 1, · · · , N.
= Ŝjz + ,
2
=
(9.112)
From now on, we assume the spin-1/2 representation, in which case
the Hilbert space Hxxz is 2N –dimensional and it is customary to label
states in Hxxz by the Casimir operator for the total spin
!2
N
X
2
Ŝ tot :=
(9.113)
Ŝ j
j=1
z
of the total spin
and the z-component Ŝtot
Ŝ tot :=
N
X
Ŝ j ,
(9.114)
j=1
2
z
as one verifies that Ŝ tot and Ŝtot
commute with Ĥ xxz . (The choice of
the quantization axis along the z direction in spin space is of course a
matter of convention.)
Hamiltonian Ĥ xxz can be diagonalized when |Jz /J⊥ | = ∞ and
|Jz /J⊥ | = 0.
9.3.2.1. Ising limit. When |Jz /J⊥ | = ∞, Ĥ xxz reduces to the Ising
model along a ring,
z
Ĥ := Jz
N
X
z
Ŝjz Ŝj+1
≡ H Ising ,
(9.115a)
j=1
where
H
Ising
:= Jz
N
X
j=1
szj szj+1 ,
sj = ±
1
eigenvalues of Ŝjz .
2
(9.115b)
538 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
When Jz > 0, this is the nearest-neighbor antiferromagnetic Ising
model along a ring (periodic boundary conditions). When Jz < 0,
this is the nearest-neighbor ferromagnetic Ising model along a ring.
The ferromagnetic Ising model (9.115) is related to the antiferromagnetic Ising model (9.115) by the transformation
Jz , Ŝjz → − Jz , (−1)j Ŝjz
(9.116)
for all sites j = 1, · · · , N . Hence, all properties in thermodynamic
equilibrium of the antiferromagnetic Ising model (9.115) are in one-toone correspondence with those of the ferromagnetic Ising model (9.115)
through the transformation (9.116).
The antiferromagnetic Ising model (9.115) supports long-range order (LRO) in the form of two degenerate ground states having all
spins parallel to the z-axis in spin space and nearest-neighbor spins
antiparallel. Any one of these two ground states is also called the onedimensional Néel state. Any one of these two Néel states breaks spontaneously the translation symmetry by one lattice spacing of Ĥ xxz . Any
one of these two Néel states also breaks spontaneously time-reversal
symmetry, defined by reversing locally all the spin directions, of Ĥ xxz .
It is essential to realize that the symmetries of Ĥ xxz that are spontaneously broken by any one of its two Néel states are discrete in the
Ising limit Jz /J⊥ → ∞. For this reason, a Néel state is separated from
all excited states by a finite-energy gap in the thermodynamic limit.
At any finite temperature, the Néel-long-range order is destroyed by
the finite-energy excitations owing to a celebrated argument by Peierls.
Correspondingly, Ising spin correlation functions decay exponentially
fast with large separations in position space at any non-vanishing temperature.
9.3.2.2. Quantum xy limit. When |Jz /J⊥ | = 0, Ĥ xxz reduces to
Ĥ xy := J⊥
N X
y
x
Ŝjx Ŝj+1
+ Ŝjy Ŝj+1
.
(9.117)
j=1
This limit, called the quantum xy limit, represents another point in
the coupling-space 0 ≤ |Jz /J⊥ | ≤ ∞ which is exactly soluble, as we
shall see shortly with the help of the Jordan-Wigner transformation.
The ground state of Ĥ xy is featureless and the finite-size gap to all excitations above this ground state collapses to 0 in the thermodynamic
limit N → ∞. Spin-spin correlation functions for the x or y components of the spins obey isotropic power laws for large separations in
position space and in the thermodynamic limit when Jz /J⊥ = 0. Furthermore, the quantum xy critical point Jz /J⊥ = 0 is the lower critical
end point of a finite segment of critical points with the upper critical
value 0 < (Jz /J⊥ )c as upper critical end point along the parametric
line 0 ≤ |Jz /J⊥ | ≤ ∞. Above (Jz /J⊥ )c a zero-temperature gap opens
9.3. APPLICATIONS
539
up. This gap evolves smoothly to the one in the antiferromagnetic
Ising limit Jz /J⊥ → ∞. This is the antiferromagnetic Ising regime
of the zero-temperature phase diagram. The critical point (Jz /J⊥ )c
belongs to the universality class of the Kosterlitz-Thouless transition.
The segment −∞ < Jz /J⊥ < 0 supports a gap. This gap is smoothly
connected to the one in the ferromagnetic Ising limit Jz /J⊥ → −∞.
The half-line −∞ ≤ Jz /J⊥ < 0 can thus be identified with the ferromagnetic regime. The existence of the lower and upper critical values
Jz /J⊥ = 0 and (Jz /J⊥ )c , respectively, can be inferred from quantumfield-theoretical arguments, although the numerical value of (Jz /J⊥ )c
is beyond quantum field theory. The key step towards a quantum
field theory is the Jordan-Wigner transformation. But before taking
advantage of the Jordan-Wigner transformation (which is thoroughly
described in appendix E.3), we identify the origin of the quantum fluctuations along 0 ≤ |Jz /J⊥ | < ∞ .
9.3.2.3. Quantum fluctuations. When |Jz /J⊥ | = ∞, Ĥ xxz becomes
the classical Ising model Ĥ z , for any local Ŝjz commutes with Ĥ xxz .
Quantum fluctuations are restored for any non-vanishing J⊥ , since any
local Ŝjz fails to commute with Ĥ xxz for any J⊥ 6= 0.
To appreciate further the role played by quantum fluctuations, observe first that the physics is invariant under transformation (9.116) in
the Ising limit J⊥ = 0. This is not true anymore when J⊥ 6= 0, as can
be verified from the fact that the two ferromagnetic states, which are
defined by the condition that they are the eigenstates with eigenvalues
±N/2 of the uniform magnetization
z
Ŝtot
:=
N
X
Ŝjz ,
(9.118)
j=1
are always eigenstates of Eq. (9.109), whereas the two (Néel) states,
which are defined by the condition that they are the eigenstates with
eigenvalues ±N/2 of the staggered magnetization
z
Ŝstag
:=
N
X
(−)j Ŝjz ,
j=1
are not eigenstates of Eq. (9.109) when J⊥ 6= 0.
(9.119)
540 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
Second, consider for simplicity the two-sites problem at the Heisenberg point defined by the Hamiltonian
ĤNxxz
=2 := J Ŝ 1 · Ŝ 2
2 3
J
=
Ŝ 1 + Ŝ 2 − J
2
4
3
= − J|0; 0ih0; 0|
4 1
+ J |1; −1ih1; −1| + |1; 0ih1; 0| + |1; +1ih1; +1|
4
(9.120)
2
with open boundary conditions. Here, eigenstates of Ŝ tot = (Ŝ 1 + Ŝ 2 )2
z
= Ŝ1z + Ŝ2z with eigenvalues s(s + 1) and sz , respectively, are
and Ŝtot
denoted |s; sz i. This basis can also be represented in terms of the
tensorial basis |1/2; sz1 i ⊗ |1/2; sz2 i ≡ |sz1 ; sz2 i 1 as
2
1
|0; 0i := √
|↑↓i 1 − |↓↑i 1 ,
2
2
2
|1; +1i := |↑↑i 1 ,
2
1
|1; 0i := √
|↑↓i 1 + |↓↑i 1 ,
2
2
2
|1; −1i := |↓↓i 1 .
(9.121a)
(9.121b)
(9.121c)
(9.121d)
2
The ferromagnetic states are given by Eqs. (9.121b) and (9.121d). Evidently, they are eigenstates of ĤNxxz
=2 and are symmetric under exchange
of the two spin labels. The Néel states are given by √12 (|0; 0i ± |1; 0i),
i.e., they are linear superpositions of eigenstates belonging to the sub2
z
space Ŝtot
= 0, but with different Ŝ tot . Evidently, they are neither
eigenstates of ĤNxxz
=2 nor eigenstates of the operator that interchanges
label 1 and 2 of the spins. Conversely, the ground state of the antiferromagnetic (J > 0) ĤNxxz
=2 is built out of an antisymmetric linear
superposition of the classical (Néel) states. By a classical state, we
thus understand a many-body state that is the tensorial product of
“single-particle” states.
9.3.2.4. Jordan-Wigner transformation. With the help of appendix E.3, Ĥ xxz can be represented solely in terms of spinless fermions,
which are called Jordan-Wigner fermions. To see this, define the nonlocal operators
iπ
K̂j := e
j−1
P
(Ŝkz + 12 )
k=1
=e
iπ
j−1
P
k=1
Ŝk+ Ŝk−
,
j = 1, · · · , N.
(9.122)
9.3. APPLICATIONS
541
The non-local operator K̂j rotates all spins to the left of site j by the
angle π around the z axis in spin space. The operator K̂j is called a
kink operator. It is shown in appendix E.3 that the operators
ĉ†j := Ŝj+ K̂j ,
ĉj := K̂j† Ŝj− ,
(9.123a)
realize the fermion algebra
{ĉk , ĉ†l } = δk,l ,
0 = {ĉk , ĉl } = {ĉ†k , ĉ†l },
k, l = 1, · · · , N.
(9.123b)
These operators create and destroy Jordan-Wigner fermions. As shown
in appendix E.3, the quantum Hamiltonian (9.109) becomes
N
N X
J⊥ X †
1
1
†
†
†
xxz
Ĥ
=+
ĉj+1 ĉj+1 −
ĉ ĉ + ĉj+1 ĉj + Jz
ĉj ĉj −
2 j=1 j j+1
2
2
j=1
N N
X
1
J⊥ X †
1
†
†
†
ĉj ĉj −
ĉ ĉ + ĉj+1 ĉj + Jz
→−
ĉj+1 ĉj+1 −
,
2 j=1 j j+1
2
2
j=1
(9.124a)
in the Jordan-Wigner representation. We have performed the local
gauge transformation
ĉj → (−)j ĉj
(9.124b)
to reach the second equality.
The total number-operator
N̂tot :=
N
X
ĉ†j ĉj
(9.125)
j=1
for Jordan-Wigner fermions commutes with Ĥ xxz . It is related to the
total spin operator (9.114) by
N
.
(9.126)
2
The boundary conditions obeyed by the Jordan-Wigner fermions dez
pend on the eigenvalue sztot of Ŝtot
through [see Eq. (E.49)]
z
N̂tot = Ŝtot
+
ĉj+N = (−)N̂tot +1 ĉj .
(9.127)
The unitary transformation
ĉj → (−)j ĉj
(9.128)
of the fermions corresponds to the unitary transformation
Ŝjx → (−)j Ŝjx ,
Ŝjy → (−)j Ŝjy ,
Ŝjz → Ŝjz ,
(9.129)
of the spins, i.e., to a local rotation by the angle π around the z axis
in spin space.
542 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
9.3.2.5. Path-integral representation. A path-integral representation
of
Z xxz := Tr e−β Ĥ
xxz
,
(9.130)
with Ĥ xxz defined in Eq. (9.109) and the trace performed over the
Hilbert space (9.110d), is
Z
xxz
Z
=
D[c∗ , c] e−Sxxz ,
(9.131a)
where the Euclidean action reads
S
xxz
Zβ
:=
dτ
0
N
X
Lxxz
j,j+1 ,
(9.131b)
j=1
with the Lagrangian density
J⊥ ∗
1
1
∗
∗
∗
:= ∂τ cj −
c c + cj+1 cj +Jz cj cj −
cj+1 cj+1 −
,
2 j j+1
2
2
(9.131c)
and the boundary conditions
Lxxz
j,j+1
c∗j
z
N
z
c∗j+N (τ + β) = (−)Ŝtot + 2 c∗j (τ ),
N
cj+N (τ + β) = (−)Ŝtot + 2 cj (τ ).
(9.131d)
9.3.2.6. Field theory. When |Jz /J⊥ | = 0, the Hamiltonian (9.117)
for interacting spin-1/2 is nothing but the non-interacting Hamiltonian
(9.87a) for spinless fermions provided one makes the identification
t→+
J⊥
.
2
(9.132)
The many-body ground state is represented by a Fermi sea for the
Jordan-Wigner fermions (see appendix E.3.2). Thus, it is featureless.
Correlation functions at long distances are controlled by the two Fermi
points and their immediate vicinity. In others words, linearization of
the spectrum around the two Fermi points gives the Dirac Hamiltonian
in the non-diagonal or diagonal representations (9.91a) and (9.92b),
respectively, from which the asymptotic decays of spin-spin correlation
functions can be calculated and shown to be power laws.
9.3. APPLICATIONS
543
A naive continuum limit of the Ising contribution (9.115) to the
spin Hamiltonian (9.109) replaces
Ĥ
z
N X
1
1
†
†
= +Jz
ĉj ĉj −
ĉj+1 ĉj+1 −
2
2
j=1
N
2 J N
Jz X †
ĉj ĉj − ĉ†j+1 ĉj+1 + z
= −
2 j=1
4
N/2 2 2 J N
Jz X ˆ† ˆ
†
† ˆ
† ˆ
ˆ
ˆ
ˆ
ˆ
+ z
fei fei − foi foi + foi foi − fe(i+1) fe(i+1)
= −
2 i=1
4
N/2 2 J N
2 Jz X ˆ† ˆ
† ˆ
† ˆ
† ˆ
z
ˆ
ˆ
ˆ
+(9.133)
= −
fei fei − foi foi + foi foi − fei fei + · · ·
2 i=1
4
by
Ĥ
z
≈ −Jz
N/2 X
fˆei† fˆei − fˆoi† fˆoi
2
+
i=1
By Eq. (9.91b)
By Eq. (9.92a)
2J
→ −vF z
J⊥
2J
= −vF z
J⊥
ZL
dx
η̂1† η̂1
dx
†
η̂−
η̂+
−
η̂2† η̂2
Jz N
4
2
+
Jz N
4
0
ZL
+
†
η̂+
η̂−
2
+
Jz N
,(9.134)
4
0
whereby it is assumed that
|Jz | J⊥
(9.135)
for linearization of the kinetic dispersion relation to make sense.
To pass to the Grassmann representation of the partition function (9.130), we need to normal order Eq. (9.134). At the operator
level, normal ordering is a highly non-trivial step as it requires a careful regularization through point-splitting of the product of fields at the
same position in space (see footnote 6). We gloss over these subtleties
and assume that we can replace the operators on the last line of Eq.
(9.134) by Grassmann-valued fields in the path-integral representation
544 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
of the partition function. The Ising interaction then becomes
∗
∗
η−
η+ + η+
η−
2
∗
∗
∗
= +2η−
η+ η+
η− + η−
η+
2
∗
+ η+
η−
2
2
2
∗
∗
∗
∗
= −2η−
η− η+
η+ + η−
η+ + η+
η−
2
2 i
1h ∗
∗
∗
∗
η η + η+ η+ − η− η− − η+ η+
= −
2| − −
{z
}
Current−current interaction
+
By Eq. (9.12)
= −
1h
2|
∗
η−
η+ +
∗
η+
η−
2
+
{z
∗
η−
η+ −
Umklapp interaction
∗
η+
η−
2 i
}
1
1
(η̄ σ1 η)2 − (η̄ σ2 η)2 + (η̄ η)2 + (η̄ γ5 η)2 .
2|
{z
} 2|
{z
}
Current−Current interaction
Umklapp interaction
(9.136)
With the help of the transformation (9.94), setting the Fermi velocity
[recall Eqs. (9.132) and (9.91b)]
vF = (2a)
J⊥
= a J⊥
2
(9.137)
to one, and ignoring the constant on the right-hand side of Eq. (9.134),
we end up with the path-integral representation of the partition function at vanishing temperature and in the thermodynamic limit L → ∞
[recall transformation (9.94)]
D[ψ̄, ψ] e−S
xxz
∝
xxz
Z+∞
Z+∞
=
dx1
dx2 Lxxz ,
Z
S
Z
xxz
−∞
L
xxz
,
−∞
2
2 i
Jz
Jz
Jz h
2
2
= ψ̄iσµ ∂µ ψ −
(ψ̄ σ1 ψ) −
(ψ̄ σ2 ψ) +
ψ̄ ψ + ψ̄ γ5 ψ
.
J⊥
J⊥
J⊥
(9.138)
9.3. APPLICATIONS
545
The Abelian bosonization rules (9.107) give 7 the bosonic representation
of the partition function
+∞
+∞
R
R
Z
−
dτ
dx L
−∞
−∞
Z = D[φ] e
,
2 2
Jz
∂φ
Jz
∂φ
1
J cos(2φ)
1
+ 2
+ 2
+
− 2z
L=
8π 4π J⊥
∂τ
8π 4π J⊥
∂x
π J⊥ (2a)2
" 2 2 #
∂φ
1
2Jz
∂φ
J cos(2φ)
=+
1+
+
− 2z
.
8π
π J⊥
∂τ
∂x
π J⊥ (2a)2
(9.139)
Equation (9.139) is the Sine-Gordon model (9.1) at the inverse temperature
4π
(9.140)
t=
1 + π2JJz
⊥
and in the magnetic field
h
J
1
= 2z ×
t
π J⊥ (2a)2
(9.141)
corresponding to vortices of charge 2 if the Sine-Gordon model is interpreted as the classical two-dimensional XY model.
The cosine interaction in Eq. (9.139) breaks the symmetry
φ → φ + const
(9.142)
of the bosonic kinetic energy down to the discrete subgroup
const = ±π
(9.143)
which is isomorphic to the multiplicative group
Z2 := {+1, −1}.
(9.144)
The transformation (9.142) corresponds to the global U (1) axial transformation (9.5) in the fermionic representation. It is a symmetry of the
Lagrangian density in the massless two-dimensional Thirring model.
The transformation (9.142) with the choice (9.143) corresponds to the
global U (1) axial transformation (9.5) with the choice α5 = ±π/2. The
latter transformation changes the sign of the Dirac and axial masses
ψ̄ ψ and ψ̄ γ5 ψ, respectively. Hence, adding a squared Dirac mass or a
squared axial mass to the Lagrangian density in the two-dimensional
massless Thirring model breaks the global U (1) axial symmetry down
to the discrete axial subgroup Z2 . From the point of view of lattice
fermions, this discrete remnant of the continuous axial symmetry is
nothing but a manifestation of the bipartite nature of the underlying
7
Use cos2 α =
cos(2α).
1
2
[1 + cos(2α)], sin2 α =
1
2
[1 − cos(2α)], and cos2 α − sin2 α =
546 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
lattice in the tight-binding microscopic model. From this microscopic
perspective, the cosine interaction in Eq. (9.139) corresponds to Umklapp processes by which the net change of momentum in a scattering
event is twice the Fermi momentum and equals the reciprocal vector
spanning the first Brillouin zone. It should be noted that linearization
of the lattice spectrum induces a much larger symmetry group (the
chiral symmetry) at the level of the field theory than the original (microscopic) sublattice symmetry. Destruction of the continuous chiral
symmetry of the field theory with forward and backward scattering only
is not generic at the microscopic level, for it requires commensuration
between the Fermi wave vector and the lattice.
The lessons from the Kosterlitz-Thouless transition is that theory (9.139) has a “low-temperature” spin-wave phase and a “hightemperature” paramagnetic phase. The spin-wave phase corresponds
here to a line of critical points characterized by power laws obeyed by
correlation functions when 0 ≤ Jz /|J⊥ | < (Jz /|J⊥ |)c . The paramagnetic phase when (Jz /|J⊥ |)c ≤ Jz /|J⊥ | ≤ ∞ is here characterized by
long-range order, a gap, and exponentially decaying correlation functions. The difference with the Kosterlitz-Thouless transition studied
in the context of the classical two-dimensional XY model is that it
is the vortices of charge 2 and not vortices of charge 1 that trigger
the continuous phase transition from a quasi-long-range-ordered phase
(quantum xy regime) to a massive phase (Ising regime). Moreover, the
terminology of long-range order in the Ising regime refers to long-range
order for Ising degrees of freedom and should not be confused with the
long-range order of the ground state of the classical O(2) Heisenberg
model. The latter order refers to a continuous symmetry, while the
former order refers to a discrete symmetry.
The critical value (Jz /|J⊥ |)c cannot be predicted reliably from the
naive continuum limit that we took. One needs to resort to a matching
of field theory and exact methods such as the Bethe Ansatz solution to
the xxz lattice model to extract the true dependence of the couplings
λ1,2,3 on the lattice coupling constants from which the critical value
(Jz /|J⊥ |)c follows by demanding that the cosine interaction is marginal.
It turns out that the critical value (Jz /|J⊥ |)c corresponds to the socalled Heisenberg point
(Jz /|J⊥ |)c = 1.
(9.145)
9.3.3. Single impurity of the mass type. As a third example of
interacting fermions, we consider the two-dimensional Thirring model
(9.2) with the mass term
m(x, τ ) := −i(2πa) V0 δ(x),
∀τ ∈ R.
(9.146)
This mass term varies only in the space direction with a delta-function
profile. It can be interpreted as a static impurity located at the origin
9.3. APPLICATIONS
547
that scatters incoming right and left movers through a delta function
potential. The strength of the impurity is measured by the dimensionful coupling constant V0 , whereby [V0 ] = (length)−1 . Upon Abelian
bosonization, the partition function at zero temperature becomes [see
Eq. (9.83)]
Z
Z = D[φ] e−S ,
Z+∞ Z+∞
S=
dτ
dx L,
−∞
(9.147a)
−∞
1
L=
(∂ φ)2 + V0 δ(x) cos φ,
4π η µ
where
η=
2
.
1 + πg
(9.147b)
Our strategy is to integrate over all the components φ(x 6= 0, τ ) of
the field in the path integral so as to induce an effective action for the
field
θ(τ ) ≡ φ(x = 0, τ ).
(9.148)
The first step towards this goal is to rewrite the integration measure
as
D[φ] = D[φ] D[θ] δ [θ(τ ) − φ(0, τ )]
+∞
R
Z
dτ λ(τ )[θ(τ )−φ(0,τ )]
i
−∞
.
= D[φ] D[θ] D[λ(τ )] e
(9.149)
On the second line, the Lagrange multiplier λ(τ ) is introduced at each
imaginary time τ to enforce the delta-function constraint on the scalar
field θ by way of a modification of the Lagrangian density.
Second, we may perform the path integrals in the following order.
We begin by integrating over φ(x, τ ) for all x and τ . Then, we integrate
over λ(τ ) for all τ . In this way we obtain an effective action in (0 + 1)–
dimensional space and (imaginary) time for θ,
Z
Zeff = D[θ] e−Seff [θ] ,
Z
−Seff [θ]
e
∝ D[λ]e−Sint [θ,λ] ,
e
−Sint [θ,λ]
−
∝e
+∞
R
−∞
Z
×
(9.150)
dτ [V0 cos θ(τ )−iθ(τ ) λ(τ )]
−
D[φ] e
+∞
R
−∞
dτ
+∞
R
−∞
1
(∂µ φ)2 (x,τ )+iδ(x)φ(x,τ )λ(τ )]
dx[ 4πη
.
548 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
The path integral over φ on the last line of Eq. (9.150) is Gaussian and
is most easily performed in Fourier space,
−
Z
D[φ]e
D[φ]e
−
e
−∞
−
e
+∞
R
+∞
R
−∞
−∞
d$
−∞
"
d$
+∞
R
+∞
R
−∞
+∞
R
dτ
−∞
−
Z
+∞
R
1
dx[ 4πη
(∂µ φ)2 (x,τ )+iδ(x)φ(x,τ )λ(τ )]
+∞
R
dq
n
−∞
$ 2 +q 2
4πη
=
φ(+q,+$)φ(−q,−$)+i 2√12π [φ(+$,+q)λ(−$)+φ(−$,−q)λ(+$)]
e
+∞
R
−∞
∝
#
πη
dq
2π $ 2 +q 2
λ(+$)λ(−$)
=
πη
d$ 2|$|
λ(+$)λ(−$)
.
(9.151)
The path integral on the second line of Eq. (9.150) is again Gaussian
and is also most easily performed in Fourier space,
+∞
R
Z
πη
d${ 2|$|
−
λ(+$)λ(−$)− 2i [λ(+$)θ(−$)+λ(−$)θ(+$)]}
−∞
D[λ]e
∝
(9.152)
−
o
|$|
d$ 2πη θ(+$)θ(−$)
.
We thus conclude that
Z+∞
Z+∞
|$|
dτ cos θ(τ ).
θ(+$) θ(−$) + V0
Seff [θ] =
d$
2πη
−∞
(9.153)
−∞
This is nothing but the effective action S1 + S0 int from Eq. (8.172a)
for a dissipative Josephson junction. It can be interpreted as a single
particle moving on the circle subject to the periodic potential V0 cos θ
and to a dissipation with friction coefficient
1
γ=
.
(9.154)
πη
We saw that the single-particle motion was either delocalized or localized depending on the strength of the friction coefficient γ. The critical
value γc at which the cosine interaction is marginal, to first non-trivial
order in a perturbative RG analysis, is given by [recall Eq. (9.147b)]
1
1 g 1
⇐⇒
1+ c =
⇐⇒ gc = 0.
(9.155)
γc =
2π
4π
π
4π
When the friction coefficient γ is larger than the critical value γc , the
cosine interaction is relevant, the particle is localized in a minimum
of the impurity potential, and translation symmetry is broken. When
the friction coefficient γ is smaller than the critical value γc , the cosine interaction is irrelevant, and the particle is in a Bloch state that
preserves the periodicity of the action. From the point of view of the
Thirring model (9.2) (and of the underlying tight-binding electronic
model), the critical value γc corresponds to the free fermionic point
9.4. PROBLEMS
549
gc = 0, γ > γc to a repulsive current-current interaction, and γ < γc
to an attractive current-current interaction. Here, the interpretation
of g > 0 (g < 0) in terms of a repulsive (attractive) interaction follows
from the identification of g with 4Jz /|J⊥ | in the fermionized lattice xxz
chain model [see Eqs. (9.2) (9.109), (9.138), and (9.139)]. Finally, we
can use the Abelian bosonization rules (9.107) to infer that a change
of θ(τ ) by 2π corresponds to the transmission of one electron (in its
spinor incarnation) through the impurity site. The localized nature of
θ(τ ) when γ > γc means that all incoming electrons on the impurity
site are reflected, i.e., total reflection by the impurity. The delocalized
nature of θ(τ ) when γ < γc means that all incoming electrons on the
impurity site are transmitted, i.e., total transmission by the impurity.
A deviation from total reflection or total transmission can only occur
at the critical value of the friction coefficient γc when the impurity potential is exactly marginal. In other words, partial transmission and
partial reflection by the impurity can only occur when the electrons are
non-interacting. The transmission and reflection probabilities depend
on V0 when γ = γc and can be computed by elementary means. This
is a unique feature of one-dimensional physics.
9.4. Problems
9.4.1. Quantum chiral edge theory.
Introduction. We have shown in section 9.2.1 that the massless
Thirring model in two-dimensional Euclidean space realizes a line of
critical points labeled by the dimensionless coupling constant of the
current-current interaction. At each critical point, the partition function factorizes into two sectors, a holomorphic and an antiholomorphic
sector, respectively. We then showed in section 9.2.3 that the massless
Thirring model could be rewritten as a free real-valued scalar field theory in two-dimensional Euclidean space through the equivalence (9.81).
We left open the question of how to demonstrate factorization of this
free real-valued scalar field theory into holomorphic and antiholomorphic sectors. We are going to provide a constructive answer to this
question that goes beyond the equivalence (9.81).
We are going to abandon imaginary time and work within the
Hamiltonian formalism of quantum field theory in one-dimensional position space. The factorization into a holomorphic and an antiholomorphic sector becomes a factorization into a right-moving and left-moving
sector. In the right-moving sector, the quantum fields depend exclusively on the linear combination x − vF t of the coordinate x in position
space and the coordinate t in time. In the left-moving sector, the
quantum fields depend exclusively on the linear combination x + vF t.
Whereas in a Lorentz-invariant quantum field theory the identification
of the velocity vF with the speed of light c would hold, as an emerging
550 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
low-energy and long-wavelength theory we interpret vF as the Fermi
velocity from an underlying lattice model.
Right-moving and left-moving quantum fields in one-dimensional
position space differ by their chirality, the choice of sign in their dependence on x±vF t. As we have shown for the Dirac Lagrangian on the
left-hand side of the equivalence (9.81), the massless and neutral KleinGordon Lagrangian on the right-hand side of the equivalence (9.81) will
be shown to decompose additively into a right-moving and a left moving
sector.
However, we will show that it is also possible to define quantum
field theories in one-dimensional position space with unequal numbers
of fields in the left-moving and right-moving sectors. These so-called
chiral quantum field theories cannot emerge from lattice fermions defined on a one-dimensional lattice due to the fermion-doubling obstruction (the Nielsen-Ninomiya no-go theorem from Ref. [115]). They can
emerge on the boundary of a two-dimensional lattice model and play
an essential role in the quantum Hall effect as emphasized by Halperin
in Ref. [61] for the IQHE and by Wen in Ref. [116] for the FQHE.
As a byproduct, we will obtain a representation of right-moving and
left-moving fermions in terms of right-moving and left-moving bosons,
respectively, that goes back to Mandelstam. [109]
Definition. Define the quantum Hamiltonian (in units with the
positron charge e > 0, the speed of light c, and ~ set to one)
ZL
Ĥ[Aµ (t, x)] :=
q
1
i
−1
dx
V (D û ) Dx ûj + A0
K
Dx ûj
(t, x),
4π ij x i
2π ij
0
Dx ûi := (∂x ûi + qi A1 ) ,
i = 1, · · · , N.
(9.156a)
The summation convention over repeated indices is implied throughout.
The N Hermitean quantum fields ûi (t, x) are postulated to obey the
equal-time commutation relations 8
ûi (t, x), ûj (t, y) := iπ Kij sgn(x − y) + Lij ,
i, j = 1, · · · , N.
(9.156b)
The function sgn(x) = −sgn(−x) gives the sign of the real variable x
and will be assumed to be periodic with periodicity L. The N × N
matrix K is integer-valued, symmetric, and invertible
Kij = Kji ∈ Z,
8
Kij−1 = Kji−1 ∈ Q,
i, j = 1, · · · , N.
(9.156c)
As we shall see, this is the algebra to be imposed on the phase operator
[recall Eq. (7.11)] of creation and annihilation operators if they are to obey the
canonical commutation relations of quantum fields. This interpretation also justifies
the definition of the covariant derivative (9.156a).
9.4. PROBLEMS
The N × N matrix L is antisymmetric

 0,
Lij = −Lji =

sgn(i − j) Kij + qi qj ,
551
if i = j,
(9.156d)
otherwise,
for i, j = 1, · · · , N . The sign function sgn(i) of any integer i is here
not made periodic and taken to vanish at the origin of Z. The external
scalar gauge potential A0 (t, x) and vector gauge potential A1 (t, x) are
real-valued functions of time t and space x coordinates. The N × N
matrix V is symmetric and positive definite
Vij = Vji ∈ R,
i, j = 1, · · · , N,
vi Vij vj > 0,
(9.156e)
for any non-vanishing vector v = (vi ) ∈ RN . The charges qi are integer
valued and satisfy
(−1)Kii = (−1)qi ,
i = 1, · · · , N.
(9.156f)
Finally, we shall impose the boundary conditions
ûi (t, x + L) = ûi (t, x) + 2πni ,
ni ∈ Z,
(9.156g)
and
(∂x ûi ) (t, x + L) = (∂x ûi ) (t, x),
(9.156h)
for any i = 1, · · · , N .
Chiral equations of motion. Exercise 1.1: We set A0 = A1 = 0.
(a) Show that, for any i = 1, · · · , N , the equations of motion are
i (∂t ûi ) (t, x) = − iKij Vjk (∂x ûk ) (t, x).
(b) The equation of motion
0 = δik ∂t + Kij Vjk ∂x ûk ,
i = 1, · · · , N,
(9.157)
(9.158)
is chiral. Show that if we define a Hamiltonian of the form
Eq. (9.156) with the substitution ûi → v̂i and if we change the
sign of the right-hand side of Eq. (9.156b), we then find the
chiral equation
0 = δik ∂t − Kij Vjk ∂x v̂k ,
i = 1, · · · , N,
(9.159)
with the opposite chirality.
Gauge invariance. Exercise 2.1: Verify that Hamiltonian (9.156a)
is invariant under the local U (1) gauge transformation
A0 (t, x) = A00 (t, x),
A1 (t, x) = A01 (t, x) − (∂x χ) (t, x),
ûi (t, x) =
û0i (t, x)
+ qi χ(t, x),
(9.160a)
i = 1, · · · , N.
for any real-valued function χ that satisfies the periodic boundary conditions
χ(t, x + L) = χ(t, x).
(9.160b)
552 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
Differentiation of Hamiltonian (9.156a) with respect to the gauge
potentials allows to define the gauge-invariant two-current with the
components
δ Ĥ
δA0 (t, x)
1
= qi Kij−1 Dx ûj (t, x)
2π
Jˆ0 (t, x) :=
(9.161a)
and
δ Ĥ
δA1 (t, x)
(9.161b)
1
1
−1
q K q A0 (t, x).
= qi Vij Dx ûj (t, x) +
2π
2π i ij j
Exercise 2.2: Verify that the two components of this gauge-invariant
two-current reduce to
1
ρ̂(t, x) :=
qi Kij−1 ∂x ûj (t, x)
(9.162a)
2π
and
1
qi Vij ∂x ûj (t, x)
(9.162b)
ĵ(t, x) :=
2π
when the external gauge fields vanish.
Exercise 2.3: Verify that
∂ µ Jˆ ≡ ∂ Jˆ + ∂ Jˆ
Jˆ1 (t, x) :=
µ
t 0
x 1
1
qi Kij−1 qj (∂t A1 )
(9.163)
2π
1
1
qi Vij qj (∂x A1 ) +
qi Kij−1 qj (∂x A0 ) .
+
2π
2π
We recall that the magnetic and electric fields are related to the gauge
fields by (remember that c = 1)
= ∂t ρ̂ + ∂x ĵ +
B = ∇ ∧ A,
E = −∇A0 − ∂t A
(9.164)
in d-dimensional position space. We also recall that the constraints
∂t A0 + ∇ · A = 0
(9.165)
and
∇·A=0
(9.166)
are called the Lorenz and Coulomb gauges, respectively. [117]
Exercise 2.4: Show that, for the one-dimensional chiral edge in
the Coulomb gauge,
1
(9.167a)
qi Kij−1 qj E,
∂ µ Jˆµ = ∂t ρ̂ + ∂x ĵ −
2π
where
E(t, x) = − (∂x A0 ) (t, x) − (∂t A1 ) (t, x)
(9.167b)
9.4. PROBLEMS
553
and
0 = (∂x A1 ) (t, x).
(9.167c)
Conserved topological charges. We turn off the external gauge potentials
A0 (t, x) = A1 (t, x) = 0
(9.168a)
and use the short-hand notation
Ĥ ≡ Ĥ[Aµ (t, x) = 0].
For any i = 1, · · · , N , define the operator
ZL
1
N̂i (t) :=
dx (∂x ûi ) (t, x)
2π
0
=
1
[û (t, L) − ûi (t, 0)] .
2π i
(9.168b)
(9.169)
554 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
Exercise 3.1:
(a) Show that N̂i (t) is conserved (i.e., time independent) if and
only if
(∂x ûi ) (t, x) = (∂x ûi ) (t, x + L),
0 ≤ x ≤ L.
(9.170)
(b) Show that, if we demand that there exists an ni ∈ Z such that
ûi (t, x + L) = ûi (t, x) + 2πni ,
(9.171)
it then follows that
N̂i = ni .
(9.172)
(c) Show that the N conserved topological charges Ni with i =
1, · · · , N commute pairwise. Hint: Make us of the fact that,
for any i, j = 1, · · · , N
h
i
N̂i , ûj (y) = iKij
(9.173)
is independent of y.
The local counterpart to the global conservation of the topological
charge is
+ ∂x ĵitop = 0,
(9.174a)
∂t ρ̂top
i
where the local topological density operator is defined by
1
ρ̂top
(∂ û ) (t, x)
(9.174b)
i (t, x) :=
2π x i
and the local topological current operator is defined by
1
ĵitop (t, x) :=
K V (∂ û ) (t, x)
(9.174c)
2π ik kl x l
for i = 1, · · · , N .
Exercise 3.2: Verify the equal-time current algebra
top
i
ρ̂i (t, x), ρ̂top
K ∂ δ(x − y),
(9.175a)
j (t, y) = −
2π ij x
i
h
i
ĵitop (t, x), ĵjtop (t, y) = − Kik Vkl Kjk0 Vk0 l0 Kll0 ∂x δ(x −(9.175b)
y),
2π
h
i
i
top
ρ̂top
(t,
x),
ĵ
(t,
y)
= − Kjk Vkl Kil ∂x δ(x − y),
(9.175c)
i
j
2π
for any i, j = 1, · · · , N .
We also introduce the local charges and currents
ρ̂i (t, x) := Kij−1 ρ̂top
j (t, x)
(9.176a)
and
ĵi (t, x) := Kij−1 ĵjtop (t, x),
(9.176b)
respectively, for any i = 1, · · · , N . The continuity equation (9.174a) is
unchanged under this linear transformation,
∂t ρ̂i + ∂x ĵi = 0,
(9.176c)
9.4. PROBLEMS
555
for any i = 1, · · · , N . The topological current algebra (9.175) transforms into
i
ρ̂i (t, x), ρ̂j (t, y) = − Kij−1 ∂x δ(x − y),
(9.177a)
2π
h
i
i
ĵi (t, x), ĵj (t, y) = − Vik Vjl Kkl ∂x δ(x − y), (9.177b)
2π
h
i
i
ρ̂i (t, x), ĵj (t, y) = − Vij ∂x δ(x − y),
(9.177c)
2π
for any i, j = 1, · · · , N .
At last, if we contract the continuity equation (9.176c) with the
integer-valued charge vector, we obtain the flavor-global continuity
equation [compare with Eq. (9.163)]
∂t ρ̂ + ∂x ĵ = 0,
(9.178a)
where the local flavor-global charge operator is [compare with Eq. (9.162a)]
ρ̂(t, x) := qi Kij−1 ρ̂top
j (t, x)
(9.178b)
and the local flavor-global current operator is [compare with Eq. (9.162b)]
ĵ(t, x) := qi Kij−1 ĵjtop (t, x).
(9.178c)
The flavor-resolved current algebra (9.177) turns into the flavor-global
current algebra
i
qi Kij−1 qj ∂x δ(x − y),
(9.179a)
[ρ̂(t, x), ρ̂(t, y)] = −
2π
h
i
i
ĵ(t, x), ĵ(t, y) = −
qi Vik Kkl Vlj qj ∂x δ(x − y),(9.179b)
2π
h
i
i
ρ̂(t, x), ĵ(t, y) = −
qi Vij qj ∂x δ(x − y).
(9.179c)
2π
Quasiparticle and electronic excitations. When Eq. (9.168a) holds,
there exist N conserved global topological (i.e., integer valued) charges
N̂i with i = 1, · · · , N defined in Eq. (9.169) that commute pairwise.
Define the N global charges
ZL
Q̂i := dx ρ̂i (t, x) = Kij−1 N̂j ,
i = 1, · · · , N.
(9.180)
0
We shall shortly interpret these charges as the elementary Fermi-Bose
charges.
Define for any i = 1, · · · , N the pair of vertex operators
−1
Ψ̂†q-p,i (t, x) := e−iKij
ûj (t,x)
(9.181a)
and
Ψ̂†f-b,i (t, x) := e−iδij ûj (t,x) ,
(9.181b)
respectively. The quasiparticle vertex operator Ψ̂†q-p,i (t, x) is multivalued under a shift by 2π of all ûj (t, x) with j = 1, · · · , N . The
556 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
Fermi-Bose vertex operator Ψ̂†f-b,i (t, x) is single valued under a shift by
2π of all ûj (t, x) with j = 1, · · · , N .
Exercise 4.1: Verify that, for any pair i, j = 1, · · · , N , the commutator (9.173) delivers the identities
h
i
h
i
N̂i , Ψ̂†q-p,j (t, x) = δij Ψ̂†q-p,j (t, x),
N̂i , Ψ̂†f-b,j (t, x) = Kij Ψ̂†f-b,j (t, x),
(9.182a)
and
h
i
h
i
Q̂i , Ψ̂†q-p,j (t, x) = Kij−1 Ψ̂†q-p,j (t, x),
Q̂i , Ψ̂†f-b,j (t, x) = δij Ψ̂†f-b,j (t, x),
(9.182b)
respectively. The quasiparticle vertex operator Ψ̂†q-p,i (t, x) is an eigenstate of the topological number operator N̂i with eigenvalue one. The
Fermi-Bose vertex operator Ψ̂†f-b,i (t, x) is an eigenstate of the charge
number operator Q̂i with eigenvalue one.
The Baker-Campbell-Hausdorff formula implies that
e eB̂ = eÂ+B̂ e+(1/2)[Â,B̂] = eB̂ e e[Â,B̂]
(9.183)
whenever two operators  and B̂ have a C-number as their commutator.
Exercise 4.2: Show that a first application of the Baker-CampbellHausdorff formula to any pair of quasiparticle vertex operators at equal
time t but two distinct space coordinates x 6= y gives
Ψ̂†q-p,i (t, x) Ψ̂†q-p,j (t, y) =
 †
−1
−1 −1
−1 −1
†
−iπ [Kii
sgn(x−y)+(Kik
Kil Kkl +qk Kik
Kil ql )sgn(k−l)]

,
Ψ̂q-p,i (t, y) Ψ̂q-p,i (t, x) e
if i = j,

−1
−1 −1
−1 −1
 †
Ψ̂q-p,j (t, y) Ψ̂†q-p,i (t, x) e−iπ [Kji sgn(x−y)+(Kik Kjl Kkl +qk Kik Kjl ql )sgn(k−l)] , if i 6= j.
(9.184)
Here and below, it is understood that
sgn(k − l) = 0
(9.185)
when k = l = 1, · · · , N . Argue that the quasiparticle vertex operators
obey neither bosonic nor fermionic statistics whenever det K 6= ±1.
Hint: Kij−1 ∈ Q has rational matrix elements.
Exercise 4.3: Show that the same exercise applied to the FermiBose vertex operators yields

†
†
K

if i = j,
(−1) ii Ψ̂f-b,i (t, y) Ψ̂f-b,i (t, x),
†
†
Ψ̂f-b,i (t, x) Ψ̂f-b,j (t, y) =

(−1)qi qj Ψ̂† (t, y) Ψ̂† (t, x), if i 6= j,
f-b,j
f-b,i
(9.186)
when x 6= y. The self statistics of the Fermi-Bose vertex operators is
carried by the diagonal matrix elements Kii ∈ Z. The mutual statistics
9.4. PROBLEMS
557
of any pair of Fermi-Bose vertex operators labeled by i 6= j is carried
by the product qi qj ∈ Z of the integer-valued charges qi and qj . Had
we not assumed that Kij with i 6= j are integers, the mutual statistics
would not be Fermi-Bose because of the non-local term Kij sgn (x − y).
Exercise 4.4: Show that a third application of the Baker-CampbellHausdorff formula allows to determine the boundary conditions
−1
Ψ̂†q-p,i (t, x + L) = Ψ̂†q-p,i (t, x) e−2πi Kij
N̂i
−1
e−πi Kii
(9.187)
and
Ψ̂†f-b,i (t, x + L) = Ψ̂†f-b,i (t, x) e−2πi N̂i e−πi Kii
(9.188)
obeyed by the quasiparticle and Fermi-Bose vertex operators, respectively.
We close this discussion with the following definitions. Introduce
the operators
−1
Ψ̂†f-b,m := e−imi δij ûj (t,x) ,
(9.189)
N
where m ∈ Z is the vector with the integer-valued components mi
for any i = 1, · · · , N . The N charges qi with i = 1, · · · , N that enter
Hamiltonian (9.156a) can also be viewed as the components of the
vector q ∈ ZN . Define the functions
Ψ̂†q-p,m := e−imi Kij
Q̂ := qi Q̂i ,
ûj (t,x)
,
q : ZN −→ Z
(9.190a)
m 7−→ q(m) := qi mi ≡ q · m
and
K : ZN −→ Z
(9.190b)
m 7−→ K(m) := mi Kij mj
On the one hand, for any distinct pair of space coordinates x 6= y, we
deduce from Eqs. (9.182b), (9.184), and (9.187) that
h
i
Q̂, Ψ̂†q-p,m (t, x) = qi Kij−1 mj Ψ̂†q-p,m (t, x),
Ψ̂†q-p,m (t, x) Ψ̂†q-p,n (t, y) = Ψ̂†q-p,n (t, y) Ψ̂†q-p,m (t, x)
−1
× e−iπ [mi Kij
−1
−1
−1
−1
nj sgn(x−y)+(mi Kik
Kkl Klj
nj +qk Kki
mi nj Kjl
ql )sgn(k−l)]
Ψ̂†q-p,m (t, x + L) = Ψ̂†q-p,m (t, x) e
−1
−2πi mi Kij
N̂j
e
−1
−πi mi Kij
mj
,
,
(9.191)
558 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
respectively. On the other hand, for any distinct pair of space coordinates x 6= y, we deduce from Eqs. (9.182b), (9.186), and (9.188) that
h
i
Q̂, Ψ̂†f-b,m (t, x) = q(m) Ψ̂†f-b,m (t, x),
Ψ̂†f-b,m (t, x) Ψ̂†f-b,n (t, y) = Ψ̂†f-b,n (t, y) Ψ̂†f-b,m (t, x)
× e−iπ [mi Kij nj sgn(x−y)+mi (Kij +qi qj )nj sgn(i−j)] ,
Ψ̂†f-b,m (t, x + L) = Ψ̂†f-b,m (t, x) e−2πi mi N̂i e−πi mi Kij mj ,
(9.192)
respectively.
Exercise 4.5: The integer quadratic form K(m) is thus seen to
dictate whether the vertex operator Ψ̂†f-b,m (t, x) realizes a fermion or a
boson. The vertex operator Ψ̂†f-b,m (t, x) realizes a fermion if and only
if
K(m) is an odd integer
(9.193)
or a boson if and only if
K(m) is an even integer.
(9.194)
Show that, because of assumption (9.156f),
(−1)K(m) = (−1)q(m) .
(9.195)
Hence, the vertex operator Ψ̂†f-b,m (t, x) realizes a fermion if and only if
q(m) is an odd integer
(9.196)
or a boson if and only if
q(m) is an even integer.
(9.197)
From the Hamiltonian to the Lagrangian formalism. What is the
Minkowski path integral that is equivalent either to the quantum theory defined by Eq. (9.156) or to the quantum theory defined with the
opposite chirality as is explained in exercise 1.1(b)? The label (+) will
be associated to the choice of chirality made in Eq. (9.158), the label
(−) to the choice of chirality made in Eq. (9.159). In other words, we
seek the path integrals
Z
(±)
(±)
Z := D[φ] eiS [φ]
(9.198a)
with the Minkowski action
S
(±)
Z+∞
Z+∞ ZL
(±)
[φ] :=
dt L [φ] ≡
dt dx L(±) [φ](t, x)
−∞
−∞
0
(9.198b)
9.4. PROBLEMS
such that one of the two Hamiltonians
ZL h
i
(±)
(±)
H := dx Πi (∂t φi ) − L(±) [φ]
559
(9.199)
0
can be identified with Ĥ in Eq. (9.156a) after elevating the classical
fields
φi (t, x)
(9.200a)
and
δL(±)
(±)
(9.200b)
Πi (t, x) :=
δ(∂t φi )(t, x)
(±)
entering L(±) [φ] to the status of quantum fields φ̂i (t, x) and Π̂j (t, y)
upon imposing the equal-time commutation relations
h
i
i
(±)
φ̂i (t, x), Π̂j (t, y) = δij δ(x − y)
(9.200c)
2
for any i, j = 1, · · · , N . The unusual factor 1/2 (instead of 1) on
the right-hand side of the commutator between pairs of canonically
conjugate fields arises because each real-valued scalar field φi with i =
1, · · · , N is chiral, i.e., it represents “one-half” of a canonical realvalued scalar field.
Exercise 5.1: We try
1 L(±) :=
∓ (∂x φi ) Kij−1 ∂t φj − (∂x φi ) Vij ∂x φj .
(9.201a)
4π
Show that there follows the chiral equations of motion
δL(±) δL(±)
−
δ∂µ φi
δφi
(9.201b)
Kij−1 =∓
∂ δ ∂ ± Kjk Vkl ∂x φl
2π x il t
for any i = 1, · · · , N .
Exercise 5.2: Show that it is only the term that mixes time t and
space x derivatives that becomes imaginary upon analytical continuation from real time t to Euclidean time τ = it.
We need to verify that the Hamiltonian density that follows from
the Lagrangian density (9.201a) is, upon quantization, Eq. (9.156) with
the gauge fields set to zero.
(±)
Exercise 5.3: The canonical momentum Πi to the field φi is
0 = ∂µ
(±)
Πi (t, x) :=
1
δL(±)
= ∓ Kij−1 ∂x φj (t, x)
δ(∂t φi )(t, x)
4π
(9.202)
for any i = 1, · · · , N owing to the symmetry of the matrix K. Show
that the Legendre transform
(±)
H(±) := Πi
(∂t φi ) − L(±)
(9.203)
560 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
delivers
1
(∂x φi ) Vij ∂x φj .
(9.204)
4π
The right-hand side does not depend on the chiral index (±).
Exercise 5.4: We now quantize the theory by elevating the classical fields φi to the status of operators φ̂i obeying either the algebra (9.156b) for the choice of chirality (+) or the one with a minus
sign on the right-hand side of Eq. (9.156b) for the choice of chirality
(−). Show that this gives a quantum theory that meets all the demands
of the quantum chiral edge theory (9.156) in all compatibility with the
canonical quantization rules (9.200c), for
h
i
i
(±)
(9.205)
φ̂i (t, x), Π̂j (t, y) = δij δ(x − y)
2
where i, j = 1, · · · , N .
Finally, analytical continuation to Euclidean time
H(±) =
τ = it
(9.206a)
allows to define the finite temperature quantum chiral theory through
the path integral
Z
(±)
(±)
Zβ := D[φ] e−S [φ] ,
S
(±)
Zβ
[φ] :=
ZL
dτ
0
dx
1 (±)i (∂x φi ) Kij−1 ∂τ φj + (∂x φi ) Vij ∂x φj .
4π
0
(9.206b)
9.4.2. Two-point correlation function in the massless Thirring
model.
Introduction. We are going to derive the two-point function (F.47)
for the massless Thirring model, a relativistic quantum field theory in
(1+1)-dimensional position space and time, that we choose to represent
through Eq. (9.210). We work in units of ~ = 1 and c = 1 (or vF =
1 if the Lorentz invariance is an emergent one at low energies and
long wavelength of some underlying lattice model). The short-distance
cutoff a is also set to one.
A pair of freely counterpropagating chiral bosons. Define the bosonic
Hamiltonian
!
ZL
1
dx
(t, x).
∂x φ̂− ∂x φ̂− + ∂x φ̂+ ∂x φ̂+
Ĥ0 :=
4π
0
(9.207a)
The dependence on t on the right-hand side refers to the Heisenberg
picture. Of course, Ĥ0 is conserved in time. The 2 Hermitean quantum
9.4. PROBLEMS
561
fields φ̂i (t, x) with i = 1, 2 ≡ −, + are postulated to obey the equal-time
commutation relations
h
i
φ̂− (t, x), φ̂− (t, y) := +iπ sgn(x − y),
h
i
(9.207b)
φ̂+ (t, x), φ̂+ (t, y) := −iπ sgn(x − y),
h
i
h
i
φ̂− (t, x), φ̂+ (t, y) = − φ̂+ (t, x), φ̂− (t, y) := −iπ.
The function sgn(x) = −sgn(−x) gives the sign of the real variable x
and will be assumed to be periodic with periodicity L. We shall impose
the boundary conditions
ni ∈ Z,
(9.207c)
∂x φ̂i (t, x + L) = ∂x φ̂i (t, x),
(9.207d)
φ̂i (t, x + L) = φ̂i (t, x) + 2πni ,
and
for any i = 1, 2 ≡ −, +.
Exercise 1.1:
(a) What are the matrices K, V , L, and the charge vector q defined in Eq. (9.156) that deliver Eq. (9.207)?
(b) Show that
(9.208a)
∂t φ̂− = −∂x φ̂−
and
∂t φ̂+ = +∂x φ̂+ .
(9.208b)
Hence, φ̂− is a right-moving bosonic field, while φ̂+ is a leftmoving bosonic field.
(c) Define the pair of operators
1
∂ φ̂ .
(9.209a)
Jˆ∓ :=
2π x ∓
Show that they obey periodic boundary conditions under x →
x + L, satisfy the equal-time (Schwinger) algebra
h
i
i
ˆ
ˆ
J− (t, x), J− (t, y) = − ∂x δ(x − y),
2π
h
i
i
(9.209b)
Jˆ+ (t, x), Jˆ+ (t, y) = + ∂x δ(x − y),
2π
h
i h
i
Jˆ− (t, x), Jˆ+ (t, y) = Jˆ+ (t, x), Jˆ− (t, y) = 0,
and
ZL
Ĥ0 = π
dx Jˆ− Jˆ− + Jˆ+ Jˆ+ (t, x),
0
(∂t ± ∂x ) Jˆ∓ (t, x) = 0.
(9.209c)
562 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
A pair of interacting counterpropagating chiral bosons. Define the
interacting Hamiltonian
Ĥ := Ĥ0 + Ĥ1 (δv) + Ĥ2 (λ)
where
ZL
Ĥ0 = π
(9.210a)
dx Jˆ− Jˆ− + Jˆ+ Jˆ+ (t, x)
(9.210b)
dx Jˆ− Jˆ− + Jˆ+ Jˆ+ (t, x)
(9.210c)
0
was defined in Eq. (9.209),
ZL
Ĥ1 (δv) = π δv
0
renormalizes additively the coefficient (the bare energy scale π ~ vF /a
if we reinstate all units) of Ĥ0 by the dimensionless real-valued number
π δv, and
ZL
Ĥ2 (λ) = λ dx Jˆ− Jˆ+ (t, x)
(9.210d)
0
mixes left with right movers for any real-valued and non-vanishing λ.
Exercise 2.1: Define the one-parameter family of currents
Jˆ−θ := cosh θ Jˆ− + sinh θ Jˆ+ ,
(9.211)
Jˆ+θ := sinh θ Jˆ− + cosh θ Jˆ+ ,
with the label θ ∈ R. Fix θ to the value
λ
1
.
(9.212)
θ̄ := arctanh
2
2π(1 + δv)
Define the finite real number
1 + δv
v̄ :=
.
(9.213)
cosh 2θ̄
Show that the currents Jˆ∓θ̄ satisfy periodic boundary conditions under
x → x + L, obey the equal-time (Schwinger) algebra
h
i
i
Jˆ−θ̄ (t, x), Jˆ−θ̄ (t, y) = − ∂x δ(x − y),
2π
h
i
i
(9.214a)
Jˆ+θ̄ (t, x), Jˆ+θ̄ (t, y) = + ∂x δ(x − y),
2π
h
i h
i
Jˆ−θ̄ (t, x), Jˆ+θ̄ (t, y) = Jˆ+θ̄ (t, x), Jˆ−θ̄ (t, y) = 0,
and
ZL
Ĥ = π v̄
dx Jˆ−θ̄ Jˆ−θ̄ + Jˆ+θ̄ Jˆ+θ̄ (t, x),
0
(∂t ± v̄ ∂x ) Jˆ∓θ̄ (t, x) = 0.
(9.214b)
9.4. PROBLEMS
563
Diagonalization of Ĥ ≡ Ĥ θ̄ . We are going to diagonalize
Ĥ0θ̄ := π v̄
ZL
dx Jˆ−θ̄ Jˆ−θ̄ + Jˆ+θ̄ Jˆ+θ̄ (t, x),
(9.215a)
0
where the currents obey the equal-time (Schwinger) algebra
i
i
Jˆ−θ̄ (t, x), Jˆ−θ̄ (t, y) = − ∂x δ(x − y),
2π
i
h
i
Jˆ+θ̄ (t, x), Jˆ+θ̄ (t, y) = + ∂x δ(x − y),
h
i h 2π
i
θ̄
θ̄
θ̄
θ̄
ˆ
ˆ
ˆ
ˆ
J− (t, x), J+ (t, y) = J+ (t, x), J− (t, y) = 0,
h
(9.215b)
and periodic boundary conditions under x → x + L. To make contact
to the representation (9.207), we also define
1 Jˆ∓θ̄ (t, x) =:
∂x φ̂θ̄∓ (t, x).
2π
(9.215c)
We observe that the Schwinger algebra (9.215) is very close to the
equal-time algebra obeyed by two Hermitean (field-valued) operators
ϕ̂θ̄− and ϕ̂θ̄+ and their canonical momenta Π̂θ̄− and Π̂θ̄+ , namely
h
i
ϕ̂θ̄− (t, x), Π̂θ̄− (t, y) = iδ(x − y),
h
i
ϕ̂θ̄+ (t, x), Π̂θ̄+ (t, y) = iδ(x − y),
h
i h
i
ϕ̂θ̄− (t, x), Π̂θ̄+ (t, y) = ϕ̂θ̄+ (t, x), Π̂θ̄− (t, y) = 0.
(9.216)
We seek to relate the pair (Jˆiθ̄ , Jˆiθ̄ ) to the pair of raising and lowering
operators associated to (ϕ̂θ̄i , Π̂θ̄i ) for i = −, +. This can be done in
momentum space.
564 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
Exercise 3.1: We do the Fourier expansions
φ̂θ̄i (t, x)
1 X +ip x θ̄
=√
φ̂i p (t),
e
L Lp
2π
∈Z
1 X +ip x ˆθ̄
Jˆiθ̄ (t, x) = √
e
Ji p (t),
L Lp
2π
ϕ̂θ̄i (t, x)
φ̂θ̄i p (t)
∈Z
1 X +ip x θ̄
=√
e
ϕ̂i p (t),
L Lp
2π
∈Z
1 X +ip x θ̄
Π̂θ̄i (t, x) = √
e
Π̂i p (t),
L Lp
2π
∈Z
1
=√
L
1
Jˆiθ̄p (t) = √
L
ϕ̂θ̄i p (t)
ZL
dx e−ip x φ̂θ̄i (t, x),
0
ZL
dx e−ip x Jˆiθ̄ (t, x),
0
1
=√
L
ZL
1
Π̂θ̄i p (t) = √
L
dx e−ip x ϕ̂θ̄i (t, x),
0
ZL
dx e−ip x Π̂θ̄i (t, x),
0
(9.217)
for i = −, +. From now on, any summation over the momenta p is be
understood as the sum over the integer n in p = 2π n/L.
(a) Verify that
i†
h
θ̄
θ̄
φ̂i (+p) (t) = φ̂i (−p) (t) ,
h
i†
θ̄
θ̄
ˆ
ˆ
Ji (+p) (t) = Ji (−p) (t) ,
(9.218)
h
i†
θ̄
θ̄
ϕ̂i (+p) (t) = ϕ̂i (−p) (t) ,
h
i†
θ̄
θ̄
Π̂i (+p) (t) = Π̂i (−p) (t) ,
for i = −, +.
(b) Show that
h
i
p
Jˆ−θ̄ (+p) (t), Jˆ−θ̄ (+p0 ) (t) = + δp,−p0 ,
2π
h
i
p
θ̄
θ̄
ˆ
ˆ
J+ (+p) (t), J+ (+p0 ) (t) = − δp,−p0 ,
h
i h 2π
i
θ̄
θ̄
θ̄
θ̄
ˆ
ˆ
ˆ
ˆ
J− (+p) (t), J+ (+p0 ) (t) = J+ (+p) (t), J− (+p0 ) (t) = 0,
(9.219)
whereas
h
i
ϕ̂θ̄i p (t), ϕ̂θ̄j p0 (t) = 0,
h
i
θ̄
θ̄
Π̂i p (t), Π̂j p0 (t) = 0,
h
i
ϕ̂θ̄i p (t), Π̂θ̄j p0 (t) = i δi,j δp,−p0 ,
for i, j = −, +.
(9.220)
9.4. PROBLEMS
565
(c) Verify that if we invert the relation
ip θ̄
Jˆiθ̄(+p) (t) =
φ̂
(t)
2π i (+p)
(9.221a)
for any p = 2π n/L with n ∈ Z \ {0} according to
r π θ̄
θ̄
φ̂− (t, x) =
b̂0 + ib̂θ̄†
0
2
i
1 X − p/2 1 h +ip x ˆθ̄
e
J− (+p) − e−ip x Jˆ−θ̄ (−p) ,
e
+ lim+ 2π √
→0
ip
L p>0
r π θ̄
θ̄
φ̂+ (t, x) =
b̂0 − ib̂θ̄†
0
2
i
1 X − p/2 1 h +ip x ˆθ̄
−ip x ˆθ̄
e
e
J+ (+p) − e
J+ (−p) ,
+ lim+ 2π √
→0
ip
L p>0
(9.221b)
for i = −, +, we then recover Eq. (9.207b) in the limit L → ∞
provided the only non-vanishing commutator
i
1 h θ̄
φ̂− 0 (t), φ̂θ̄+ 0 (t) = −iπ
(9.221c)
L
of the zero-mode operators
r r 1 θ̄
π θ̄
1 θ̄
π θ̄
θ̄†
θ̄†
√ φ̂− 0 (t) =:
√ φ̂+ 0 (t) =:
b̂ + ib̂0 ,
b̂ − ib̂0 ,
2 0
2 0
L
L
(9.221d)
follows from the zero-mode algebra
i
h
i
h
i h
θ̄† θ̄†
θ̄ θ̄
θ̄ θ̄†
(9.221e)
b̂0 , b̂0 = b̂0 , b̂0 = 0.
b̂0 , b̂0 = 1,
Hint: Use the integral representation
2
sgn(x) = lim+
→0 π
Z∞
dp e− p
sin p x
.
p
(9.222)
0
(d) Verify that
Ĥ0θ̄ = π v̄
X X
Jˆiθ̄(+p) Jˆiθ̄(−p) + Jˆiθ̄(−p) Jˆiθ̄(+p) (t),
(9.223)
i=−,+ p>0
whereby one must also explain why the contribution Jˆiθ̄0 Jˆiθ̄0 (t)
is not accounted for.
566 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
Exercise 3.2: From now on, p = 2π n/L is strictly positive, i.e.,
n = 1, 2, 3, · · · . Inspection of exercise 3.1(d) and exercise 3.1(b) suggests that we define the operators
r
r
p θ̄
p θ̄ †
θ̄
θ̄
ˆ
ˆ
b̂− (+p) (t),
J− (−p) (t) =:
b̂
J− (+p) (t) =:
(t),
2π
2π − (+p)
(9.224a)
and
r
r
p
p θ̄
θ̄†
b̂+ (+p) (t),
Jˆ+θ̄ (−p) (t) =:
b̂
(t).
Jˆ+θ̄ (+p) (t) =:
2π
2π + (+p)
(9.224b)
(a) Verify that the only possible non-vanishing commutators originate from
h
i
L p L p0
b̂θ̄i (+p) (t), b̂θ̄†
(t)
=
δ
δ
,
i,
j
=
−,
+,
,
= 1, 2, 3, · · · .
0
i,j p,p
j (+p0 )
2π 2π
(9.225)
(b) Verify that
v̄ X X θ̄†
Ĥ0θ̄ =
p b̂i (+p) b̂θ̄i (+p) + b̂θ̄i (+p) b̂θ̄†
i (+p) (t)
(9.226)
2 i=−,+ p>0
and show that the ground state of Ĥ0θ̄ is the state |0i annihilated by b̂θ̄i (+p) for i = −, + and L2πp = 1, 2, 3, · · · .
(c) Verify that
r π θ̄
θ̄
φ̂− (t, x) =
b̂0 + ib̂θ̄†
0
2
r
i
−i X − p/2 2π h +ip x θ̄
+ lim+ √
e
e
b̂− (+p) − e−ip x b̂θ̄†
− (+p) ,
→0
p
L p>0
r π θ̄
θ̄†
θ̄
b̂ − ib̂0
φ̂+ (t, x) =
2 0
r
i
−i X − p/2 2π h +ip x θ̄†
+ lim+ √
b̂+ (+p) − e−ip x b̂θ̄+ (+p) .
e
e
→0
p
L p>0
(9.227)
Exercise 3.3: We define the additive decomposition
θ̄+
φ̂θ̄− (t, x) = φ̂θ̄−
− (t, x) + φ̂− (t, x),
(9.228a)
where
r
r
π θ̄
−i X − p/2 2π +ip x θ̄
:= +
b̂ + lim √
e
e
b̂− (+p) ,
2 0 →0+ L p>0
p
r
r
π θ̄†
+i X − p/2 2π −ip x θ̄†
θ̄+
ib̂ + lim+ √
e
e
b̂− (+p) ,
φ̂− (t, x) := +
→0
2 0
p
L p>0
φ̂θ̄−
− (t, x)
(9.228b)
9.4. PROBLEMS
567
in the right-moving sector. Similarly, we define the additive decomposition
θ̄+
φ̂θ̄+ (t, x) = φ̂θ̄−
(9.229)
+ (t, x) + φ̂+ (t, x),
where
r
r
π θ̄
+i X − p/2 2π −ip x θ̄
:= +
b̂+ (+p) ,
b̂ + lim √
e
e
2 0 →0+ L p>0
p
r
r
−i X − p/2 2π +ip x θ̄†
π θ̄†
θ̄+
φ̂+ (t, x) := −
ib̂ + lim+ √
e
b̂+ (+p) ,
e
→0
2 0
p
L p>0
φ̂θ̄−
+ (t, x)
(9.230a)
in the left-moving sector. These additive decompositions are advantageous in that
θ̄+†
h0|φ̂θ̄†
(t, x),
i (t, x) = h0|φ̂i
φ̂θ̄i (t, x)|0i = φ̂θ̄+
i (t, x)|0i,
(9.231)
for i = −, +.
(a) Verify that
i
h
iπ
2π X − p 1 −ip (x−x0 )
θ̄−
0
φ̂θ̄+
(t,
x),
φ̂
(t,
x
)
=
−
−
lim
e
e
. (9.232)
−
−
→0+ L
2
p
p>0
(b) Verify that
h
i
iπ
2π X − p 1 +ip (x−x0 )
θ̄−
0
φ̂θ̄+
(t,
x),
φ̂
(t,
x
)
=
+
−
lim
e
e
. (9.233)
+
+
→0+ L
2
p
p>0
(c) Verify that
i
i h
h
θ̄+
θ̄+
θ̄−
0
0
φ̂
(t,
x),
φ̂
(t,
x
)
= 0,
φ̂θ̄−
(t,
x),
φ̂
(t,
x
)
=
−
+
−
+
i h
i
h
(9.234)
π
θ̄+
0
θ̄+
θ̄−
0
(t,
x),
φ̂
(t,
x
)
=
−i
.
φ̂
φ̂θ̄−
(t,
x),
φ̂
(t,
x
)
=
−
+
−
+
2
(d) Verify that
nh
i h
io
− ij (x − x0 )
θ̄+
θ̄−
θ̄+
θ̄−
0
lim
φ̂j (t, x), φ̂j (t, x ) − φ̂j (t, 0), φ̂j (t, 0) = lim+ log
L→∞
→0
(9.235)
for j = −, +.
(e) Verify that
θ̄+
e−i j φ̂j (t,x) e+i j φ̂j (t,x ) = e+i j (−φ̂j
θ̄
θ̄
0
θ̄+
× e[φ̂j
0
(t,x)+φ̂θ̄+
j (t,x ))
θ̄+
θ̄−
0
(t,0),φ̂θ̄−
j (t,0)]−[φ̂j (t,x ),φ̂j (t,x)]
θ̄−
× e+i j (−φ̂j
(9.236)
0
(t,x)+φ̂θ̄−
j (t,x ))
for j = −, +. Hint: Use twice the Baker-Campbell-Hausdorff
formula (9.183).
568 9. ABELIAN BOSONIZATION IN TWO-DIMENSIONAL SPACE AND TIME
(f) Verify that, at any unequal two points in position space and
in the thermodynamic limit L → ∞,
1 D −i j φ̂θ̄j (t,x) +i j φ̂θ̄j (t,x0 ) E
1
lim+
0 e
e
,
j = −, +.
0 =
→0 ij (x − x0 )
(9.237)
Lorentz covariance then dictates that, at any two unequal
points in position space and time,
−i
1 D +i φ̂θ̄− (t,x) −i φ̂θ̄− (t0 ,x0 ) E
0 e
e
,
lim+
0 =
0
→0 (t − t ) − (x − x0 )
(9.238)
1 D −i φ̂θ̄+ (t,x) +i φ̂θ̄+ (t0 ,x0 ) E
−i
lim
0 e
e
.
0 =
→0+ (t − t0 ) + (x − x0 )
Exercise 3.4: Let |FSi be the Fermi sea obtained from the linear
dispersion relation ε− (p) = +p for the right movers and ε+ (p) = −p
for the left movers when the Fermi energy equals zero, i.e., when the
Fermi point is p = 0. Let n−(p) := Θ(−p) be the occupation number of
all single-particle states with negative momentum and negative energy.
Let n+(p) := Θ(+p) be the occupation number of all single-particle
states with positive momentum and negative energy. Show that, up to
the multiplicative prefactor 1/(2π),
X
1
0
lim+ lim
e−|p|−ip (x−x ) nj(p) ,
j = −, +,
(9.239)
→0 L→∞ L
L p/(2π)∈Z
is nothing but the right-hand side (9.237).
Fermionic two-point functions. Exercise 4.1: We may define the
pair of adjoint vertex operators
r
r
1
1 +iφ̂− (t,x)
() †
()
ψ̂q-p− (t, x) :=
e−iφ̂− (t,x) ,
ψ̂q-p− (t, x) :=
e
,
2π 2π (9.240a)
and
r
r
1 +iφ̂+ (t,x)
1 −iφ̂+ (t,x)
() †
()
e
,
ψ̂q-p+ (t, x) :=
e
.
ψ̂q-p+ (t, x) :=
2π 2π (9.240b)
Verify that these vertex operators obey the fermion algebra.
Exercise 4.2: Alternatively, we may define the pair of adjoint
vertex operators
r
r
1 −iφ̂− (t,x)
1 +iφ̂− (t,x)
() †
()
ψ̂f− (t, x) :=
e
,
ψ̂f− (t, x) :=
e
,
2π 2π (9.241a)
and
r
r
1 −iφ̂+ (t,x)
1 +iφ̂+ (t,x)
() †
()
ψ̂f+ (t, x) :=
e
,
ψ̂f+ (t, x) :=
e
.
2π 2π (9.241b)
9.4. PROBLEMS
569
Verify that these vertex operators also satisfy the fermion algebra.
Exercise 4.3: Define the parameter
g := sinh2 θ̄ ≥ 0.
(9.242)
Show that
E
D ()
() †
0 ψ̂f− (t, x) ψ̂f− (t0 , x0 ) 0 ∝
−i
2g
,
(t − t0 ) − (x − x0 ) [(t − t0 )2 − (x − x0 )2 ]g
E
D −i
2g
() †
()
0 ψ̂f+ (t, x) ψ̂f+ (t0 , x0 ) 0 ∝
,
(t − t0 ) + (x − x0 ) [(t − t0 )2 − (x − x0 )2 ]g
(9.243)
in the ground state of Hamiltonian (9.210). Hint: Invert Eq. (9.211),
i.e.,
Jˆ− = + cosh θ Jˆ−θ − sinh θ Jˆ+θ ,
(9.244)
Jˆ+ = − sinh θ Jˆ−θ + cosh θ Jˆ+θ ,
and make use of Eq. (9.238). Comment on the presence of the factor
2g on the right-hand side of Eq. (9.243). Fourier transformation of
Eq. (9.243) to frequency and momentum space delivers Eq. (F.47).
APPENDIX A
The harmonic-oscillator algebra and its coherent
states
A.1. The harmonic-oscillator algebra and its coherent states
A.1.1. Bosonic algebra. The quantum Hamiltonian for the harmonic oscillator is
1
†
,
(A.1)
Ĥ = ~ω â â +
2
when represented in terms of the lowering (annihilation) and raising
(creation) operators â and ↠, respectively. This pair of operators obeys
the bosonic algebra
[â, ↠] = 1,
[â, â] = [↠, ↠] = 0.
(A.2)
A complete, orthogonal, and normalized basis of Ĥ is given by
1
(↠)n
Ĥ|ni = ~ω n +
|ni,
n = 0, 1, 2, · · · ,
|ni = √ |0i,
2
n!
(A.3)
where the ground state (vacuum) |0i is annihilated by â,
â|0i = 0.
(A.4)
For Ĥ to be Hermitean, annihilation â and creation ↠operators must
be adjoint to each other, i.e., represented by
√
√
â |ni = n |n − 1i,
↠|ni = n + 1 |n + 1i,
(A.5)
√
√
hm|â|ni = n δm+1,n ,
hm|↠|ni = n + 1 δm−1,n .
The single-particle Hilbert space H(1) of twice differentiable and square
integrable functions on the real line for the harmonic oscillator can be
reinterpreted as the Fock space F for the annihilation and creation
operators â and ↠, respectively, since the number operator
N̂ := ↠â
(A.6)
commutes with the Hamiltonian and the Fock space F is, by definition,
the direct sum of the energy eigenspaces:
H(1) ∼
= F :=
∞
M
λ|ni|λ ∈ C .
n=0
571
(A.7)
572
A. THE HARMONIC-OSCILLATOR ALGEBRA AND ITS COHERENT STATES
One possible resolution of the identity 1 on H(1) ∼
= F is
1=
∞
X
|nihn|.
(A.8)
n=0
More informations on the harmonic oscillator can be found in chapter V of Ref. [118].
A.1.2. Coherent states. Define the uncountable set of coherent
states for the harmonic oscillator, in short bosonic coherent states, by
†
|αics := eα â |0i :=
∞
X
αn
√ |ni,
n!
n=0
α ∈ C.
(A.9a)
The adjoint set is (α∗ denotes the complex conjugate of α ∈ C)
â α∗
cs hα| := h0|e
:=
∞
X
(α∗ )n
hn| √ ,
n!
n=0
α ∈ C.
(A.9b)
Properties of bosonic coherent states are:
• Coherent state |αics is a right eigenstate with eigenvalue α of
the annihilation operator â, 1
†
â|αics = â eα â |0i
∞
X
αn
√ â|ni
=
n!
n=0
∞
X αn √
√
=
n|n − 1i
n!
n=1
∞
X
αn−1
p
=α
|n − 1i
(n − 1)!
n=1
(A.10)
= α|αics .
• Coherent state cs hα| is a left eigenstate with eigenvalue α∗ of
the creation operator ↠,
â|αics = α|αics =⇒
1
cs hα|â
†
=
cs hα|α
∗
.
(A.11)
Non-Hermitean operators need not have the same left and right eigenstates.
A.1. THE HARMONIC-OSCILLATOR ALGEBRA AND ITS COHERENT STATES
573
• The action of creation operator ↠on coherent state |αics is
differentiation with respect to α,
†
↠|αics = ↠eα â |0i
∞
X
αn †
√
=
â |ni
n!
n=0
∞
X
αn √
√
n + 1|n + 1i
=
n!
n=0
!
∞
X
d
αn+1
p
=
|n + 1i
dα
(n
+
1)!
n=0
=
d
|αi .
dα cs
• The action of creation operator â on coherent state
differentiation with respect to α∗ ,
↠|αics =
(A.12)
d
|αi =⇒
dα cs
cs hα|â
=
d
dα∗
cs hα|.
cs hα|
is
(A.13)
• The overlap cs hα|βics between two coherent states is exp(α∗ β),
cs hα|βics =
hm|ni = δm,n
=
∞
X
(α∗ )m β n
√ |ni
hm| √
m! n!
m,n=0
∞
X
(α∗ β)n
n=0
α∗ β
=e
(A.14)
n!
.
• There exists a resolution of the identity in terms of bosonic
coherent states,
Z
dz ∗ dz −z∗ z
1=
e
|zics cs hz|
2πi
Z+∞
Z+∞
(A.15)
1
−z ∗ z
dRe z
dIm z e
|zics cs hz|.
:=
π
−∞
−∞
Proof. Write
Z
dz ∗ dz −z∗ z
Ô :=
e
|zics cs hz|.
2πi
(A.16)
By construction, Ô belongs to the algebra of operators generated by â and ↠.
574
A. THE HARMONIC-OSCILLATOR ALGEBRA AND ITS COHERENT STATES
– Step 1: With the help of Eqs. (A.10) and (A.13),
[â, |zics cs hz|] = â|zics cs hz| − |zics cs hz|â
d
= z|zics cs hz| − |zics
hz|
(A.17)
dz ∗ cs
d
= z − ∗ |zics cs hz|.
dz
Hence, after making use of integration by parts,
Z
d
dz ∗ dz −z∗ z
z − ∗ |zics cs hz|
e
[â, Ô] =
2πi
dz
=0.
(A.18)
– Step 2: By taking the adjoint of Eq. (A.18), [↠, Ô] = 0.
– Step 3:
Z
dz ∗ dz −z∗ z
h0|Ô|0i =
e
h0|zics cs hz|0i
2πi
Z
dz ∗ dz −z∗ z
(A.19)
e
hm|ni = δm,n
=
2πi
= 1.
– Step 4: Any linear operator from F to F belongs to the
algebra generated by â and ↠. Since Ô commutes with
both â and ↠by steps 1 and 2, Ô commutes with all
linear operators from F to F. By Schur’s lemma, Ô must
be proportional to the identity operator. By Step 3, the
proportionality factor is 1.
• For any operator â : F → F,
Tr â :=
∞
X
hn|â|ni
n=0
∞
dz ∗ dz −z∗ z X
=
e
hn|zics cs hz|â|ni
2πi
n=0
!
Z
∞
∗
X
dz dz −z∗ z
=
e
|nihn| |zics
cs hz|â
2πi
n=0
Z
dz ∗ dz −z∗ z
=
e
cs hz|â|zics .
2πi
Z
By Eq. (A.15)
By Eq. (A.8)
(A.20)
• Any operator â : F → F is some linear combination of products of â’s and ↠’s. Normal ordering of â, which is denoted
: â :, is the operation of moving all creation operators to the
A.2. PATH-INTEGRAL REPRESENTATION OF THE ANHARMONIC OSCILLATOR
575
left of annihilation operators as if all operators were to commute. For example,
â = ↠ââ↠+ ↠â↠=⇒: â := ↠↠ââ + ↠↠â = â − 2↠â − ↠. (A.21)
The matrix element of any normal ordered operator : â(↠, â) :
between any two coherent states cs hz| and |z 0 ics follows from
Eqs. (A.10), (A.11), and (A.14),
cs hz|
: â(↠, â) : |z 0 ics =
: A(z ∗ , z 0 ) : |z 0 ics = ez
∗ z0
: A(z ∗ , z 0 ) : .
(A.22)
Here, : A(z ∗ , z 0 ) : is the complex-valued function obtained from
the normal ordered operator : â(↠, â) : by substituting ↠for
the complex number z ∗ and â for the complex number z 0 .
• Define the continuous family of unitary operators
cs hz|
† −α∗ â
D(α) := eαâ
From Glauber formula
α ∈ C.
,
(A.23)
2
|α|2
2
e+αâ e−α â ,
D(α)|0i = e−
|α|2
2
e+αâ e−α â |0i
= e−
|α|2
2
e+αâ |0i
= e−
|α|2
2
D(α) = e−
†
∗
†
∗
(A.25)
which implies that
†
(A.26)
|αics .
Hence, D(α) is the unitary transformation that rotates the
vacuum |0i into the coherent state |αics , up to a proportionality constant.
More informations on bosonic coherent states can be found in complement GV of Ref. [118].
A.2. Path-integral representation of the anharmonic
oscillator
Define the anharmonic oscillator of order n = 2, 3, 4, · · · by
2n
X
m
1
†
Ĥ = Ĥ0 +Ĥn ,
Ĥ0 := ~ω â â +
,
Ĥn :=
λm ↠+ â .
2
m=3
(A.27)
Of the real-valued parameters λm , m = 3, 4, · · · , 2n, it is only required
that λ2n > 0. This insures that there exists a vacuum |0i annihilated
by â. With the help of the bosonic algebra (A.2), it is possible to move
2
Let A and B be two operators that both commute with their commutator
[A, B]. Then,
1
eA eB = eA+B e 2 [A,B] .
(A.24)
576
A. THE HARMONIC-OSCILLATOR ALGEBRA AND ITS COHERENT STATES
all annihilation operators to the right of the creation operators in the
interaction Ĥn . This action generates many terms that can be grouped
by ascending order in the combined number of creation and annihilation
operators. The monomials of largest order are all contained
in : Ĥn :.
†
3
† † †
† †
For example, : (â + â) : = â â â + 3â â â + H.c. . Evidently, : Ĥn :
cannot be written anymore as a polynomial in x̂ ∝ (↠+ â) of degree
2n.
After normal ordering of Ĥ, the canonical partition function on the
Hilbert space F in Eq. (A.7) becomes
Z := e
−βE0
−β:Ĥ:
Tr e
=e
−βE0
∞
X
hn|e−β:Ĥ: |ni,
(A.28)
n=0
where E0 is the normal ordering energy, i.e., the expectation value
h0|Ĥ|0i. We will now give an alternative representation of the canonical
partition function that relies on the use of coherent states. We begin
with the trace formula (A.20)
Z
dϕ∗0 dϕ0 −ϕ∗0 ϕ0
Z = exp(−βE0 )
e
cs hϕ0 | exp(−β : Ĥ :)|ϕ0 ics . (A.29)
2πi
For M a large positive integer, write
M −1
β X
exp(−β : Ĥ :) = exp −
: Ĥ :
M j=0
!
(A.30)
" #
M −1
2
β X
β
: Ĥ : +O
.
=1 −
M j=0
M
To the same order of accuracy,
−β:Ĥ:
e
h
iM
−β:Ĥ:/M
= e
.
(A.31)
Insert the resolution of identity (A.15) (M − 1)-times,
!
Z
1
∗
Y
dϕ
dϕ
β
β
∗
j
j −ϕj ϕj
e−β:Ĥ: = e− M :Ĥ:
e
|ϕj ics cs hϕj | e− M :Ĥ: .
2πi
j=M −1
(A.32)
Equation (A.22) together with Eq. (A.30) gives
cs hϕ0 |
β
−M
e
:Ĥ:
|ϕM −1 ics = e
β
+ϕ∗0 ϕM −1 − M
:H(ϕ∗0 ,ϕM −1 ):
"
+O
β
M
2 #
,
(A.33a)
and
cs hϕj |
β
−M
e
:Ĥ:
β
+ϕ∗j ϕj−1 − M
|ϕj−1 ics = e
:H(ϕ∗j ,ϕj−1 ):
"
+O
β
M
2 #
, (A.33b)
A.2. PATH-INTEGRAL REPRESENTATION OF THE ANHARMONIC OSCILLATOR
577
for j = M − 1, M − 2, · · · , 1. The operator-valued function : Ĥ : of â
and ↠has been replaced by a complex-valued function : H : of ϕ and
ϕ∗ , respectively. Altogether, a M -dimensional integral representation
of the partition function has been found,
!
Z MY
−1
dϕ∗j dϕj
Z = exp(−βE0 )
2πi
j=0
!
M X
β
(A.34a)
× exp −
ϕ∗j ϕj − ϕj−1 +
: H(ϕ∗j , ϕj−1 ) :
M
j=1
" #
2
β
+O
,
M
whereby
ϕM := ϕ0 ,
ϕ∗M := ϕ∗0 .
(A.34b)
It is customary to write, in the limit M → ∞, the functional path
integral representation of the partition function
Z
∗
−βE0
Z=e
D[ϕ∗ , ϕ]e−SE [ϕ ,ϕ] ,
(A.35a)
where the so-called Euclidean action SE [ϕ∗ , ϕ] is given by
SE [ϕ∗ , ϕ] =
Zβ
dτ {ϕ∗ (τ )∂τ ϕ(τ )+ : H[ϕ∗ (τ ), ϕ(τ )] :} ,
(A.35b)
0
and the complex-valued fields ϕ∗ (τ ) and ϕ(τ ) obey the periodic boundary conditions
ϕ∗ (τ ) = ϕ∗ (τ + β),
ϕ(τ ) = ϕ(τ + β).
Hence, their Fourier transform are
1 X ∗ +i$l τ
1X
ϕ∗ (τ ) =
ϕl e
,
ϕ(τ ) =
ϕ e−i$l τ .
β l∈Z
β l∈Z l
(A.35c)
(A.36a)
The frequencies
2π
l,
l ∈ Z,
(A.36b)
β
are the so-called bosonic Matsubara frequencies.
Convergence of the (functional) integral representing the partition
function is guaranteed by the contribution λ2n (ϕ∗ +ϕ)2n to the interaction : Hn (ϕ∗ , ϕ) :. Thus, convergence of an integral is the counterpart
in a path integral representation to the existence of a ground state in
operator language.
$l :=
578
A. THE HARMONIC-OSCILLATOR ALGEBRA AND ITS COHERENT STATES
Quantum mechanics at zero temperature is recovered from the partition function after performing the analytical continuation (also called
a Wick rotation)
τ = +it,
dτ = +idt,
∂τ = −i∂t ,
under which
−SE → + iS
Z+∞
= +i
dt {ϕ∗ (t)i∂t ϕ(t)− : H[ϕ∗ (t), ϕ(t)] :} .
(A.37)
(A.38)
−∞
The path-integral representation of the anharmonic oscillator relies
solely on two properties of bosonic coherent states: Equations (A.15)
and (A.22). Raising, ↠, and lowering, â, operators are not unique to
bosons. As we shall see, one can also associate raising and lowering
operators to fermions. Raising and lowering operators are also well
known to be involved in the theory of the angular momentum. In general, raising and lowering operators appear whenever a finite (infinite)
set of operators obey a finite (infinite) dimensional Lie algebra. Coherent states are those states that are eigenstates of lowering operators in
the Lie algebra and they obey extensions of Eqs. (A.15) and (A.22).
Hence, it is possible to generalize the path-integral representation of
the partition function for the anharmonic oscillator to Hamiltonians
expressed in terms of operator obeying a fermion, spin, or any type
of Lie algebra. Due to the non-vanishing overlap of coherent states,
a first-order imaginary-time derivative term always appears in the action. This term is called a Berry phase when it yields a pure phase
in an otherwise real-valued Euclidean action as is the case, say, when
dealing with spin Hamiltonians. 3 It is the first-order imaginary-time
derivative term that encodes quantum mechanics in the path-integral
representation of the partition function. A reference on generalized
coherent states is the book in Ref. [119].
3
By writing [compare with Eq. (1.62a)]
r
r
1
1
∗
ϕ(τ ) =
[x(τ ) + ip(τ )] ,
ϕ (τ ) =
[x(τ ) − ip(τ )] ,
2
2
(A.39)
we can derive the path-integral representation of the (an)harmonic oscillator in
terms of the coordinate and momentum of the single particle of unit mass m =
1, unit characteristic frequency ω = 1, and with ~ = 1. The first-order partial
derivative term becomes purely imaginary
Zβ
∗
Zβ
dτ (ϕ ∂τ ϕ)(τ ) = i
0
dτ (x∂τ p)(τ ).
0
(A.40)
A.3. HIGHER DIMENSIONAL GENERALIZATIONS
579
A.3. Higher dimensional generalizations
The path-integral representation of the partition function for a single anharmonic oscillator is a functional integral over the exponential
of the Euclidean classical action (A.35b) in (0 + 1)-dimensional (position) space and (imaginary) time. The path-integral representation
of the quantum field theory of a d-dimensional continuum of coupled
anharmonic oscillators is a functional integral over the exponential of
the Euclidean classical action in (d + 1)-dimensional (position) space
and (imaginary) time of the form
Zβ
∗
SE [ϕ , ϕ] ≡
Z
dτ
dd rLE
0
Zβ
=
Z
dτ
dd r {ϕ∗ (r, τ )∂τ ϕ(r, τ )+ : H[ϕ∗ (r, τ ), ϕ(r, τ )] :} .
0
(A.41a)
The classical fields ϕ∗ (r, τ ), and ϕ(r, τ ) obey periodic boundary conditions in imaginary time τ ,
ϕ(r, τ ) = ϕ(r, τ + β),
ϕ∗ (r, τ ) = ϕ∗ (r, τ + β).
(A.41b)
At zero temperature, analytical continuation τ = +it of the action
yields
Z+∞ Z
∗
S[ϕ , ϕ] ≡
dt dd rL
−∞
Z+∞ Z
=
dt dd r {ϕ∗ (r, τ )i∂t ϕ(r, τ )− : H[ϕ∗ (r, τ ), ϕ(r, τ )] :} .
−∞
(A.42)
The classical canonical field conjugate to ϕ(r, t) is
δL
= iϕ∗ (r, t).
(A.43)
π(r, t) :=
δ[∂t ϕ(r, t)]
Canonical quantization is obtained by replacing the classical fields
ϕ(r, t) and ϕ∗ (r, t) with quantum fields ϕ̂(r, t) and ϕ̂† (r, t) that obey
the equal-time algebra
[ϕ̂(r, t), ϕ̂† (r 0 , t)] = δ(r−r 0 ),
[ϕ̂(r, t), ϕ̂(r 0 , t)] = [ϕ̂† (r, t), ϕ̂† (r 0 , t)] = 0.
(A.44)
APPENDIX B
Some Gaussian integrals
B.1. Generating function
Path integrals are generalizations of multi-dimensional Riemann integrals. Integrands of path integrals for non-interacting bosons are exponentials of quadratic forms. Hence, for any positive real-valued a,
their evaluations require generalizations to path integrals of the two
Gaussian integrals
Z
dz ∗ dz −z∗ az+j ∗ z+jz∗
∗
Za (j , j) :=
e
2πi
Z
dz ∗ dz −z∗ az+j ∗ z+jz∗ −j ∗ a−1 j+j ∗ a−1 j
e
=
2πi
Z
dz ∗ dz −(z−a−1 j)∗ a(z−a−1 j)
+j ∗ a−1 j
= e
e
2πi
Z
dz ∗ dz −z∗ az
+j ∗ a−1 j
= e
e
2πi
∗ −1 Z
e+j a j
dz ∗ dz −z∗ z
e
=
a
2πi
∞
∗ −1 Z
e+j a j
2π −r2
=
dr r
e
a
π
0
=
e
+j ∗ a−1 j
a
,
(B.1a)
and
hz ∗ zia :=
Z
dz ∗ dz ∗ −z∗ az
z ze
2πi
,
Za (j ∗ , j)|j ∗ =j=0
1
∂ 2 Za (j ∗ , j) =
Za (j ∗ , j) ∂j∂j ∗ j ∗ =j=0
1
=
.
(B.1b)
a
The function Za (j ∗ , j) is called a generating function. From it all moments of the form
2n
∗
1
∂
Z
(j
,
j)
a
h(z ∗ z)n ia :=
,
n = 0, 1, 2, · · · , (B.2)
Za (j ∗ , j) ∂j n ∂j ∗n j ∗ =j=0
581
582
B. SOME GAUSSIAN INTEGRALS
can be calculated.
Generalization of Eqs. (B.1) and (B.2) to N -dimensional Riemann
integrals is straightforward. Replace the complex conjugate pair z ∗
and z by N -dimensional vectors z † and z, respectively. Replace the
complex number a with a strictly positive real part by the N × N
positive definite Hermitean matrix A. Define the generating functional
Z N † N
d z d z −z† Az+j † ·z+z† ·j
†
ZA (j , j) :=
e
,
(B.3)
(2πi)N
from which all moments
n
†
2
Y
∂
Z
(j
,
j)
1
A
∗
h
zm zm iA :=
†
∗
ZA (j , j) m=1 ∂jm ∂jm m=1
n
Y
,
n = 0, 1, 2, · · · ,
j ∗ =j=0
(B.4)
can be calculated. Since the measure of the generating functional is
invariant under any unitary transformation of CN , we can choose a
basis of CN that diagonalizes the positive definite Hermitean matrix
A, in which case Eq. (B.1) can be used for each independent integration
over the N normal modes. Thus,
†
−1
e+j A j
,
ZA (j , j) =
det A
†
∗
hzm
zn iA = A
−1
mn
(B.5)
m, n = 1, · · · , N.
,
Imposing periodic boundary conditions for continuous systems results
in having a countable infinity of normal modes. In this case Eq. (B.5) is
generalized by replacing A, whose determinant is made of a finite product of eigenvalues, by a kernel, whose determinant is made of a countable product of eigenvalues. For infinite dimensional vector spaces, I
use the notation Det · · · for the determinant of the kernel · · · . After
taking the thermodynamic limit, the number of normal modes is uncountable. The logarithm of the determinant of the kernel becomes an
integral instead of a sum.
B.2. Bose-Einstein distribution and the residue theorem
The Bose-Einstein distribution
fBE (z) :=
eβz
1
−1
(B.6)
is analytic in the complex plane except for the equidistant first-order
poles
2πi
zl =
l,
l ∈ Z,
(B.7)
β
B.2. BOSE-EINSTEIN DISTRIBUTION AND THE RESIDUE THEOREM 583
on the imaginary axis. Each pole zl = (2πi/β)l of fBE (z) has the
residue 1/β since
exp [β(zl + z)] − 1 = βz + O(z 2 ),
∀l ∈ Z.
(B.8)
Let g(z) be a complex function such that:
• g(z) decreases sufficiently fast at infinity,
lim |z|fBE (z)g(z) = 0.
|z|→∞
(B.9)
• g(z) is analytic everywhere in the complex planes except for
two poles on the real axis away from the origin, say at z =
±x 6= 0.
Let Γ be a closed path infinitesimally close to the imaginary axis and
running antiparallel (parallel) to the imaginary axis when Re z < 0
(Re z > 0). Let ∂U±x be circular paths running clockwise and centered
about ±x. Then, path Γ can be deformed into path ∂U−x ∪ ∂U+x by
Cauchy theorem, and the residue theorem yields
Z
X
dz
g(zl ) = + β
f (z)g(z)
2πi BE
l
Γ


Z
Z
(B.10)

 dz
= +β
+
fBE (z)g(z)

2πi
∂U−x
∂U+x
= − β [Res (fBE g)(−x) + Res (fBE g)(+x)] .
APPENDIX C
Non-Linear-Sigma-Models (NLσM) on
Riemannian manifolds
C.1. Introduction
We have seen in section 3.2.2 that the O(N ) NLσM is an example
of a NLσM on a Riemannian manifold. The goal of this appendix is
to derive the one-loop RG equations obeyed by the metric tensor gab
that enters the action of a generic NLσM on a Riemannian manifold.
These equations were derived up to two loops by Friedan in Ref. [120].
In terms of the short-distance cutoff a ≡ Λ−1 and up to one loop, they
are given by
1
∂
R ,
infinitesimal,
(C.1)
a gab = gab −
∂a
2π ab
when the Euclidean base space is d = (2 + )-dimensional, while Rab
represents the Ricci tensor and Rapqr represents the curvature tensor
of the (Riemannian) target manifold. Summation convention over repeated indices is here implied. Equation (C.1) generalizes Eq. (3.224).
To this end, we shall employ the background-field method. [121]
This method dictates how to separate fields into slow and fast modes
in such a way that the action can be expanded in a Taylor series in
powers of the fast modes which is covariant under reparametrization of
the Riemannian manifold. Corrections to the action of the slow modes
are then computed to any desired order in a cumulant expansion by
integration over the fast modes in d = (2 + ) dimensions.
C.2. A few preliminary definitions
We begin with a collection of mathematical definitions needed to
make precise the concept of a Riemannian manifold. This section can
be ignored if one is not interested in this level of rigor.
A Riemannian manifold is a smooth manifold endowed with a metric. We thus need to define a smooth manifold and a metric. In turn,
a smooth manifold is a special type of topological space.
Topological space: Let X be a set. A topology on X is a set T
of subsets of X such that T contains:
(1) The empty set and X itself.
(2) The union of any subset of T .
(3) The intersection of any finite subset of T .
585
586C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
A topological space X, T is a set X with a topology T on X.
Homeomorphism: Let X, T and X 0 , T 0 be two topological
spaces. A mapping f : X −→ X 0 is called a homeomorphism if
(1) f is one-to-one and onto.
(2) U ∈ T =⇒ f (U ) ∈ T 0 .
(3) U 0 ∈ T 0 =⇒ f −1 (U 0 ) ∈ T .
Open sets and neighborhoods: Let X, T be a topological
space. Elements of the topology T are called open sets. A neighborhood of x ∈ X is a subset of X that includes an open set to which x
belongs to.
Hausdorff topological space: A topological space X, T is
called Hausdorff if any two distinct points possess disjoint neighborhoods.
Topological manifold: A N -dimensional topological manifold is
a Hausdorff topological space such that every point has a neighborhood
homeomorphic to RN .
Chart:
A chart (U, ϕ) of a N -dimensional topological manifold
X, T is an open set U of X, called the domain of the chart, together
with a homeomorphism ϕ : U −→ V onto an open set V in RN .
The coordinates (x1 , · · · , xN ) of the image ϕ(x) ∈ RN of the point
x ∈ U ⊂ X are called the coordinates of x in the chart (U, ϕ) or, in
short, local coordinates of x. Here, the N coordinates x1 , · · · , xN of
the point ϕ(x) ∈ RN are short-hand notations for the mappings
ai : RN −→ R,
x1 , · · · , xN −→ ai x1 , · · · , xN = xi .
(C.2)
A chart (U, ϕ) is also called a local coordinate system.
Atlas:
An atlas of class C k of a N -dimensional topological manifold
X, T is a set of charts {(Uα , ϕα )} such that:
S
(1) Uα = X.
α
T T (2) The maps ϕβ ◦ ϕ−1
Uβ −→ ϕβ Uα Uβ are maps
α : ϕα Uα
of open sets of RN into RN of class C k , i.e., k-times continuously differentiable.
Equivalent atlases: Two C k atlases {(Uα , ϕα )} S
and {(Uα0 , ϕα0 )}
are equivalent if and only if the
set
of
domains
{U
}
{Uα0 } and the
α
S
k
set of homeomorphisms {ϕα } {ϕα0 } is again a C atlas.
C k manifold: A N -dimensional topological manifold X, T together with an equivalence class of C k atlases is a C k structure on X.
It is also said that X is a C k manifold. When k = ∞ the manifold is
said to be smooth.
Differentiable functions: Charts make it possible to extend the
notion of differentiability of functions f : RN −→ R to functions whose
C.2. A FEW PRELIMINARY DEFINITIONS
587
domain of definitions are C k manifolds. Let f be a real-valued function
with the C k manifold X as domain of definition. Hence, we associate
to any x ∈ X the image f (x) ∈ R. Let (U, ϕ) be a chart at x, i.e.,
x ∈ U . The function f ◦ ϕ−1 : ϕ(U ) −→ R is a mapping from an open
set of RN into R. Just as the coordinates of ϕ(x) represent x in the
local chart (U, ϕ), the mapping f ◦ ϕ−1 represents f in the local chart.
The function f is of class C j at x with j ≤ k if f ◦ ϕ−1 is of class C j
at ϕ(x).
Tangent vector v x : A tangent vector to a C k manifold X at a
point x ∈ X is a function v x from the space of functions defined and
differentiable on some neighborhood of x ∈ X into R, that satisfies
(1) v x (αf + βg) = αv x (f ) + βv x (g)
(linearity),
(2) v x (f g) = f v x (g) + gv x (f )
(Leibniz rule),
for all α and β in R and for all functions f and g on X that are differentiable at x. In the chart (U, ϕ), the local coordinates
(components)
of a tangent vector v x are the N numbers v 1 , · · · , v N where
v i := v x (ϕi ),
(C.3a)
ϕi ≡ ai ◦ ϕ. Here, the N coordinates functions a1 , · · · , aN are defined
by
ai : RN −→ R,
u1 , · · · , uN −→ ai u1 , · · · , uN = ui .
(C.3b)
The tangent vector v x is also called a derivation.
Tangent vector as a directional derivative: Let f be a function
defined on some neighborhood of x ∈ X into R that is differentiable.
The directional derivative of f along v x is the image v x (f ) of f . The
rational for this terminology follows from the following argument. Define F : RN −→ R through the composition F := f ◦ ϕ−1 and assume
that f is a C ∞ function in U , the domain of the local chart (U, ϕ).
Taylor expansion gives (for f a C 1 function, one uses the mean value
theorem of analysis)
N ∂F X
i
i
f (x) = F ϕ(y) +
ϕ (x) − ϕ (y)
+ ···
(C.4a)
i
∂x
ϕ(y)
i=1
for any pair x and y in U . By definition, the directional derivative of
f along v x is
N
X
i ∂F vx ϕ
+ ··· ,
(C.4b)
i
∂x
ϕ(y)
i=1
i.e., it reduces to
N
X
i=1
vy
N
X
∂F ∂F
i
=
v
ϕ
i
∂xi ϕ(y)
∂x
ϕ(y)
i=1
i
(C.4c)
588C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
when x = y.
Tangent vector space: The set of all tangent vectors to the C k
manifold X at x ∈ X together with the addition and scalar multiplication defined by
(1) αv x + βwx (f ) := αv x (f ) + βwx (f )
is a vector space called the tangent vector space and denoted Tx X.
According to Eq. (C.4) the vectors of Tx X can be
represented as linear
combinations of the basis ∂/∂x1 , · · · , ∂/∂xN , which is also called
the natural basis of the tangent vectorspace.
The natural or coordinate basis of Tx X is also denoted by ea where ea ≡ ∂a ≡ ∂/∂xa
for a = 1, · · · , N . A chart (U, ϕ) has thus induced an isomorphism
between Tx X and RN . The basis of Tx X need not be the natural or
the coordinate one. We may also choose the basis êâ defined by
êâ := Aâa ea
(C.5)
where summation over repeated
upper and lower indices is implied and
a
the N × N matrix A ≡ Aâ belongs to the group GL(N, R) of N × N
real-valued and invertible matrices. The basis (C.5) is known as the
non-coordinate basis of Tx X.
Cotangent vector space: The cotangent space Tx∗ X is the vector
space dual to the tangent space Tx X, i.e., it is the vector space of all
linear functions f : Tx X −→ R. The basis {e∗a } of the cotangent
space Tx∗ X dual to the basis {ea } of the tangent space Tx X is defined
by the condition
e∗a eb = δ ab .
(C.6)
Tensors and tensor
fields: A tensor T of a smooth manifold
m
X at x ∈ X of type n is a multilinear mapping that maps m dual
vectors and n vectors into R,
T : (Tx∗ X × · · · × Tx∗ X) × (Tx X × · · · × Tx X) −→ R.
|
{z
} |
{z
}
m−times
(C.7)
n−times
The set of all tensors of a smooth manifold X at x ∈ X of type m
n
is called
the tensor space of a smooth manifold X at x ∈ X of type
m
m
1
0
X where T0,x
X ≡ Tx X and T1,x
X ≡ Tx∗ X. By
and
denoted
by Tn,x
n
defining a linear combination of two tensors of the same type by the
same linear combination of their point-wise values,
the tensor space
is
endowed
with the
of a smooth manifold X at x ∈ X of type m
n
structure of a vector space. A smooth assignment of an element of
m
Tn,x
X at each point x ∈ X defines a smooth tensor field on the smooth
is denoted Tnm X.
manifold X. The set of all tensor fields of type m
n
Riemannian manifold: A Riemannian manifold is a smooth manifold M together with a smooth tensor field g : T M × T M −→ R of
type 02 such that:
(1) g is symmetric.
C.3. DEFINITION OF A NLσM ON A RIEMANNIAN MANIFOLD
589
(2) For each p ∈ M, the bilinear form gp is positive definite.
C.3. Definition of a NLσM on a Riemannian manifold
Consider the NLσM defined by the partition function
Z
Z := D[φ] exp (−S[φ]) ,
(C.8a)
the Euclidean action
Z
S[φ] :=
dd x
L(φ),
ad−2
(C.8b)
the Lagrangian density
1
L(φ) := gab (φ)∂µ φa ∂µ φb ,
2
(C.8c)
and the measure
D[φ] :=
Yp
||g(φ)||
x∈Rd
Y
dφa (x).
(C.8d)
a
d
At each point x ∈ R , the N × N φ-dependent matrix g(φ) with realvalued matrix elements gab (φ) is positive definite and symmetric with
determinant ||g(φ)||. 1 Some few words about the conventions we are
using in Eq. (C.8). We reserve the Greek alphabet to denote the coordinates of x ∈ Rd . We reserve the Latin alphabet to denote the N × N
real-valued entries gab (φ) in the defining representation of the Riemannian metric g, which, for each point p [represented by φ(x) ∈ RN ] in the
Riemannian manifold (M, g), is the bilinear mapping
gp : Tp M × Tp M −→ R
(C.9a)
(U, V ) −→ gp (U, V )
(Tp M the tangent space to p ∈ M) that obeys the condition for symmetry
gp (U, V ) = gp (V, U ),
∀U, V ∈ Tp M,
(C.9b)
the condition for positivity
gp (U, U ) ≥ 0,
∀U ∈ Tp M,
(C.9c)
and the condition for non-degeneracy
gp (U, U ) = 0 =⇒ U = 0.
(C.9d)
There is no distinction between upper and lower Greek indices and we
will always choose them to be lower indices. Summation over repeated
Greek indices is always implied. There is a distinction between upper
and lower Latin indices. Summation over repeated upper and lower
Latin indices is implied. Raising and lowering Latin indices is done
with the metric
φa ≡ gab φb
(C.10a)
1
If A is a matrix, ||A|| := |det A|.
590C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
where the convention
(C.10b)
gab gbc ≡ δca
is used to denote the matrix with entries gab which is the inverse of
the matrix with entries gab . A partial derivative only acts on the first
object to its right. For example,
∂φa b
φ,
∂xµ
∂φa b
φ,
∂φc
∂µ φa φb ≡ ∂µ φa φb + φa ∂µ φb ,
∂c φa φb ≡ ∂c φa φb + φa ∂c φb .
∂µ φa φb ≡
∂ c φa φb ≡
(C.11)
In this appendix, we are going to derive the change in the action (C.8b)
evaluated at a solution ϕ of its equations of motion due to fluctuations
arising from the path integral. This will be done perturbatively up to
one loop in the so-called loop expansion. For conciseness, we shall abusively call the saddle-point solution a classical solution, while referring
to the fluctuations about it as quantum fluctuations.
C.4. Classical equations of motion for NLσM:
Christoffel symbol and geodesics
Since the plan of action is to expand about some classical solution
of the equations of motion, we need to derive them. First, for any
a = 1, · · · , N , we choose arbitrarily small functional variations of the
independent fields φa and ∂µ φa ,2
φa −→ φa + δφa ,
∂µ φa + δ ∂µ φa = ∂µ φa + ∂µ δφa , (C.12a)
up to the condition that they vanish when x is at infinity. Variations
(C.12a) induce for the action (C.8b) the change
#
Z d "
d x
δL
δL
− a δφa
δS := −
∂µ
ad−2
δφ
δ ∂ µ φa
(C.12b)
Z d d x
1
=−
∂µ gab ∂µ φb − ∂a gbc ∂µ φb ∂µ φc δφa .
ad−2
2
The chain rule for differentiation delivers for the right-hand side
Z d d x
1
b
b
c
b
c
gab ∂µ ∂µ φ + ∂c gab ∂µ φ ∂µ φ − ∂a gbc ∂µ φ ∂µ φ δφa .
ad−2
2
(C.12c)
In turn, relabeling of summation indices delivers for the right-hand side
Z d d x
1
1
1
b
b
c
b
c
b
c
gab ∂µ ∂µ φ + ∂c gab ∂µ φ ∂µ φ + ∂b gac ∂µ φ ∂µ φ − ∂a gbc ∂µ φ ∂µ φ δφa.
d−2
a
2
2
2
(C.12d)
2The
equality δ ∂µ φa = ∂µ δφa is only true for infinitesimal variations.
C.4. CLASSICAL EQUATIONS OF MOTION FOR NLσM: CHRISTOFFEL SYMBOL AND GEODESICS
591
Since the infinitesimal δφa is arbitrarily chosen, there follows, with the
help of Eq. (C.10), the N functional derivatives
n o
δS
(C.13a)
= − ∂µ ∂µ φa + abc ∂µ φb ∂µ φc ,
δφa
n o
where abc is called the Christoffel symbol and defined to be 3
n o
1 ad a
:=
∂
g
+
∂
g
−
∂
g
(C.13b)
g
bc
b dc
c db
d bc .
2
The metric is flat whenever it is independent of φ, in which case
the Christoffel symbol vanishes. The classical equations of motion are
obtained by demanding that S be extremal, i.e, are given by the saddlepoint equations
n o
a
0 = ∂µ ∂µ φ + abc ∂µ φb ∂µ φc .
(C.14)
For a flat metric they are just the equations of motion of N independent, massless, and free bosonic fields.
Next, we consider the curve C in Rd parametrized by
x : [0, 1] −→ Rd
(C.15a)
t −→ x(t)
between the end points x(0) and x(1). We then associate to the
curve (C.15a) the curve CM in M between the end points λ(0) and
λ(1) through
dλa
λa (t) := φa x(t) ,
λ̇a ≡
,
a = 1, · · · , N.
(C.15b)
dt
The arclength L[CM ] of the curve CM is then defined to be
Z1 q
Z
a
b
L[CM ] := dt gab λ̇ λ̇ ≡
ds
(C.16a)
0
CM
where the arclength line element ds is defined by
2
2
ds
:= gab λ̇a λ̇b ⇐⇒ ds := gab dλa dλb .
dt
(C.16b)
The arclength (C.16a) can also be thought of as a functional restricted
to any smooth path with given end points, in which case the extremal
paths satisfy the N geodesic differential equations
n o
0 = λ̈a + abc λ̇b λ̇c ,
a = 1, · · · , N.
(C.17)
As we shall see next, the arclength L[CM ] of the curve CM is invariant
under reparametrization of the manifold M. In this sense, the arclength
L[CM ] is a geometrical invariant.
3
Observe that any of these components of the Christoffel symbol is unchanged
under gab → −gab .
592C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
C.5. Riemann, Ricci, and scalar curvature tensors
In Eq. (C.8), we have chosen a specific parametrization of the Riemannian manifold M in terms of the N coordinates φa . In this section,
we are going to investigate the consequences of demanding that the
theory (C.8) be invariant under the reparametrization
φa = φa (φ0 )
(C.18)
0b
in terms of N coordinates φ .
The transformation law of ∂µ φa under the reparametrization (C.18)
is
∂φa
∂µ φa = Tca ∂µ φ0c ,
Tca (φ0 ) :=
.
(C.19)
∂φ0c
Invariance of the Lagrangian (C.8c) under the reparametrization (C.18),
L(φ) = L(φ0 ),
(C.20)
is achieved if and only if the metric transforms as
e
f
gab (φ) = T −1 a T −1 b g0ef (φ0 ),
(C.21)
where
Tba (φ0 )
∂φa
,
=
∂φ0b
Tba
b
T −1 c (φ0 )
=
δca
=⇒
b
T −1 c (φ0 )
∂φ0b
.
=
∂φc
(C.22)
p
||g(φ)|| transforms as
p
p
||g(φ)|| =
||T −1 (φ0 )|| × ||T −1 (φ0 )|| × ||g0 (φ0 )||
p
||g0 (φ0 )||
=
,
||T (φ0 )||
Q
while a dφa transforms as
Y
Y
dφa = ||T (φ0 )||
dφ0b .
If so,
a
(C.23)
(C.24)
b
We conclude that the measure (C.8d) transforms as
D[φ] = D[φ0 ]
(C.25)
under the reparametrization (C.18). We have thus proved that the
transformation laws (C.19) and (C.21) under the reparametrization
(C.18) guarantee both the (classical) invariance (C.20) and the (quantum) invariance (C.25). As a byproduct we have also proved that
the infinitesimal and finite arclengths (C.16) are invariant under the
reparametrization (C.18).
Transformation laws (C.20) and (C.25) define scalar quantities under the reparametrization (C.18). Transformation law (C.19) defines
a contravariant vector under the reparametrization (C.18). From a
contravariant vector V a , which transforms as
V a = Tāa V 0ā
(C.26)
C.5. RIEMANN, RICCI, AND SCALAR CURVATURE TENSORS
593
under the reparametrization (C.18), one defines a covariant vector
Vb := gbc V c ,
(C.27)
which must then transform as
Vb =
=
=
T −1
b̄
b
b̄
T −1 b
b̄
T −1 b
T −1
c̄
c
¯
Tc̄¯c g0b̄c̄ V 0c̄
g0b̄c̄ V 0c̄
Vb̄0
(C.28)
under the reparametrization (C.18). An example of a covariant vector
is ∂a L since it transforms as
∂a L = T −1
b
a
∂b0 L,
T −1
b
a
:=
∂φ0b
,
∂φa
∂b0 L :=
∂L
,
∂φ0b
(C.29)
under the reparametrization (C.18). As a corollary of transformation
laws (C.26) and (C.28), it follows that
V a Wa = V a gab W b = Va W a
(C.30)
is a scalar for any pair V a and Wa of contravariant and covariant vectors. Transformation law (C.21) defines a covariant tensor of rank
a ···a
2 under the reparametrization (C.18). An object Vb11···bnm which transforms like the tensor product of m contravariant vectors and n covariant
vectors,
d
d 0c1 ···cm
a ···a
Vb11···bnm = Tca11 · · · Tcamm T −1 b 1 · · · T −1 b n Vd1 ···d
(C.31)
n
n
1
defines a tensor of rank m
under the reparametrization (C.18) [see
n
Eq. (C.7)].
The derivative ∂a Vb of a contravariant vector V b does not transform
as a tensor of rank 11 under the reparametrization (C.18). Rather it
transforms as
ā
∂a V b = T −1 a ∂ā0 Tb̄b V 0b̄
ā
ā
= T −1 a Tb̄b ∂ā0 V 0b̄ + T −1 a ∂ā0 Tb̄b V 0b̄ .
(C.32)
The covariant derivative ∇a Vb is defined by the condition that it transforms like a tensor of type 11 under the reparametrization (C.18),
ā
∇a V b := T −1 a Tb̄b ∇0ā V 0b̄ .
(C.33)
To verify that such an object does indeed exist, write
∇a V b ≡ ∂a V b + Γbac V c .
(C.34)
The object Γbac is called a linear or affine connection when it exists.
The transformation law obeyed by the affine connection Γbac (when
594C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
it exists) under the reparametrization (C.18) is deduced in two steps.
First, Eqs. (C.33) and (C.34) deliver
ā
∇a V b = T −1 a Tb̄b ∇0ā V 0b̄
ā (C.35)
≡ T −1 a Tb̄b ∂ā0 V 0b̄ + Γ0b̄āc̄ V 0c̄ .
Second, Eq. (C.32) delivers
∇a V b ≡ ∂a V b + Γbac V c
ā
ā
= T −1 a Tb̄b ∂ā0 V 0b̄ + T −1 a ∂ā0 Tb̄b V 0b̄ + Γbac Tc̄c V 0c̄(. C.36)
Comparing the right-hand sides of Eqs. (C.35) and (C.36) gives
ā
ā
T −1 a Tb̄b Γ0b̄āc̄ = T −1 a ∂ā0 Tc̄b + Γbac Tc̄c
(C.37)
since V b is arbitrary. Multiplication of Eq. (C.37) by Tā¯a T −1
summation over a and b gives the final transformation law
b̄
b̄
Γ0b̄āc̄ = T −1 b ∂ā0 Tc̄b + Tāa T −1 b Tc̄c Γbac
=
¯b̄
b
and
(C.38)
∂φ0b̄ ∂ 2 φb
∂φa ∂φ0b̄ ∂φc b
+
Γ
∂φb ∂φ0ā ∂φ0c̄ ∂φ0ā ∂φb ∂φ0c̄ ac
obeyed by the affine connection, provided it exists. The inhomogeneous transformation law (C.38) immediately implies two important
properties of affine connections:
ea are two affine connections obeying the transfor(1) If Γabc and Γ
bc
mation law (C.38) under the reparametrization
(C.18), then
their difference is a tensor of type 12 .
(2) If Γabc is an affine connection obeying the transformation law
(C.38) under the
reparametrization (C.18), and if tabc is a ten
sor of type 12 , then Γabc + tabc is an affine connection obeying the transformation law (C.38) under the reparametrization
(C.18).
On the way to proving the existence of the affine connection and
thus of the covariant derivative of a contravariant vector we need to extend the definition of the action of the covariant derivative to arbitrary
linear combinations of tensors. The actionof the covariant derivative
a ···a
on an arbitrary tensor Vb11···bnm of type m
is to produce a tensor of
n
m
type n+1 given by
a ···a
a ···a
∇a Vb11···bnm := ∂a Vb11···bnm
+
m
X
i=1
ai
aā
Γ
a ···a
Vb11···bni−1
ā ai+1 ···am
−
n
X
a ···am
Γb̄abj Vb 1···b
1
j−1 b̄ bj+1 ···bn
,
j=1
(C.39a)
C.5. RIEMANN, RICCI, AND SCALAR CURVATURE TENSORS
595
where it is understood that the covariant derivative is simply the usual
partial derivative
∇a f = ∂a f
(C.39b)
when applied to a scalar function f . Second, Eq. (C.39) is supplemented by the condition that it remains valid if indices are contracted.
Third, Eq. (C.39) is supplemented by the condition that the covariant derivative acts linearly on linear combinations of tensors. The
claim that the transformation law of the covariant derivative (C.39)
under the reparametrization (C.18), if it exists, is that of a tensor of
m
type n+1
then follows from the fact that an arbitrary tensor of type
m
is nothing but the direct product of m contravariant vectors and
n
n covariant vectors together with Eq. (C.33) and its counterpart for
covariant vectors. Finally, we must make the affine connection compatible with the metric by demanding that the scalar product defined
in Eq. (C.56a) transforms covariantly, i.e., as in Eq. (C.56b). This
condition is achieved if
0 = ∇a gbc
=
0 =
=
0 =
∂a gbc − Γb̄ab gb̄c − Γc̄ac gbc̄ ,
∇b gca
∂b gca − Γc̄bc gc̄a − Γāba gcā ,
∇c gab
= ∂c gab − Γāca gāb − Γb̄cb gab̄ .
(C.40a)
(C.40b)
(C.40c)
All three equations are here related by cyclic permutations of a, b, c.
The compatibility condition (C.40) can be used to construct the
affine connection explicitly. The combination −(C.40a) +(C.40b) +(C.40c)
yields, owing to the symmetry of the metric tensor,
0 = −∂a gbc + Γdab gdc + Γdac gdb
+∂b gca − Γdbc gda − Γdba gdc
+∂c gba − Γdca gdb − Γdcb gda .
Introducing the notation
1 a
1 a
a
a
a
a
Γ {bc} :=
Γ bc + Γ cb ,
Γ [bc] :=
Γ bc − Γ cb
2
2
we can regroup underlined terms in Eq. (C.41) to get
(C.41)
(C.42)
0 = −∂a gbc + ∂b gca + ∂c gba + 2Γd[ab] gdc + 2Γd[ac] gdb − 2Γd{bc} gda(C.43)
.
Multiplication by gaā and summation over a turns Eq. (C.43) into
Γā{bc} = ābc + gāa Γd[ab] gdc + Γd[ac] gdb ,
(C.44)
596C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
where we already encountered the Christoffel symbol ābc in Eq. (C.13b).
We conclude that an affine connection compatible with the metric is
given by
Γabc = Γa{bc} + Γa[bc]
= abc + gaā Γd[āb] gdc + Γd[āc] gdb + Γa[bc] .
(C.45)
The antisymmetric part
T abc := 2 Γa[bc]
(C.46)
of the affine connection is called the torsion tensor. The terminology
tensor is here justified, for Tbca transforms like a tensor of type 12 as
antisymmetrization of Eq. (C.38) kills the inhomogeneous term.
An affine connection whose torsion tensor vanishes everywhere on
the manifold is called the Levi-Civita connection, in which case
Γabc = abc
1 ad =
g ∂b gdc + ∂c gdb − ∂d gbc .
(C.47)
2
The term
K abc := gaā Γd[āb] gdc + Γd[āc] gdb + Γa[bc]
(C.48)
in Eq. (C.45) is called the contorsion tensor. With the help of Eq.
(C.46), it turns into
1 aā d
K abc =
g T āb gdc + gaā T dāc gdb + T abc
2
1 a
a
a
≡
T + Tb c + T bc .
(C.49)
2 cb
By construction, the contorsion tensor is of type 12 and it vanishes if
the torsion tensor vanishes. A symmetric affine connection is an affine
connection with vanishing torsion tensor.
To sum up, we have proved the fundamental theorem of Riemannian
geometry. On a Riemannian manifold (M, g), there exists a unique
symmetric connection which is compatible with the metric g. It is
given by the Levi-Civita connection (C.47).
We now return to the definition (C.34) of the covariant derivative
∇a V b of an arbitrary contravariant vector V b . Let W a be another
arbitrary contravariant vector from which we construct
W a ∇a V b = W a ∂a V b + W a Γbac V c .
(C.50)
Next, we choose the contravariant vector W a to be the tangent vector
to the curve
φ : [0, 1] −→ M
(C.51a)
t −→ φ(t),
C.5. RIEMANN, RICCI, AND SCALAR CURVATURE TENSORS
597
i.e.,
dφa
.
dt
With this choice, Eq. (C.50) becomes
Wa =
(C.51b)
dφa c
dφa ∂V b
b
V
+
Γ
ac
dt ∂φa
dt
dV b
dφa c
=
+ Γbac
V .
(C.52)
dt
dt
The contravariant vector V a is said to be parallel transported along the
curve (C.51a) with tangent vector (C.51b) when the N equations
W a ∇a V b =
0 = W a ∇a V b
dV b
dφa c
=
+ Γbac
V
(C.53)
dt
dt
are satisfied. If it is the tangent vector W b itself which is parallel
transported along the curve φ(t), it must satisfy
0 = W a ∇a W b
dW b
dφa c
b
=
+ Γ ac
W
dt
dt
d 2 φb
dφa dφc
b
+
Γ
.
(C.54)
=
ac
dt2
dt dt
These are nothing but the N geodesic equations (C.17) when the affine
connection is restricted to the Levi-Civita connection. A geodesic can
thus be interpreted as a curve with the property that its tangent vector
is parallel transported along itself in accordance with the intuition of a
straight line being the shortest path between two points in Euclidean
space.
The metric compatibility condition (C.40) is related to parallel
transport in the following manner. Let W a be the tangent vector to the
arbitrarily chosen curve (C.51) and let X b and Y c be arbitrarily chosen contravariant vectors which are parallel transported with respect
to W a ,
0 = W a ∇a X b ,
0 = W a ∇a Y c .
(C.55)
We now require that the scalar product
X b Yb = X b gbc Y c
(C.56a)
is covariantly constant as defined by the condition
0 = W a ∇ a X b Yb
= gbc Y c W a ∇a X b + W a X b Y c ∇a gbc + X b gbc W a ∇a Y c
= W a X b Y c ∇a gbc .
(C.56b)
598C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
Condition (C.55) was used to reach the last line, while the penultimate line is a consequence of the definition of the covariant derivative.
The metric compatibility condition (C.40) is seen to follow from the
arbitrariness of W a , X b , and Y c .
If a vector is parallel transported along different curves between the
same initial and final points the resulting vectors are curve dependent
in general. This is most evidently seen by considering two antipodal
points on the equator of the sphere and connecting them along the
parallel or the meridian passing through them. The Riemann curvature
tensor is a covariant measure of this difference. The Riemann curvature
tensor is defined by the action
[∇a , ∇b ]V c = Rcdab V d − T dab ∇d V c
(C.57a)
on an arbitrary contravariant vector V c , i.e., in components,
Rcdab = ∂a Γcbd − ∂b Γcad + Γcae Γebd − Γcbe Γead .
(C.57b)
Equation (C.57a) implies that:
(1) The Riemann curvature tensor Rcdab is a tensor of type 13 .
(2) The Riemann curvature tensor Rcdab is antisymmetric in the
indices a and b,
Rcdab = −Rcdba .
(C.58)
(3) For the Levi-Civita connection, the Jacobi identity for commutators [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 implies the
Bianchi identity
0 = ∇a Redbc + ∇b Redca + ∇c Redab .
(C.59)
For the Levi-Civita connection, the Riemann curvature takes the explicit form
Rcdab = gcc̄ Rc̄dab ,
(C.60a)
with
Rc̄dab =
1
∂a ∂d gc̄b − ∂a ∂c̄ gbd − ∂b ∂d gc̄a + ∂b ∂c̄ gad
2
1
− ∂a gmc̄ + ∂c̄ gma − ∂m gac̄ gmn ∂b gnd + ∂d gnb − ∂n gbd
4
1
+ ∂b gmc̄ + ∂c̄ gmb − ∂m gbc̄ gmn ∂a gnd + ∂d gna − ∂n gad .
4
(C.60b)
Contraction of the pair c· a · of indices in the Riemann curvature
tensor defines the Ricci tensor,
Rdb = δ ac Rcdab
= ∂a Γabd − ∂b Γaad + Γaae Γebd − Γabe Γead .
(C.61)
C.6. NORMAL COORDINATES AND VIELBEINS FOR NLσM
599
Of course, we could have equally well chosen to contract the pair
of indices in the Riemann curvature tensor to obtain
δ bc Rcdab = Rcdac
= ∂a Γccd − ∂c Γcad + Γcae Γecd − Γcce Γead
= −R da .
c
· · b
(C.62)
This would give a Ricci tensor with the opposite sign convention. The
choice (C.61) is made so that the Ricci tensor of the surface of a unit
sphere can be chosen locally to be the unit matrix up to a positive
normalization constant.
Contracting the remaining two indices of the Ricci tensor defines
the scalar curvature
R := Rab gba .
(C.63)
One verifies that the Ricci tensor is a symmetric tensor for the LeviCivita connection. Observe that the Riemann tensor, the Ricci tensor,
and the scalar curvature transform like
Rcdab = +Rcdab ,
Rdb → +Rdb ,
R → −R,
(C.64)
respectively, under gab → −gab , see footnote 3.
C.6. Normal coordinates and vielbeins for NLσM
C.6.1. The background-field method. Only quantities intrinsic to the Riemannian manifold (M, g) that defines the target space of
the NLσM (C.8) are physical. Any choice of local coordinate system
can introduce unphysical degrees of freedom since the theory
is invari
a
ant under reparametrization whereas the coordinates φ are not. The
background-field method is aimed at handling this complication.
The background-field method applied to the NLσM consists in decomposing the components φa (x), with a = 1, · · · , N , of the contravariant vector field φ in the action (C.8b) into two fields ψ and π according
to the additive rule
Z
dd k +ikx a
a
φ (x) =
e
φ (k)
(2π)d
|k|<Λ
Z
=
|k|<Λ−dΛ
|
Z
dd k +ikx a
dd k +ikx a
e
φ
(k)
+
e
φ (k)
(2π)d
(2π)d
Λ−dΛ<|k|<Λ
{z
} |
{z
}
=:ψ a (x)
a
=:π a (x)
= ψ a (x) + π (x),
(C.65)
whereby ψ a (x) is assumed to be a slowly varying solution to the classical equations of motion (C.14) that transforms like a contravariant
vector and π a (x) represents fast degrees of freedom. Here, Λ plays
600C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
the role of an ultraviolet cutoff.
Having identified the contravariant
a
a
vectors ψ (x) and φ (x) with two points on the Riemannian manifold (M, g), say p and q, respectively, we cannot in general interpret
their difference
π a (x) = φa (x) − ψ a (x),
a = 1, · · · , N,
(C.66)
N
as the coordinates in R of some contravariant vector. The best we
can do is to assume
that the points p and q are close enough, i.e., the
fields π a (x) are “small” enough, for there to be a unique geodesic
that connects them.
The renormalization program in the background-field method is carried out in two steps. First, the metric, Lagrangian, or, more generally,
any function of the fields φa are Taylor expanded in powers of the
fast degrees of freedom πa (x) . Second, an integration over the fast
degrees of freedom π a (x) in the partition function or in correlation
functions is performed order by order in this expansion. For example,
when carried on the partition function (C.8), this program gives
Z
Z = D[ψ] exp (−S[ψ]) ,
Z d
d x
S[ψ] =
L(ψ),
ad−2
(C.67)
i
1h
(1)
(2)
a
b
L(ψ) =
g (ψ) + Tab (ψ) + Tab (ψ) + · · · ∂µ ψ ∂µ ψ ,
2Y ab
Y
p
dψ a (x).
||g(ψ)||
D[ψ] =
x∈Rd
a
Here, the object
(1)
(2)
Tab (ψ) := gab (ψ) + Tab (ψ) + Tab (ψ) + · · · ,
(C.68)
is a symmetric tensor of type 02 which, as we shall verify explicitly to
first order in the expansion, is an algebraic function of the curvature
tensor and the covariant derivative for the Levi-Civita connection. Had
we not chosen ψ to satisfy the classical equations of motion, we would
need to account for the additional contribution
n o
δL(ψ) = Γa (ψ) ∂µ ∂µ ψ a + abc ∂µ ψ b ∂µ ψ c
(C.69)
to the renormalization of the action [see Eq. (C.13a)] where Γa (ψ) is
typically noncovariant under reparametrization and can be expressed
in terms of the Christoffel connection (C.13b).
However, integration over the fast fields π quickly becomes very
tedious as the expansion of covariant quantities in powers of the components π a is not manifestly covariant anymore. This difficulty
can be
a
overcome by expanding
the fast degrees of freedom π (x) as a power
a
series of fields ξ (x) that transform like contravariant vectors. This
intermediate step can be done in a unique way once it is guaranteed
C.6. NORMAL COORDINATES AND VIELBEINS FOR NLσM
601
that there is a unique geodesic connecting p ∼ ψ a (x) to q ∼ φa (x) .
The existence of a unique geodesic connecting p to q is closely related
to the existence of normal coordinates, which we are going to define
from a purely geometrical point of view in the following.
C.6.2. A mathematical excursion. Let p ∈ M be an arbitrary
point of the Riemannian manifold (M, g) defined by Eq. (C.8). Let
U ⊂ M be an open set of the manifold that contains p. Let ξ : U −→ U
be a smooth homeomorphism between the open set U ⊂ M in the
manifold and the open set U ⊂ RN such that
ξ −1 (0) = p.
(C.70)
The pair (U, ξ) is said to be a coordinate system on M with ξ −1 (0) = p.
This coordinate system is said to be normal with respect to p if the
inverse image under ξ of straight lines through the origin in RN are
geodesics on M with respect to the Levi-Civita connection (C.47).
To explore the usefulness of this definition, let w be an arbitrary
vector in RN . The set of points t w with t ∈ R defines a straight line
through the origin of RN . We assume that the homeomorphic mapping
ξ : U −→ RN from the open set U that contains p ∈ M into RN realizes
a normal coordinate system. By definition, there must be a > 0 such
that any curve Cw : [−, ] −→ M through p defined by
Cw (t) = ξ −1 ◦ w1 t, · · · , wN t
(C.71)
is a geodesic whose components in RN labeled by a = 1, · · · , N obey
the generic geodesic equations
n o
0 = C̈ a + Γabc Ċ b Ċ c ,
Γabc = Γacb = abc .
(C.72)
However, since the curve tw in RN is linear, insertion of Eq. (C.71)
into Eq. (C.72) brings about the simplification
0 = Γabc Cw (t) wb wc ,
∀ w1 , · · · , wN ∈ ξ(U) ⊂ RN .
(C.73)
Equation (C.73) restricted to t = 0 implies
0 = Γabc (p) wb wc ,
∀ w1 , · · · , wN ∈ ξ(U) ⊂ RN .
(C.74)
Since w ∈ RN is arbitrary, there follows
0 = Γabc (p)
(C.75)
for the Levi-Civita connection. In the normal coordinates with respect
to p, the Riemann curvature tensor (C.57b) takes the simpler form
Rcdab (p) = ∂a Γcbd (p) − ∂b Γcad (p).
(C.76)
Evidently a non-vanishing curvature tensor at p implies a non-vanishing
derivative of the Levi-Civita connection at p in the normal coordinates
with respect to p. The Levi-Civita connection is, by this argument, not
expected to vanish at a generic q 6= p in M when represented in the
602C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
normal coordinates with respect to p. The covariant derivative (C.34)
also takes the simpler form
∇a V b (p) = ∂a V b (p)
(C.77)
when evaluated at p in the normal coordinates with respect to p.
Conversely, any geodesic through p must be of the form (C.71) in
the coordinate system normal with respect to p. To see this we need
the so-called exponential map. For any p ∈ M there must be a > 0
and an open neighborhood Up of 0 ∈ RN ∼
= Tp M such that there is, for
any w ∈ Up , a unique solution to Eq. (C.72)
Cw : [−, ] −→ M
(C.78a)
t −→ Cw (t)
with
Cw (0) = p,
Ċw (0) = w,
(C.78b)
and a smooth dependence of Cw on w. The curve Cw is the geodesic
through p with tangent vector w ∈ Tp M at p. Since Eq. (C.72) is
invariant under the affine (Galilean boost) transformation
t = A + B t0
∀A, B ∈ R,
(C.79)
it follows that:
(1) The curve
0
Cw
: [−( + A)/B, ( − A)/B]] −→ M
0
t0 −→ Cw
(t0 ) = Cw (t)
(C.80a)
is a geodesic with
0 dCw
dCw dCBw =B
= Bw =
.
= p,
dt0 t0 =−A/B
dt t=0
dt t=0
(C.80b)
(2) The domain of definition of Cw can always be extended to the
interval [−1, 1] after proper rescaling of the open neighborhood
Up ⊂ Tp M.
Define the exponential mapping by
EXP : Up −→ M
(C.81a)
w −→ EXP(w) = Cw (1),
0
Cw
(−A/B)
where we note that, with the help of Eq. (C.80),
dEXP(tw) EXP(tw) = Ctw (1) = Cw (t),
= w,
dt
0 ≤ t ≤ 1.
t=0
(C.81b)
The exponential mapping can be used to define a normal coordinate
system on an open neighborhood of p ∈ M through the definition
ξ EXP(w) = w.
(C.82)
C.6. NORMAL COORDINATES AND VIELBEINS FOR NLσM
603
By this definition,
ξ Cw (t) = t w
(C.83)
parametrizes a straight line passing through the origin in Tp M when
t = 0, i.e., Cw (0) = p. According to Eq. (C.83) the normal coordinates
on an open neighborhood of p ∈ M of a geodesic passing through p
are the coordinates of the vector tangent to this geodesic at p with a
magnitude t |w| increasing linearly with t.
Normal coordinates are used in proving the following theorem: Any
point of a Riemannian manifold has a neighborhood U such that for
any two points in U there is a unique geodesic that joins the points and
lies in U.
C.6.3. Normal coordinates for NLσM. We want to integrate
over the fields π a (x) with a = 1, · · · , N in the partition function
Z
Z = D[ψ, π] exp (−S[ψ, π]) ,
Z d
d x
S[ψ, π] =
L(ψ, π),
ad−2
(C.84a)
1
L(ψ, π) = gab (ψ, π)∂µ ψ a + π a ∂µ ψ b + π b ,
2Y
Y
p
D[ψ, π] =
d ψ a (x) + π a (x) ,
||g(ψ, π)||
a
x∈Rd
where we assume that:
(1) The coordinates ψ a and ψ a + π a describe two points p and
q, respectively, from the Riemannian manifold (M, g) which
can be connected in a unique way by the geodesic
0 = λ̈a + Γabc λ̇b λ̇c ,
λa (0) = ψ a ,
λa (1) = ψ a + π a ,
a = 1, · · · , N,
(C.84b)
that lies in some open set U ⊂ RN homeomorphic to the open
neighborhood Up of p.
(2) The field ψ : Rd −→ M defined by x −→ ψ(x) satisfies the
classical equations of motion (C.14).
Given ψ a (x) let ψ a (x)+π a (x) be some arbitrary point belonging
to the open neighborhood U ⊂ RN homeomorphic to Up and in which
the geodesic (C.84b) lies. We begin by performing a Taylor expansion
in powers of 0 ≤ t ≤ 1 of Eq. (C.84b),
1 a
1
1 ...a
λ̇ (0) t + λ̈a (0) t2 + λ (0) t3 + · · · ,
1!
2!
3!
λa (1) = ψ a + π a ,
a = 1, · · · , N.
λa (t) = λa (0) +
(C.85)
604C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
From the geodesic equations of motion (C.84b), we may express all
coefficients of order n > 1 in terms of linear combinations of the LeviCivita connection or its derivatives evaluated at t = 0. For examples,
λ̈a = − Γabc λ̇b λ̇c ,
...a
a
b c
a
b c
a
b c
λ = − Γ̇ bc λ̇ λ̇ − Γ bc λ̈ λ̇ − Γ bc λ̇ λ̈
∂Γabc d b c
=−
λ̇ λ̇ λ̇ + Γabc Γbde λ̇d λ̇e λ̇c + Γabc λ̇b Γcde λ̇d λ̇e
d
∂λ
∂Γabc d b c
=−
λ̇ λ̇ λ̇ + Γaec Γedb λ̇d λ̇b λ̇c + Γabe Γedc λ̇b λ̇d λ̇c
d
∂λ
a
∂Γ bc
a
e
a
e
=−
− Γ ec Γ db − Γ be Γ dc λ̇d λ̇b λ̇c
d
∂λ
{z
}
|
(C.86)
=:Γadbc
= − Γadbc λ̇d λ̇b λ̇c .
If we introduce the tangent vector
v a := λ̇a (0),
(C.87a)
we have found the Taylor expansion
∞
X
1 a
λ (t) = ψ + v t −
Γ a1 a2 ···an (ψ)v a1 v a2 · · · v an tn
n!
n=2
a
a
a
∞
X
1 a
=ψ + v t −
Γ (a1 a2 ···an ) (ψ)v a1 v a2 · · · v an tn ,
n!
n=2
a
a
λa (1) = ψ a + π a ,
(C.87b)
a = 1, · · · , N.
The coefficient Γaa1 a2 ···an (ψ) on the first line of this Taylor expansion is
defined recursively by
Γaa1 a2 ···an (ψ) := ∇a1 Γaa2 a3 ···an (ψ)
= ∇a1 · · · ∇an−2 Γaan−1 an (ψ)
∂Γa
(C.87c)
for n = 2, 3, · · · with the seed Γabc and the rule Γadbc := ∂λdbc −Γaec Γedb −
Γabe Γedc . The operation ∇a1 Γaa2 a3 ···an (ψ) resembles the action of the covariant derivative defined in Eq. (C.39a) on tensor fields with, however,
the caveat that only the lower indices of the symbols Γaa2 ···an (they are
not tensor fields) are operated on, i.e., the first summation is omitted
on the right-hand side of Eq. (C.39a). The coefficient Γa(a a ···an ) (ψ) on
1 2
the second line of this Taylor expansion is defined by the symmetrization
1 X a
Γ a a ···a (ψ).
(C.87d)
Γa(a1 a2 ···an ) (ψ) :=
P(1) P(2)
P(n)
n! P∈S
n
C.6. NORMAL COORDINATES AND VIELBEINS FOR NLσM
605
The permutation group of n objects is here denoted by Sn . Equation
(C.87) for t = 1 gives the Taylor expansion
∂µ (ψ a + π a ) = ∂µ ψ a + ∂µ v a
∞
X
1 h
−
∂µ ψ b ∂b Γa(a1 a2 ···an ) (ψ)v a1 v a2 · · · v an
n!
n=2
+ Γa(a1 a2 ···an ) (ψ)∂µ v a1 v a2 · · · v an + · · ·
i
+Γa(a1 a2 ···an ) (ψ)v a1 v a2 · · · ∂µ v an
(C.88)
that we will use shortly.
Next, we choose to parametrize
the geodesic (C.84b) in terms of
a
the normal coordinates ξ with respect to p ≡ ψ(x) as defined in
section C.6.3. We also add an overline on the expansion coefficients in
Eq. (C.87), Christoffel-symbol, and tensors, etc., when using normal
coordinates to represent them. If we introduce the tangent vector
ξ a := λ̄˙ a (0),
(C.89a)
then Eq. (C.87), when expressed in the normal coordinates with respect
to the classical solution ψ of the equations of motion evaluated at x,
becomes 4
∞
X
1 a
a
a
a
λ (t) = ψ + ξ t −
Γ a1 a2 ···an (ψ)ξ a1 ξ a2 · · · ξ an tn
n!
n=2
∞
X
1 a
Γ (a1 a2 ···an ) (ψ)ξ a1 ξ a2 · · · ξ an tn ,
=ψ + ξ t −
n!
n=2
a
a
λa (1) = ψ a + π a ,
(C.89b)
a = 1, · · · , N,
on the one hand. On the other hand, the defining property of the normal coordinates with respect to a point in the manifold is to represent
any geodesics through this point by a straight line in the tangent space
to this point. For Eq. (C.89b) to be a straight line,
a
(a1 a2 ···an ) (ψ),
n = 2, 3, · · · ,
(C.90)
must hold. In the normal coordinate system with respect to the point
ψ(x) defined by Eq. (C.89b), the covariant derivative ∇a defined by its
action Eq. (C.39a) on tensor fields reduces to a partial derivative
0=Γ
a, a1 , a2 , · · · an = 1, · · · , N,
∇a = ∂a ,
4
a = 1, · · · , N,
(C.91)
The point ψ(x) ∈ M is represented by the point ψ a (x) ∈ RN in an arbitrary
local coordinate system. To emphasize that the local coordinate system is chosen
to be the
normal coordinate system with respect to ψ(x), we use the notation
ψ a (x) ∈ RN to represent ψ(x) ∈ M.
606C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
at ψ(x). If so, Eq. (C.90) is nothing but
a
an−1 an ) (ψ),
a, a1 , a2 , · · · an = 1, · · · , N, n = 2, 3, · · · .
(C.92)
The Taylor expansion (C.88) is also simplified when represented in
terms of the normal coordinates with respect to ψ(x),
0 = ∂(a1 ∂a2 · · · ∂an−2 Γ
a
∂µ ψ + π
a
∞
X
1
a
= ∂µ ψ + ∂µ ξ − ∂µ ψ
∂b Γ (a1 a2 ···an ) (ψ)ξ a1 ξ a2 · · · ξ an
n!
n=2
a
a
b
∞
X
1
a
= ∂µ ψ + Dµ ξ − ∂µ ψ
∂b Γ (a1 a2 ···an ) (ψ)ξ a1 ξ a2 · · · ξ an
n!
n=2
(C.93)
a
a
b
where a second covariant derivative defined by
a
Dµ ξ a := ∂µ ξ a + Γ bc ξ b ∂µ ψ c
= ∂µ ξ a ,
µ = 1, · · · , d,
a = 1, · · · , N,
(C.94)
has been introduced. 5 Expansion (C.93) is not expressed in an optimal way since the right-hand side explicitly depends on derivatives of
the Levi-Civita connection and thus is not manifestly covariant under
reparametrization of ψ(x). This problem can be fixed by an iterative
use of Eqs. (C.92) and (C.76) as we now illustrate with the second
order term in the expansion. Condition (C.92) with n = 3 gives (see
footnote 5)
a
0 = ∂(a1 Γ a2 a3 )
1 a
a
a
a
a
a
=
∂a1 Γ a2 a3 + ∂a2 Γ a3 a1 + ∂a3 Γ a1 a2 + ∂a1 Γ a3 a2 + ∂a2 Γ a1 a3 + ∂a3 Γ a2 a1
3!
1
a
a
a
∂a1 Γ a2 a3 + ∂a2 Γ a3 a1 + ∂a3 Γ a1 a2 .
(C.95)
=
3
By Eq. (C.76), the Riemann curvature tensor in the normal coordinates
with respect to ψ simplifies to
a
a1 a2 a3
a
R a3 a2 a1
a
R a1 a2 a3
R
a
a
a1 a3 − ∂a3 Γ a1 a2 ,
a
a
= ∂a2 Γ a3 a1 − ∂a1 Γ a3 a2 ,
a
a
+ R a3 a2 a1 = 2∂a2 Γ a1 a3 −
= ∂a2 Γ
(C.96)
a
∂a3 Γ a1 a2
−
a
∂a1 Γ a3 a2 ,
at ψ(x). Here, the symmetry of the Levi-Civita connection with respect to interchange of its two lower indices was used. Combining Eqs.
(C.95) and (C.96) allows to express the derivative of the Levi-Civita
5
Recall that Γabc = Γacb holds for the Levi-Civita connection.
C.6. NORMAL COORDINATES AND VIELBEINS FOR NLσM
connection in terms of the Riemann curvature tensor,
1 a
a
a
∂a2 Γ a1 a3 = +
R a1 a2 a3 + R a3 a2 a1
3
1 a
a
=−
R a1 a3 a2 + R a3 a1 a2 .
3
607
(C.97)
The antisymmetry in the interchange of the last two lower indices of
the Riemann curvature tensor was used to reach the last line. Since
the right-hand side is here symmetric under interchange of the lower
indices a1 and a3 , we conclude that the Taylor expansion (C.93) up to
second order in the normal coordinates with respect to ψ(x) is given
by
1 a
∂µ ψ a + π a = ∂µ ψ a + Dµ ξ a + R a1 a2 b ξ a1 ξ a2 ∂µ ψ b .
3
(C.98)
Although expressed in normal coordinates with respect to ψ(x), Eq.
(C.98) for an arbitrary chart containing ψ(x) is simply obtained by removing the overline as Eq. (C.98) is manifestly covariant under reparametrization of ψ(x).
In the Taylor expansion
a ···a
Tb11···b k (ψ, π) =
l
∞
X
1 a ···a
∂c1 · · · ∂cn Tb11···b k (ψ) π c1 · · · π cn
l
n!
n=0
(C.99)
a ···a
of the tensor Tb11···b k of type kl neither the expansion coefficients
l
a1 ···ak
∂c1 · · · ∂cn Tb1 ···b (ψ) nor the expansion variables π c transform col
variantly under reparametrization of ψ a . By choosing
to parame
a
a
trize π in terms of the normal coordinates
ξ with respect to
c
ψ, both the expansion
variables ξ and the expansion coefficients
a1 ···ak
∂c1 · · · ∂cn T b1 ···bl (ψ) transform covariantly under reparametrization
of ψ a in the Taylor expansion
a ···a
T b11···blk (ψ, π)
∞
X
1 a ···a
=
∂c1 · · · ∂cn T b11···blk (ψ) ξ c1 · · · ξ cn .
n!
n=0
(C.100)
a ···a
In other words, the expansion coefficients ∂c1 · · · ∂cn T b11···blk (ψ) can be
expressed solely in terms of covariant derivatives of T and the Riemann
curvature tensor of the manifold by a direct extension of the method
used to reach Eq. (C.97).
As an illustration we prove the identities, valid at ψ(x),
∂c T b1 ···bl = ∇c T b1 ···bl
(C.101a)
608C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
and
l
1 X b
b
∂c1 ∂c2 T b1 ···bl = ∇c1 ∇c2 T b1 ···bl −
R c2 bi c1 + R bi c2 c1 T b1 ···bi−1 bbi+1 ···bl ,
3 i=1
(C.101b)
that we will apply shortly to the covariant Taylor expansion of the
metric gab (ψ, π). Equation (C.101a) is a direct consequence of definition (C.39) together with Eq. (C.90). The proof of Eq. (C.101b)
starts
0
from the observation that ∇c2 Tb1 ···bl is a tensor of type l+1 . We first
implement Eq. (C.39) for ∇c1 ,
∇c1 ∇c2 Tb1 ···bl = ∂c1 ∇c2 Tb1 ···bl
− Γac1 c2 ∇a Tb1 ···bl −
l
X
Γac1 bi ∇c2 Tb1 ···bi−1 abi+1 ···bl
i=1
(C.102)
at ψ(x). We follow up by implementing Eq. (C.39) for ∇c2 ,
∇c1 ∇c2 Tb1 ···bl = ∂c1
∂c2 Tb1 ···bl −
l
X
!
Γac2 bi Tb1 ···bi−1 abi+1 ···bl
i=1
−
Γac1 c2 ∇a Tb1 ···bl
−
l
X
Γac1 bi ∇c2 Tb1 ···bi−1 abi+1 ···bl
i=1
(C.103)
at ψ(x). The partial derivative ∂c1 is then distributed by the product
rule,
∇c1 ∇c2 Tb1 ···bl = ∂c1 ∂c2 Tb1 ···bl
−
l
X
∂c1 Γac2 bi Tb1 ···bi−1 abi+1 ···bl
i=1
−
l
X
Γac2 bi ∂c1 Tb1 ···bi−1 abi+1 ···bl
i=1
− Γac1 c2 ∇a Tb1 ···bl −
l
X
Γac1 bi ∇c2 Tb1 ···bi−1 abi+1 ···bl
i=1
(C.104)
at ψ(x). The right-hand side, when restricted to the normal coordinates
with respect to ψ(x), reduces to
∇c1 ∇c2 T b1 ···bl = ∂c1 ∂c2 T b1 ···bl −
l
X
a
c2 bi T b1 ···bi−1 abi+1 ···bl
∂ c1 Γ
i=1
l
= ∂c1 ∂c2 T b1 ···bl
1 X b
b
+
R c2 bi c1 + R bi c2 c1 T b1 ···bi−1 bbi+1 ···bl
3 i=1
(C.105)
C.6. NORMAL COORDINATES AND VIELBEINS FOR NLσM
609
at ψ(x). Equation (C.97) was used to reach the last line. The Taylor
expansion (C.99) simplifies for the metric tensor in view of the compatibility condition (C.40),
gb1 b2 (ψ, π) = gb1 b2 (ψ)
−
1 1
2! 3
h
R
b
c2 b1 c1
+R
b
b1 c2 c1
i
b
b
gbb + R c2 b2 c1 + R b2 c2 c1 gbb (ψ)ξ c1 ξ c2
2
1
+···
= gb1 b2 (ψ)
−
1 1
2! 3
h
i
Rb2 c2 b1 c1 + Rb2 b1 c2 c1 + Rb1 c2 b2 c1 + Rb1 b2 c2 c1 (ψ)ξ c1 ξ c2
+···
1
= gb1 b2 (ψ) − Rb1 c1 b2 c2 (ψ)ξ c1 ξ c2 + · · · .
3
To reach the last line, Eq. (C.60b) was used to deduce from
Rabcd = −Rabdc ,
Rabcd = Rcdab ,
a, b, c, d = 1, · · · , N, (C.107)
that underlined curvature tensors cancel out and that the remaining
two curvature tensors, when contracted with ξ c1 ξ c2 , are equal.
C.6.4. Gaussian expansion of the action. With the help of
normal coordinates with respect to a solution ψ of the classical equations of motion of the NLσM, we have found in section C.6.3 the covariant expansions
1
∂µ (ψ a + π a ) = ∂µ ψ a + Dµ ξ a + Rac1 c2 c3 ξ c1 ξ c2 ∂µ ψ c3 + · · · , (C.108a)
3
and
1
gab (ψ, π) = gab (ψ) − Rac1 bc2 (ψ)ξ c1 ξ c2 + · · · .
(C.108b)
3
The validity of this expansion is conditioned by the existence of a
unique geodesic that connects the points p ≡ ψ(x) and q ≡ ψ(x)+π(x)
in the Riemannian manifold (M, g). Here the geodesic lies in an open
neighborhood of the domain U from the chart (U, ξ) at the point ψ(x).
We are going to prove that the corresponding expansion for the action
is given by
Z d
1
d x a
b
a
b
c d
S[ψ, π] = S[ψ] +
g
(ψ)D
ξ
D
ξ
−
∂
ψ
∂
ψ
R
ξ
ξ
+ ···
ab
µ
µ
µ
µ
acbd
2
ad−2
Z d
d x 1
a
b
a
b
c d
= S[ψ] +
g
(ψ)D
ξ
D
ξ
+
∂
ψ
∂
ψ
R
ξ
ξ
+ ··· .
µ
µ
µ
µ
acdb
2
ad−2 ab
(C.108c)
(The second equality was reached with the help of Racbd = −Racdb .)
Recall that we have introduced the covariant derivative
Dµ ξ a = ∂µ ξ a + ∂µ ψ b Γabc ξ c ,
µ = 1, · · · , d,
a = 1, · · · , N,
(C.108d)
(C.106)
610C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
where Γabc denotes the Levi-Civita connection and Rabcd ≡ gaā Rābcd
denotes the associated curvature tensor.
Proof. Step 1: Choose an arbitrary pair (µ, ν) with µ, ν = 1, · · · , d,
1
l1 l2
gij (ψ, π)∂µ (ψ + π) ∂ν (ψ + π) = gij (ψ) − Ril1 jl2 (ψ)ξ ξ + · · ·
3
1
× ∂µ ψ i + Dµ ξ i − ∂µ ψ m Ri l1 ml2 (ψ)ξ l1 ξ l2 + · · ·
3
1
j
j
n j
l1 l2
× ∂ν ψ + Dν ξ − ∂ν ψ R l nl (ψ)ξ ξ + · · · .
1 2
3
(C.109)
i
j
Distribution of the products gives
gij (ψ, π)∂µ (ψ + π)i ∂ν (ψ + π)j = gij (ψ)∂µ ψ i ∂ν ψ j
+ gij (ψ)∂µ ψ i Dν ξ j + gij (ψ)Dµ ξ i ∂ν ψ j
+ gij (ψ)Dµ ξ i Dν ξ j
1
− Ril1 jl2 (ψ)∂µ ψ i ∂ν ψ j ξ l1 ξ l2
3
1
− gij (ψ)∂µ ψ m Ri l1 ml2 (ψ)∂ν ψ j ξ l1 ξ l2
3
1
− gij (ψ)∂µ ψ i ∂ν ψ n Rj l1 nl2 (ψ)ξ l1 ξ l2
3
+ ··· .
(C.110)
Contraction of the metric tensor lowers indices on the curvature tensor
of the last two lines,
gij (ψ, π)∂µ (ψ + π)i ∂ν (ψ + π)j = gij (ψ)∂µ ψ i ∂ν ψ j
+ gij (ψ)∂µ ψ i Dν ξ j + gij (ψ)Dµ ξ i ∂ν ψ j
+ gij (ψ)Dµ ξ i Dν ξ j
1
− ∂µ ψ i ∂ν ψ j Ril1 jl2 (ψ)ξ l1 ξ l2
3
1
− ∂µ ψ i ∂ν ψ j Rjl1 il2 (ψ)ξ l1 ξ l2
3
1
− ∂µ ψ i ∂ν ψ j Ril1 jl2 (ψ)ξ l1 ξ l2
3
+ ··· .
(C.111)
C.6. NORMAL COORDINATES AND VIELBEINS FOR NLσM
611
With the help of Rabcd = Rcdab , we conclude that
gij (ψ, π)∂µ (ψ + π)i ∂ν (ψ + π)j = gij (ψ)∂µ ψ i ∂ν ψ j + gij (ψ)∂µ ψ i Dν ξ j
+ gij (ψ)Dµ ξ i ∂ν ψ j + gij (ψ)Dµ ξ i Dν ξ j
− ∂µ ψ i ∂ν ψ j Ril1 jl2 (ψ)ξ l1 ξ l2
+ ··· .
(C.112)
When µ = 1, · · · , d,
gij (ψ, π)∂µ (ψ + π)i ∂µ (ψ + π)j = gij (ψ)∂µ ψ i ∂µ ψ j
+ 2gij (ψ)∂µ ψ i Dµ ξ j
+ gij (ψ)Dµ ξ i Dµ ξ j − ∂µ ψ i ∂µ ψ j Ril1 jl2 (ψ)ξ l1 ξ l2
+ ··· .
(C.113)
Step 2: Summation over µ = 1, · · · , d, gives
1
g (ψ, π)∂µ (ψ + π)i ∂µ (ψ + π)j
2 ij
1
=
g (ψ)∂µ ψ i ∂µ ψ j
2 ij
+gij (ψ)∂µ ψ i Dµ ξ j
1
1
+ gij (ψ)Dµ ξ i Dµ ξ j − ∂µ ψ i ∂µ ψ j Ril1 jl2 (ψ)ξ l1 ξ l2
2
2
+···
= L(ψ)
+gij (ψ)∂µ ψ i Dµ ξ j
i
1h
+ gij (ψ)Dµ ξ i Dµ ξ j − ∂µ ψ i ∂µ ψ j Ril1 jl2 (ψ)ξ l1 ξ l2
2
+··· .
(C.114)
L(ψ, π) =
Step 3: By assumption ψ is a solution of the classical equations of
motion, i.e., ψ is an extremum of the action. Hence the term linear in
ξ j must vanish in the action constructed from Eq. (C.114),
Z d
i
d x 1h
i
j
i
j
l1 l2
S[ψ, π] = S[ψ]+
g (ψ)Dµ ξ Dµ ξ − ∂µ ψ ∂µ ψ Ril1 jl2 (ψ)ξ ξ +· · · .
ad−2 2 ij
(C.115)
612C. NON-LINEAR-SIGMA-MODELS (NLσM) ON RIEMANNIAN MANIFOLDS
C.6.5. Diagonalization of the metric tensor through vielbeins. The covariant Gaussian expansion (C.108c) about an extremum
ψ of the action is still not practical for computations because of the
presence of the ψ-dependent metric tensor. It is desirable to perform
a GL(N, R) transformation of the tangent space at ψ(x) that diagonalizes the metric tensor at ψ(x). To this end, in any chart (U, ξ) for
which it is possible to find an open neighborhood such that therein lies
a unique geodesic that connects ψ(x) to ψ(x) + π(x), introduce the fast
degree of freedom ζ(x) by the linear transformation
ζ â := êâa (ψ)ξ a ,
ξ a = êâa (ψ)ζ â ,
(C.116a)
where êâa(ψ) ∈ GL(N, R), which is called the vielbeins, is the inverse
of êâa (ψ) ∈ GL(N, R),
êâa (ψ)êb̂a (ψ) = δb̂â ,
êâa (ψ)êâb (ψ) = δba ,
(C.116b)
by demanding that
(C.116a)
ξ a gab (ψ) ξ b = ζ â δâb̂ ζ b̂ ⇐⇒ êâa (ψ) gab (ψ) êb̂b (ψ) = δâb̂ ,
(C.116b) â
⇐⇒ ê a (ψ) δâb̂ êb̂b (ψ)
(C.116c)
= gab (ψ).
From now on, latin letters with a hat refer to the coordinates of the
Riemannian manifold in the vielbein basis (C.116). Since the metric
tensor is diagonal in the vielbein basis (C.116), we will not distinguish
between upper and lower indices in this basis with the exception of the
vielbein matrices (C.116a). We now show that under transformation
(C.116) the covariant expansion (C.108c) becomes
Z d h
ĉ dˆi
1
d x b â b â
a
b
c d
D
ζ
D
ζ
+
∂
ψ
∂
ψ
R
ê
ê
S[ψ, π] = S[ψ]+
µ
µ
µ
µ
acdb ĉ dˆ ζ ζ +· · · ,
2
ad−2
(C.117a)
where yet another covariant deriva