Lecture 6: Non-Interacting fermions Abstract

Lecture 6: Non-Interacting fermions
Christopher Mudry∗
Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland.
(Dated: March 29, 2010)
Abstract
The physics of non-interacting fermions is reviewed.
∗
Electronic address: [email protected]; URL: http://people.web.psi.ch/mudry
1
Contents
I.Introduction
3
II.
Second quantization for fermions
3
III.
The non-interacting Jellium model
8
A.
Thermodynamics without magnetic field
9
B.
Sommerfeld semiclassical theory of transport
19
C.
Pauli paramagnetism
24
D.
Landau levels in a magnetic field
27
IV.
Time-ordered Green functions
29
A.
Introduction
29
B.
Time-ordered Green functions in imaginary time
32
C.
Time-ordered Green functions in real time
37
D.
Time-ordered single-particle Green functions: non-interacting jellium model
38
1.Momentum-space representation
38
2.Real-space representation
40
3.At equal times
42
A.
Grassmann coherent states
45
B.
Path integral representation for fermions
49
C.
Jordan-Wigner fermions
51
1.Introduction
51
2.Nearest-neighbor quantum xy limit in one-dimensional space
54
D.
The ground state energy and the single-particle time-ordered Green function 61
65
E.
Linear response
1.Introduction
65
2.The Kubo formula
67
3.Kubo formula for the conductivity
70
4.Kubo formula for the dc conductance
74
2
5.Kubo formula for the dielectric function
76
6.Fluctuation-dissipation theorem
78
References
81
3
I.
INTRODUCTION
This lecture is a (partial) review of material taught in undergraduate quantum mechanics
and solid state physics at ETHZ. It is 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 lecture and, thus, will be omitted entirely. It will play an important
role in the next two lectures when the Coulomb interaction between electrons in the jellium
model is accounted for.
After a quick summary of second quantization for fermions (Sec. II), the notions of the
Fermi sea and the Fermi surface will be reviewed (Sec. III). 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 non-interacting jellium model
(Sec. IV), the Grassmann coherent states (Appendix A), fermion path integrals (Appendix B), Jordan-Wigner fermions (Appendix C), the electronic correlation energy (Appendix D), and the fluctuations dissipation theorem (Appendix E) do not belong to the
cursus devoted to undergraduate solid state physics at ETHZ.
II.
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. I will 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
4
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 C).
Assume that the single-particle Hamiltonian (from now on, ~ = 1 unless specified)
H=−
∆
+ U(r)
2m
(2.1a)
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 ′ ) = δ(r − r ′ ), (2.1b)
n
V
in the single-particle Hilbert space H(1) of square integrable and twice differentiable functions
on R3 . I also assume that the single-particle potential U(r) is bounded from below, i.e.,
there exists a single-particle nondegenerate1 ground-state energy, say ε0 . Hence, the energy
eigenvalue index n can be chosen to run over the positive integers, n ∈ N. The time evolution
of any solution of Schrödinger equation
i∂t Ψ(r, t) = HΨ(r, t),
Ψ(r, t = 0) given,
(2.2a)
can be written as
Ψ(r, t) =
X
−iεn t
Cn ψn (r) e
,
n
Cn =
Z
dd r ψn∗ (r)Ψ(r, t = 0).
(2.2b)
V
The formalism of second quantization starts with two postulates:
• There exists a set of pairs of adjoint operators ĉn (annihilation operator) and ĉ†n (creation operator) labeled by the energy eigenvalue index n and obeying the fermion
algebra 2
{ĉm , ĉ†n } = δm,n ,
{ĉm , ĉn } = {ĉ†m , ĉ†n } = 0,
∀m, n.
(2.3)
• There exists a vacuum state |0i that is annihilated by all annihilation operators,
ĉn |0i = 0,
1
2
∀n.
(2.4)
By hypothesis fermions are spinless and there is no Kramer degeneracy associated to the spin-1/2 degrees
of freedom of real electrons.
My 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
With these postulates in hand, it becomes possible to define the Heisenberg representation
for the operator-valued field (in short, quantum field):
X
ψ̂(r, t) :=
ĉn ψn (r) e−iεnt ,
(2.5)
n
together with its adjoint
X
ψ̂ † (r, t) :=
ĉ†n ψn∗ (r) e+iεnt .
(2.6)
n
The fermion algebra (2.3) endows the quantum fields ψ̂(r, t) and ψ̂ † (r, t) with the equal-time
algebra3
{ψ̂(r, t), ψ̂ †(r ′ , t)} = δ(r − r ′ ),
{ψ̂(r, t), ψ̂(r ′ , t)} = {ψ̂ † (r, t), ψ̂ †(r ′ , t)} = 0. (2.10)
The quantum fields ψ̂(r, t) and ψ̂ † (r, t) act on the “big” many-particle space
(N
)
∞
M
^
F :=
H(1) .
(2.11)
N =0
Here, each
VN
H(1) is spanned by states of the form
Y † mi
|m0 , · · · , mi−1 , mi , mi+1 , · · · i :=
ĉi
|0i,
mi = 0, 1,
(2.12a)
i
with the condition on mi = 0, 1 that
X
mi = N.
(2.12b)
i
The algebra obeyed by the ĉ’s and their adjoints ensures that
VN
H(1) is the N-th antisym-
metric power of H(1) , i.e., that the state |m0 , · · · , mi−1 , mi , mi+1 , · · · i made of N identical
3
Alternatively, if we start from the classical Lagrangian density
L := (ψ ∗ i∂t ψ)(r, t) −
|∇ψ|2 (r, t)
− |ψ|2 (r, t)U (r),
2m
(2.7)
we can elevate the field ψ(r, t) and its momentum conjugate
π(r, t) :=
δL
= iψ ∗ (r, t)
δ(∂t ψ)(r, t)
(2.8)
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 ′ , t)} = iδ(r − r ′ ),
{ψ̂(r, t), ψ̂(r ′ , t)} = {π̂(r, t), π̂(r ′ , t)} = 0.
6
(2.9)
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 (2.11) is the sum over the subspaces
over wave functions for N identical particles that are antisymmetric under 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:
• The second-quantized representation Ĥ of the single-particle Hamiltonian (2.1a) is
Z
Ĥ :=
dd r ψ̂ † (r, t)H ψ̂(r, t)
V
=
X
εn ĉ†n ĉn .
(2.13)
n
As it should be it is explicitly time-independent.
• The second-quantized total particle number operator Q̂ is
Z
Q̂ :=
dd r ψ̂ † (r, t)11ψ̂(r, t)
V
=
X
ĉ†n ĉn .
(2.14)
n
It is explicitly time-independent as follows from the continuity equation
0 = (∂t ρ)(r, t) + (∇ · J )(r, t),
(2.15a)
ρ(r, t) := |Ψ(r, t)|2 ,
1
J (r, t) :=
[Ψ∗ (r, t) (∇Ψ) (r, t) − (∇Ψ∗ ) (r, t)Ψ(r, t)] ,
2mi
(2.15b)
(2.15c)
obeyed by Schrödinger equation (2.2a). 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, a global gauge transformation on the Fock space space is implemented by the operation
|m0 , · · · , mi−1 , mi , mi+1 , · · · i → e+iqQ̂ |m0 , · · · , mi−1 , mi , mi+1 , · · · i,
∀q ∈ R,
(2.16)
4
An odd permutation is made of an odd product of pairwise exchanges.
7
on states, or, equivalently,5
ĉn → e+iqQ̂ ĉn e−iqQ̂ = e−iq ĉn ,
(2.19)
ĉ†n → e+iqQ̂ ĉ†n e−iqQ̂ = e+iq ĉ†n ,
(2.20)
and
for all pairs of annihilation and creation operators. Equation (2.19) tells us that
annihilation operators carry particle number −1. Equation (2.20) tells us that creation
operators carry particle number +1.
• The second-quantized local particle number density operator ρ̂ and the particle number
current density Jˆ are
ρ̂(r, t) = ψ̂ † (r, t)11ψ̂(r, t),
(2.21a)
i
1 h †
Jˆ(r, t) :=
ψ̂ (r, t) ∇ψ̂ (r, t) − ∇ψ̂ † (r, t)ψ̂(r, t) ,
2mi
(2.21b)
and
respectively. The continuity equation
0 = (∂t ρ̂)(r, t) + (∇ · Jˆ)(r, t)
(2.21c)
that follows from evaluating the commutator between ρ̂ and Ĥ is obeyed as an operator
equation.
The operators Ĥ, Q̂, ρ̂, and Jˆ all act on the Fock space F . They are thus distinct from
their single-particle counterparts H, Q, ρ, and J whose actions are restricted to the Hilbert
V
space H(1) . By construction, the action of Ĥ, Q̂, ρ̂ and Jˆ on the subspace 1 H(1) of F , say,
coincide with the action of H, Q, ρ, and J on H(1) . For examples,
5
I 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,
(2.17)
and, similarly,
[ĉ† ĉ, ĉ† ] = +ĉ† .
8
(2.18)
• A single-particle wave function is recovered by defining the single-particle state
|mi := ĉ†m |0i,
(2.22)
h0|ψ̂(r, t)|mi = ψm (r) e−iεmt .
(2.23)
and calculating the overlap
• 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.
This state is called the Fermi-sea. The overlap
−iε
t
−iε
t
−iε
t
1
2
N
ψ1 (r1 ) e
ψ2 (r1 ) e
· · · ψN (r1 ) e
+
* N
−iε
t
−iε
t
−iε
t
Y
1
2
N
ψ1 (r2 ) e
ψ2 (r2 ) e
· · · ψN (r2 ) e
0 ψ̂(rj , t) Φ0 = .
.
.
.
.
.
.
.
···
.
j=1
−iε1 t
−iε2 t
−iεN t ψ2 (rN ) e
· · · ψN (rN ) e
ψ1 (rN ) e
ψ1 (r1 ) ψ2 (r1 ) · · · ψN (r1 ) ! N
X
ψ1 (r2 ) ψ2 (r2 ) · · · ψN (r2 ) = exp −i
εj t .
..
.. ,
.
.
.
···
. j=1
ψ1 (rN ) ψ2 (rN ) · · · ψN (rN )
(2.24)
(2.25)
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 HartreeFock approximation to the quantum many-body problem seeks the best trial function among
decomposable states.
III.
THE NON-INTERACTING JELLIUM MODEL
The non-interacting jellium model describes non-interacting electrons with the mass m
and the electrical charge −e 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
9
chemical potential µ is defined by the grand canonical partition function
Z(L3 , β, µ) := TrF e−β (Ĥ−µN̂ ) ,
XX
XX †
Ĥ :=
εσ,k ĉ†σ,kĉσ,k,
N̂ :=
ĉσ,kĉσ,k,
σ
F := span
k
(
Y
ĉ†ι
ι≡(σ,k)
mι
σ
|0iσ =↑, ↓,
εσ,k :=
k
L
k ∈ Z3 ,
2π
~2 k 2
,
2m
(3.1)
mι = 0, 1,
)
ĉι |0i = 0, {ĉι , ĉ†ι′ } = δι,ι′ , {ĉ†ι , ĉ†ι′ } = {ĉι , ĉι′ } = 0 .
The choice of periodic boundary conditions does not affect bulk properties in the thermodynamic limit L → ∞. 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 an external periodic potential,
the non-interacting electronic dispersion is a periodic function of momenta belonging to the
Brillouin zone, i.e., the momenta are bounded from above and below in magnitude.
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
semiclassical 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.
A.
Thermodynamics without magnetic field
Evaluation of the grand canonical partition function (3.1) yields


X
Z(L3 , β, µ) = TrF exp −β
(ει − µ) ĉ†ι ĉι 

Fι := span
(
“
ĉ†
ι
”m
˛
)
˛
ι |0i˛m = 0, 1
˛ ι
˛
= TrF 
=
Y
ι=(σ,k)
=
Y
ι=(σ,k)
Y
ι=(σ,k)

exp −β (ει − µ) ĉ†ι ĉι 
†
TrFι e−β(ει −µ)ĉι ĉι
X
e−β(ει −µ)mι
ι=(σ,k) mι =0,1
=
Y
ι=(σ,k)
1 + e−β(ει −µ) ,
10
σ =↑, ↓,
k=
2π
n,
L
n ∈ Z3 .
(3.2a)
In terms of the Fermi-Dirac distribution
fF (ει ) :=
1
eβ(ει −µ) + 1
⇐⇒ 1 − fF (ει ) :=
Eq. (3.2a) becomes
Z(L3 , β, µ) =
Y
ι=(σ,k)
eβ(ει −µ)
,
eβ(ει −µ) + 1
1
.
1 − fF (ει )
The internal energy U of the non-interacting jellium model is
−β (Ĥ−µN̂ )
Ĥ
TrF e
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
=
fF (ει ) ει .
(3.2b)
(3.2c)
(3.3a)
ι=(σ,k)
The free energy 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
(3.3b)
X
ln 1 − fF (ει ) .
ι=(σ,k)
The entropy S of the non-interacting jellium model is
∂F (L3 , β, µ)
S(L , β, µ) := −
∂T
∂F (L3 , β, µ)
= − kB
∂β −1
i
X h
fF (ει ) ln fF (ει ) + 1 − fF (ει ) ln 1 − fF (ει ) .
= − kB
3
ι=(σ,k)
11
(3.3c)
The pressure P of the non-interacting jellium model is
∂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)
∂ει
∂(L3 )−2/3
∝
= −(2/3)(L3 )−2/3−1
∂L3
∂L3
∂ε
e−β(ει −µ)
(−)β ι3
−β(ε
−µ)
ι
1+e
∂L
X e−β(ει −µ) ε
2
ι
= + L−3
3
1 + e−β(ει −µ)
ι=(σ,k)
2 −3 X
=+ L
fF (ει ) ει
3
(3.3d)
ι=(σ,k)
Eq. (3.3a)
2
= L−3 × U(L3 , β, µ).
3
The average number of electrons is
Ne (L3 , β, µ) :=
TrF e−β (Ĥ−µN̂ ) N̂
TrF e−β (Ĥ−µN̂ )
∂ ln Z(L3 , β, µ)
= β −1
∂µ
−β(ει −µ)
X
e
=
1 + e−β(ει −µ)
ι=(σ,k)
X
=
fF (ει )
(3.3e)
ι=(σ,k)
while the average occupation number of the single-particle level ι = (σ, k) is
TrF e−β (Ĥ−µN̂ ) ĉ†ι ĉι
hĉ†ι ĉι iL3 ,β,µ :=
TrF e−β (Ĥ−µN̂ )
(3.4)
= fF (ει ).
We now take the thermodynamic limit L → ∞ in which 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)
12
δ (ε − ει )
(3.5)
becomes the continuous function6
ν(ε) =
XZ
σ=↑,↓
4π is the area of the unit sphere
2 k2
,
ω := ~2m
k=
√
2mω
,
~
dk = dω
q
m
2~2 ω
= 2 × 4π
=2×
=2×
d3 k
~2 k 2
δ ε−
(2π)3
2m
Z+∞
0
1 m
2π 2 ~2
1 m
2π 2 ~2
dk k 2
~2 k 2
δ ε−
8π 3
2m
r
r
Z+∞
2m
dω ω 1/2 δ (ε − ω)
~2
(3.7)
0
2mε
Θ(ε).
~2
We have introduced the Heaviside function



1, if x > 0,


Θ(x) :=



 0, if x < 0.
(3.8)
In the thermodynamic limit L → ∞, the internal internal energy per unit volume u, the free
energy per unit volume f , the entropy per unit volume s, the pressure p, and the average
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 .
13
(3.6)
kz
kF
kF
ky
kF
kx
FIG. 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.
number of electrons per unit volume ne are given by
u(β, µ) := lim L−3 U(L3 , β, µ)
L→∞
Z
=
dε ν(ε) fF(ε) ε,
(3.9a)
R
f (β, µ) := lim L−3 F (L3 , β, µ)
L→∞
Z
−1
dε ν(ε) ln 1 − fF (ε) ,
= β
(3.9b)
R
s(β, µ) := lim L−3 S(L3 , β, µ)
L→∞
Z
h
i
= −kB dε ν(ε) fF (ε) ln fF (ε) + 1 − fF (ε) ln 1 − fF (ε) ,
(3.9c)
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) fF(ε),
(3.9d)
=
(3.9e)
R
respectively. Equations (3.9a-3.9e) 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 non-interacting jellium
14
model is the Fermi sea with the Fermi wave vector kF . The Fermi sea is the sphere with the
radius
kF
(3.10a)
4πkF3 /3
(3.10b)
and the volume
obtained by filling all single-particle levels ει where ι = (σ, k) with the wave vectors satisfying
Since there is a total of
0 ≤ |k| ≤ kF .
(3.10c)
4πkF3 /3
kF3
=
(2π)3
6π 2
(3.11)
single-particle wave vectors available per unit volume, the Fermi wave vector is given by
ne = 2 ×
1/3
kF3
⇐⇒ kF = 3π 2 ne
2
6π
(3.12)
when the number of electron per unit volume is given by ne . The Fermi energy εF is the
largest single-particle energy that is occupied in the Fermi sea,
εF = ει ,
ι = (σ, kF ) ⇐⇒ εF = ει =
~2 kF2
.
2m
(3.13)
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
εF =
~2 kF2
e2 aB kF2
e2
=
=
(kF aB )2 = Ry × (kF aB )2
2m
2
2aB
(3.14)
when expressed in term of the Bohr radius
~2
me2
(3.15)
e2
Ry :=
2aB
(3.16)
aB :=
and the ground-state binding energy
of the hydrogen atom, i.e., 13.6 eV. As good metals have
kF aB ≈ 1
15
(3.17)
fF
kB T
1
µ
ε
−1
is an analytic function of the energy
FIG. 2: The Fermi-Dirac function fF (ε) := eβ(ε−µ) + 1
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 red box in the figure.
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
vF :=
~kF
m
(3.18)
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,
(3.19)
[vF ] = 0.005 c.
At zero temperature, the Fermi-Dirac distribution (3.2b) is the step function
lim fF (ε) = Θ (µ − ε)
β→∞
(3.20)
whose derivative with respect to energy is the delta function
dfF (ε)
= −δ (µ − ε) .
β→∞
dε
lim
This suggests a Taylor expansion about the chemical potential µ of the function
Zε
dh(ε)
:= g(ε)
h(ε) :=
dε′ g(ε′) ⇐⇒
dε
−∞
16
(3.21)
(3.22)
that appears in the integral
Z
dε g(ε)fF(ε) =
R
dfF (ε)
dε h(ε) −
dε
Z
R
(3.23)
provided g vanishes as ǫ → −∞ and diverges no faster than polynomially for ǫ → +∞. The
so-called Sommerfeld expansion
Z
Z
dfF (ε)
dε g(ε)fF(ε) = dε h(ε) −
dε
R
R
=
Z
dε
R
∞
X
(ε − µ)m dm h(µ)
h(µ) +
m!
dµm
m=1
Z
∞
X
(ε − µ)2m
d2m h(µ)
= h(µ) +
dε
dµ2m
(2m)!
m=1
R
=
Zµ
dε g(ε) +
−∞
∞
X
am
m=1
!
df (ε)
− F
dε
dfF (ε)
−
dε
(3.24)
d2m−1 g(µ)
(kB T )2m
dµ2m−1
follows. To reach the third equality, we used the fact that
dfF (ε)
β/4
−
=
2
dε
cosh β2 (ε − µ)
(3.25)
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)
to write
Z
R
(ε − µ)2m
dε
(2m)!
df (ε)
− F
dε
= (kB T )
2m
(3.26)
1
×
(2m)!
Z
2m
dx x
R
= (kB T )2m × 2
|
∞
X
i=1
= am (kB T )2m ,
(−)i+1
{z
≡am
d
1
−
dx ex + 1
1
i2m
}
(3.27)
m = 1, 2, · · · .
If g varies significantly on the energy scale of µ, i.e.,
d2m−1 g(µ)
g(µ)
≈
,
dµ2m−1
µ2m−1
17
m = 1, 2, · · · ,
(3.28)
then the ratio of two successive terms in the Sommerfeld expansion is of the order O (kB T /µ)2
so that
Z
dε g(ε)fF(ε) =
Zµ
−∞
R
π2 ′
7π 4 ′′′
dε g(ε) +
g (µ) (kB T )2 +
g (µ) (kB T )4 + O
6
360
kB T
µ
6
. (3.29)
The Sommerfeld expansion (3.29) applied to the internal energy density (3.9a) and the
average occupation number density (3.9e) yields
Z∞
u(T, µ) =
dε ν(ε) ǫ +
−∞
Z∞
ne (T, µ) =
π2
(k T )2 [µν ′ (µ) + ν(µ)] + · · · ,
6 B
(3.30)
2
dε ν(ε) +
π
(k T )2 ν ′ (µ) + · · · ,
6 B
−∞
respectively.
We now assume that
µ = εF + O (kB T )2 ,
(3.31)
an assumption whose consistency we shall shortly verify. Under this assumption and owing
to the vanishing of the density of state for negative energies,
u(T, µ) =
ZεF
0
ne (T, µ) =
dε ν(ε) ǫ + εF (µ − εF) ν(εF ) +
ZεF
π2
[εF ν ′ (εF ) + ν(εF )] (kB T )2 + · · · ,
6
(3.32)
2
π ′
ν (εF ) (kB T )2 + · · · .
6
dε ν(ε) + (µ − εF ) ν(εF ) +
0
We also assume that the electronic density is temperature independent
ne (T, µ) =
ZεF
0
dε ν(ε) ≡ ne ,
∀T, µ.
(3.33)
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 ′ (εF)
(kB T )2 + · · ·
6 ν(εF )
(3.34)
while the internal energy density reduces to
u(T ) =
ZεF
0
dε ν(ε) ǫ +
π2
ν(εF ) (kB T )2 + · · · .
6
18
(3.35)
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 the support
kB T of the Fermi-Dirac distribution over which it varies significantly at finite temperature
times the density of states at the Fermi energy ν(εF ) times the characteristic excitation
energy kB T measured relative to the Fermi energy, i.e.,
u(T ) −
ZεF
0
dε ν(ε) ǫ ∝ ν(εF ) (kB T )2 .
(3.36)
At last, the specific heat of the non-interacting jellium model at fixed electronic concentration is
∂u(T ) Ĉv (T ) :=
∂T ne
π2
= ν(εF )kB2 T + · · · .
3
(3.37)
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
Cv (T ) =
2
kB T
εF
+··· .
(3.38)
For comparison, a classical ideal gas has the constant volume specific heat
3kB ne
.
2
(3.39)
The Fermi-Dirac statistics depresses the classical result by the factor
π 2 kB T
.
3 εF
(3.40)
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
(3.41)
T
linearly, i.e., as a function of T 2 . For good metals, the linear term to the specific heat become
comparable to the cubic one at a few degrees Kelvin.
19
z
y
x
Vx
B
−e V x B
Ex
+ + + + + + + + + + + +
Ey
− − − − − − − − − − − −
jx
b pointing along the
FIG. 3: The set up for Hall’s experiment is the following. A dc electric field Ex x
b along
positive x-cartesian axis is applied on a metallic wire. It induces a steady state current jx x
the positive x-cartesian axis. A dc magnetic field of magnitude B is pointing along the positive
b ∧ zb = − evcx B yb is balanced by the force induced
z-cartesian axis. The Lorentz force − e(−vcx )B x
by the electric field Ey yb that points along the negative y-cartesian axis and 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 electrical field Ey yb pointing along the positive
y-cartesian axis. Changing the sign of the charge carrier leaves jx = (∓e)(∓vx ) unchanged but
reverses the sign of Ey and thus of the the Hall coefficient RH :=
B.
Ey
jx B .
Sommerfeld semiclassical theory of transport
The semiclassical theory of transport in metal 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 metal assumes that electricity is carried by
small (point-like) hard sphere quantized in the units of e that undergo elastic and instantaneous scattering events with a probability per unit time 1/τ while they move freely (ballis-
20
tically) between the collisions. Assuming isotropy in space, let
j = −ne ev
(3.42)
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=−
eEτ
m
(3.43)
induced by a dc (static) electric field E between the collisions with probability per unit time
1/τ . The linear relation
j = σD E,
σD =
ne e2 τ
,
m
(3.44)
defines the Drude conductivity σD and the Drude resistivity
E = ρD j,
ρD =
ne e2 τ
m
−1
.
(3.45)
Isotropy of space is broken by a dc (static) magnetic field B = B zb perpendicular to a
rectangular metallic sample. One defines the magnetoresistance
Ex
jx
(3.46a)
Ey
jx B
(3.46b)
ρ(B) :=
and the Hall coefficient
RH :=
induced by solving the steady state equation
(mv)
eB
(mv) ∧ zb −
0 = − eE +
mc
τ
(3.46c)
with
Multiplication by
b + Ey y,
b
E := Ex x
−
b.
v := vx x
n eτ
σD
=− e
e
m
(3.46d)
(3.47)
of the steady state equation and the introduction of the cyclotron frequency
ωc :=
21
eB
mc
(3.48)
yields

 
Ex
j
1 +ωc τ
  x  ⇐⇒ ρ(B) = ρD ,
σD   = 
Ey
0
−ωc τ
1


RH = −
ωc
1
=−
.
σD B
ne ec
(3.49)
The Drude magnetoresistance is independent of the applied magnetic field. The Drude Hall
coefficient depends only the electronic density and 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
ℓD := veqp τ
(3.50a)
with the characteristic (equipartition) velocity
1 2
3
mveqp := kB T.
2
2
(3.50b)
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 is given by
j = −κ∇T
(3.51a)
with
1
κ = v 2 τ Cv ,
3
1 2 3
mv = kB T,
2
2
3
Cv = ne kB .
2
(3.51b)
Drude thus predicts the universal ratio
3
κD
=
σD T
2
kB
e
2
(3.52)
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 the Maxwell-Boltzmann distribution
3/2
1
mβ
2
fB (v) := ne
e− 2 mv β .
2π
(3.53)
Thus
fB (v)d3 v
22
(3.54)
z
y
x
x−vxτ
x+v xτ
Temperature gradient along x axis
FIG. 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 red vectors emerging from a scattering event at x − vx τ longer
than the blue 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 the right that can be
modeled by jx = n2e × vx × E T (x− vx τ ) − n2e × vx × E T (x+ vx τ ) ≈ ne × vx2 τ × ∂E
− dT
∂T
dx where
ne is the electronic density, vx is the velocity at x, E T (x ∓ vx τ ) is the thermal energy at the last
scattering event. Equation (3.51) follows with the identifications vx2 → 13 v 2 , ne × ∂E
∂T → Cv , and
− dT
dx → (−∇T ).
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 (3.53) by the Fermi-Dirac distribution
1
1 m 3
.
(3.55)
1
2
(2π)3 ~
e( 2 mv −µ)β + 1
This approximation is justified if positions and momenta of electrons can be specified as
fFD (v) := 2 ×
accurately as necessary without violating the uncertainty principle. Since the typical momentum of an electron in a metal is
~kF
(3.56)
we must demand that the momentum uncertainty ∆p satisfies
∆p ≪ ~kF .
23
(3.57)
As the uncertainty in the electronic position ∆x is given by
∆x ∼
~
,
∆p
(3.58)
∆x ≫
1
.
kF
(3.59)
1
aB
(3.60)
it follows that
However, for a good metal
kF ∼
so that
∆x ≫ aB .
(3.61)
We conclude that a classical descriptions 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
(3.62)
for a semiclassical treatment à la Sommerfeld to hold. Second, there is the mean free path
ℓS which must therefore also satisfy
ℓS ≫ kF−1
(3.63)
for a semiclassical treatment à la Sommerfeld to hold.
The replacement by the Fermi-Dirac distribution (3.55) of the Maxwell-Boltzmann distribution (3.53) 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 (3.50) is changed to
ℓS = vF τ
24
(3.64)
which can be larger than ℓD by two order of magnitude at room temperature. Similarly, the
thermal velocity
r
q
(kB T ) /εF × (εF /m)
∼ kB T /m =
(3.65)
in the thermal conductivity (3.51) must be replaced by the Fermi velocity
∼
q
εF /m
(3.66)
while the Drude specific heat
∼ ne kB
(3.67)
must be replaced by the smaller specific heat
kB T
∼
× ne kB .
(3.68)
εF
ε
The enhancement factor k FT induced by the use of the Fermi velocity cancels the reduction
B
kB T
factor ε
induced by the use of the Fermi gas specific heat. The empirical law of
F
Wiedemann and Franz (3.52) is thus also satisfied in the model of Sommerfeld albeit with
the universal coefficient
C.
κS
π2
=
σD T
3
kB
e
2
.
(3.69)
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. This model is defined by
the grand canonical partition function
Z(L3 , β, µ, B) := TrF e−β (Ĥ−µN̂ ) ,
XX †
XX
ĉσ,kĉσ,k,
εσ,k ĉ†σ,kĉσ,k,
N̂ :=
Ĥ :=
σ
F := span
k
(
Y
ι≡(σ,k)
ĉ†ι
mι
σ
|0iσ =↑, ↓,
k
L
k ∈ Z3 ,
2π
εσ,k :=
~2 k 2
− σµB B,
2m
(3.70a)
mι = 0, 1,
)
ĉι |0i = 0, {ĉι , ĉ†ι′ } = δι,ι′ , {ĉ†ι , ĉ†ι′ } = {ĉι , ĉι′ } = 0 .
25
We have introduced the Bohr magneton
µB :=
e~
.
2mc
(3.70b)
We want to compute the magnetization per unit volume
MP (L3 , β, µ, B) := −L−3 β −1
∂ ln Z(L3 , β, µ, B)
∂B
(3.71a)
and the corresponding spin susceptibility
χP (L3 , β, µ, B) :=
∂M(L3 , β, µ, B)
∂B
(3.71b)
in the thermodynamic limit L → ∞ holding the electronic density ne fixed. Each electron
with spin parallel to B contributes
− L−3 × µB
(3.72)
to the magnetization density. Each electron with spin anti-parallel to B contributes
+ L−3 × µB
(3.73)
ne± (β, µ, B)
(3.74)
to the magnetization density. If
denotes the density of electrons with spin parallel (+) and anti-parallel (−) to B in the
thermodynamic limit, then the magnetization density is
MP (β, µ, B) = −µB ne+ (β, µ, B) − ne− (β, µ, B)
(3.75)
in the thermodynamic limit. Of course, the constraint
ne = ne+ (β, µ, B) + ne− (β, µ, B)
(3.76)
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
(3.77)
with ν(ε) defined in Eq. (3.7). When B 6= 0,
1
ν± (ε) = ν(ε ∓ µB B)
2
26
(3.78)
so that
ne± =
Z
dε ν± (ε) fF (ε).
(3.79)
R
We shall assume that
µB B ≪ εF ,
(3.80)
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 ′ (ε) + · · · ,
2Z
2
ne± =
dε ν± (ε) fF(ε)
R
1
=
2
Z
1
dε ν(ε)fF (ε) ∓ µB B
2
R
Z
dε ν ′ (ε) fF(ε) + · · · ,
R
(3.81)
MP = − µB ne+ − ne−
Z
2
= + µB B dε ν ′(ε) fF (ε) + · · ·
R
=+
µ2B B
Z
R
′
dfF (ε)
dε ν(ε) −
+··· ,
dε
subject to the constraint that
ne =
Z
dε ν(ε) fF(ε) + · · · .
(3.82)
R
We can then use Eq. (3.35) to solve for the chemical potential
µ = εF + · · · .
(3.83)
At zero temperature
MP = µ2B ν(εF )B,
χP = µ2B ν(εF )
(3.84)
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
gµB
(gµB )2 J(J + 1)
(3.85)
+O
χP = ni
3
kB T
kB T
for non-interacting ions with density ni , total angular momentum quantum number J, and
Landé factor g.
27
D.
Landau levels in a magnetic field
We take the jellium model in the presence of the magnetic field
 
 
0
0
 
 
 
 
B =  0  = rot Bx ≡ rot A.
 
 
0
B
(3.86a)
The relevant single-particle Hamiltonian is the Pauli Hamiltonian
2
1
~
e
H=
∇ − A σ0 − µB σ3 B
2m i
c
"
#
2
~
e
1
−~2 ∂x2 +
∂ − Bx − ~2 ∂z2 σ0 − µB σ3 B.
=
2m
i y c
(3.86b)
The eigenvalue problem
HΨ(r) = εΨ(r),
Ψ(r) := eikz z × eiky y × φ(x) ξσ ,
with ξσ ∈ C2 a two-component spinor and
L
k
2π z
= mz ∈ Z,
0 ≤ x, y, z ≤ L,
L
k
2π y
(3.87)
= 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
x − ky ℓ2c
2
2 2
n
2 −1/2
0 ≤ x ≤ L,
× e−(x−ky ℓc ) /(2ℓc ) ,
φn,ky (x) = 2 n!2πℓc
× Hn
ℓ
and
εn,kz ,σ
with
n ∈ N,
~2 kz2
1
=
− µB Bσ,
+ ~ωc n +
2m
2
σ = ±,
|eB|
,
ωc :=
mc
ℓc :=
s
(3.88a)
(3.88b)
~c
,
|eB|
(3.88c)
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
L2
2πℓ2c
(3.89)
as follows from the constraint
0≤
2πmy 2
ℓ ≤ L,
L c
my ∈ Z
⇐⇒
28
0 ≤ my ≤
L2
,
2πℓ2c
my ∈ Z.
(3.90)
In the thermodynamic limit L → ∞, the density of states per unit energy, per unit volume,
and per spin is
X
L2
δ ε − εn,kz ,σ
×
2
L→∞
2πℓc
kz
Z
~2 kz2
1
1
× dkz δ ε −
+ µB Bσ
− ~ωc n +
=
2πℓ2c
2m
2
R
3/2
(2m) ωc Θ ε − ~ωc n + 12 + µB Bσ
q
.
=
8π 2
1
ε − ~ωc n + 2 + µB Bσ
ν(ε, n, σ) := lim L−3 ×
(3.91)
For a fixed n ∈ N 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 (3.7) 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. (3.2a)
with the identifications
ι → (n, kz , σ),
~2 kz2
1
− µB Bσ,
ει →
+ ~ωc n +
2m
2
n ∈ N,
2π
k ∈ Z,
L z
σ = ±.
(3.92)
The magnetization per unit volume M can be calculated in closed form with the help of the
Poisson formula. It is

M = χP B 1 −
The susceptibility
1 πkB T
+
3
µB B
r

πεF m
π
−
sin
4
µ B
εF
1
2 B .
√
µB B m=1 m sinh π kB T m
∞
X
∂M
∂B
reduces to the difference between the Pauli susceptibility
χ :=
(3.93)
µB B
(3.94)
χP = µ2B ν(εF )
(3.95)
1
χL = − χP
3
(3.96)
kB T
≫ 1.
µB B
(3.97)
and the Landau susceptibility
in the limit
29
In the opposite limit
kB T
≪ 1.
µB B
(3.98)
of very low temperatures, the dependence of χ on 1/B oscillates with the dominant period
∆(1/B) given by
πεF
∆(1/B) = 2π,
µB
(3.99)
i.e.,
∆(1/B) =
2µB
εF
2m
e~
× 2 2
2mc ~ kF
2πe 1
)
=
~c A(kF
=2×
(3.100)
where
A(kF ) = πkF2
(3.101)
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
non-interacting jellium model shortly after qualitatively similar oscillations were measured
in metals by 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.
IV.
A.
TIME-ORDERED GREEN FUNCTIONS
Introduction
Before specializing to the case of the non-interacting jellium model, we consider the
generic case of a conserved many-body Hamiltonian Ĥµ acting on a Z2 -graded Fock space
F . The Fock space
(
hh
ii
Y
n
F := span
â†ι ι |0i âι , â†ι′ = δι,ι′ ,
ι
ii
ii hh
hh
âι , âι′ = â†ι , â†ι′ = 0,
deg (âι ) = 0 ⇒ nι ∈ N,
30
âι |0i = 0,
deg (âι ) = 1 ⇒ nι = 0, 1
) (4.1a)
kz
B
kF
ky
kx
FIG. 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.
is Z2 -graded because any pair âι , â†ι carries, through its degree
deg â†ι ≡ deg âι = 0, 1,
the bosonic or fermionic algebra
ii
hh
†
âι , âι′ := âι â†ι′ − (−)deg(âι )deg(âι′ ) â†ι′ âι = δι,ι′ ,
ii
hh
â†ι , â†ι′ := â†ι â†ι′ − (−)deg(âι )deg(âι′ ) â†ι′ â†ι = 0,
ii
hh
âι , âι′ := âι âι′ − (−)deg(âι )deg(âι′ ) âι′ âι = 0,
(4.1b)
(4.1c)
whenever
deg (âι ) = 0
(4.1d)
deg (âι ) = 1,
(4.1e)
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 (4.1) in such a way that it
can be decomposed into the sum of two non-commuting and conserved Hermitean operators
Ĥ0,µ and Ĥ1 ,
Ĥµ = Ĥ0,µ + Ĥ1 ,
31
h
i
Ĥ0,µ , Ĥ1 6= 0
(4.2)
whereby Ĥ0,µ is the quadratic form
Ĥ0,µ =
X
ι
(ει − µ) â†ι âι
(4.3)
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 .
(4.4)
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 .
(4.5)
Ĥµ ≡ Ĥµ (λ = 1)
(4.6)
We shall use the notation
so that Ĥµ (λ) interpolates between Ĥ0,µ and Ĥµ as λ varies between 0 and 1.
The free energy in the grand canonical ensemble is defined by
1
1
F (β, µ; λ) := U(β, µ; λ) − T S(β, µ; λ) ≡ − ln TrF e−β Ĥµ (λ) ≡ − ln Z(β, µ; λ).
β
β
(4.7)
The thermal expectation value of the interaction is
−β Ĥµ (λ)
D E
Tr
e
Ĥ
F
1
∂F (β, µ; λ)
=
≡ Ĥ1
.
∂λ
β,µ;λ
TrF e−β Ĥµ (λ)
(4.8)
The change in the free energy induced by switching on the interaction adiabatically is
F (β, µ, 1) − F (β, µ, 0) =
Z1
∂F (β, µ; λ)
dλ
=
∂λ
0
Z1
0
E
dλ D
λĤ1
.
λ
β,µ;λ
(4.9)
It turns out (see appendix D) that the grand canonical expectation value
D
λĤ1
E
β,µ;λ
(4.10)
can be related to the so-called time-ordered single-particle Green function. One important
physical meaning of the time-ordered single-particle Green function is thus that it encodes
the correlation energy (4.9). With this motivation in mind, we first define time-ordered
Green functions in the grand canonical ensemble.
32
B.
Time-ordered Green functions in imaginary time
Let Â be any operator acting on the Fock space (4.1) on which the grand canonical
partition function (4.4). is defined. Examples of fermionic operators for the non-interacting
jellium model are
ĉ†σ,k
1
=√
V
Z
3
+ik·r
d re
ψ̂σ† (r),
V
1 X −ik·r †
ψ̂σ† (r) = √
e
ĉσ,k,
V k
ĉσ,k
1
=√
V
Z
d3 r e−ik·rψ̂σ (r),
(4.11a)
V
1 X +ik·r
ψ̂σ (r) = √
e
ĉσ,k.
V k
(4.11b)
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 non-interacting jellium model are
Z
XX †
ρ̂q ≡
ĉk,σ ĉk+q,σ = d3 r e−iq·r ρ̂(r),
σ
ρ̂(r) ≡
k
X
V
ψ̂σ† (r)ψ̂σ (r)
σ
1 X X +iq·r
=
e
ρ̂q .
V σ q
(4.12)
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 direct
space.
Operator Â, as any operator in 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 Â with the explicit
dependence on imaginary time
ÂH (τ, τ0 ) := e+(τ −τ0 )Ĥµ Â(τ0 ) e−(τ −τ0 )Ĥµ ,
Â(τ0 ) ≡ Â.
(4.13)
The index H stands for the Heisenberg picture. Alternatively, we endow the operator Â 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.
(4.14)
The index I stands for the interacting picture.
The equations of motion obeyed by Â 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,
33
(4.15)
in the Heisenberg picture and
∂τ ÂI
h
i
(τ, τ0 ) = Ĥ0,µ , ÂI (τ, τ0 ) ,
ÂI (τ0 ) = Â,
(4.16)
in the interacting picture.
In the Heisenberg 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


ÛH (τ, τ0 ) ≡ Tτ exp − dτ ′ Ĥµ (τ ′ , τ0 )
τ0
:= 1 +
∞
X
(−)n
n=1
Zτ
dτn · · ·
τ0
Zτ3
dτ2
τ0
Zτ2
(4.17a)
dτ1 Ĥµ (τn , τ0 ) · · · Ĥµ (τ2 , τ0 )Ĥµ (τ1 , τ0 )
τ0
= e−(τ −τ0 )Ĥµ
as their imaginary-time evolution is governed by the imaginary-time Schrödinger equation
∂τ ÛH (τ, τ0 ) = −Ĥµ ÛH (τ, τ0 ) ⇐⇒ ∂τ Ψ(τ, τ0 ) = −Ĥµ Ψ(τ, τ0 ),
Ψ(τ0 ) given.
(4.17b)
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 ′ Ĥ1I (τ ′ , τ0 )
τ0
Zτ
Zτ3
Zτ2
∞
X
:= 1 +
(−)n dτn · · · dτ2
dτ1 Ĥ1I (τn , τ0 ) · · · Ĥ1I (τ2 , τ0 )Ĥ1I (τ1 , τ0 )
n=1
τ0
τ0
τ0
(4.18a)
as their imaginary-time evolution is governed by the imaginary-time first-order differential
equation
∂τ ÛI (τ, τ0 ) = −Ĥ1I (τ )ÛI (τ, τ0 ) ⇐⇒ ∂τ ΨI (τ, τ0 ) = −Ĥ1I (τ ) ΨI (τ, τ0 ),
ΨI (τ0 ) given.
(4.18b)
The operation of imaginary-time ordering used in Eqs (4.17a) and (4.18a) is defined for any
34
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 )



Â(τ1 , τ0 )B̂(τ2 , τ0 ),
when τ1 > τ2 ,


≡



 (±)B̂(τ , τ )Â(τ , τ ), when τ > τ ,
2 0
1 0
2
1
(4.19)
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
(4.20)
depends explicitly on imaginary time in the Schrödinger-like equation (4.18b). Hence, the
integration over imaginary time cannot be performed explicitly in Eq. (4.18a). Neither
ÛH (τ, τ0 ) nor ÛI (τ, τ0 ) are unitary, but they share the composition law
ÛH (τ, τ ′ )ÛH (τ ′ , τ0 ) = ÛH (τ, τ0 ) =⇒ ÛH−1 (τ, τ ′ ) = ÛH (τ ′ , τ ),
(4.21)
ÛI (τ, τ ′ )ÛI (τ ′ , τ0 ) = ÛI (τ, τ0 ) =⇒ ÛI−1 (τ, τ ′ ) = ÛI (τ ′ , τ ),
(4.22)
and
for all triplets (τ, τ ′ , τ0 ), respectively. Either ÛH (τ, τ0 ) or ÛI (τ, τ0 ) become unitary under
the analytical continuation
τ ∈ R → +it,
t ∈ R.
(4.23)
The relation between the imaginary-time evolution in the Heisenberg and interaction pictures
is
ÛI (τ, τ0 ) = e+(τ −τ0 )Ĥ0,µ ÛH (τ, τ0 ) e−(τ −τ0 )Ĥ0,µ ⇐⇒ ΨH (τ, τ0 ) = e−(τ −τ0 )Ĥ0,µ ΨI (τ, τ0 ).
(4.24)
Let Â and B̂ be any pair of operator with the degrees deg(Â) and deg(B̂), respectively,
from the Fock space (4.1) on which the grand canonical partition function (4.4) is defined.
The time-ordered correlation function in imaginary time between Â and B̂ is the expectation
35
value
Cβ,µ;Â,B̂ (τ1 , τ2 ) = −
h
i
TrF e−β Ĥµ Tτ ÂH (τ1 , τ0 )B̂H (τ2 , τ0 )
Tr e−β Ĥµ
D F
E
≡ − Tτ ÂH (τ1 , τ0 )B̂H (τ2 , τ0 )
β,µ
(4.25)
.
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 (4.25) as
Cβ,µ;Â,B̂ (τ1 , τ2 ) = Cβ,µ;Â,B̂ (τ1 − τ2 ).
(4.26a)
(ii) The correlation function (4.25) decays exponentially fast with |τ1 − τ2 | only if
|τ1 − τ2 | < β.
(4.26b)
It grows exponentially fast with |τ1 − τ2 | otherwise.
(iii) The correlation function (4.25) can be extended to all values |τ1 − τ2 | provided it is
either periodic if Â and B̂ are bosonic or antiperiodic if Â and B̂ are fermionic with
period β,
− β < τ < 0 =⇒ Cβ,µ;Â,B̂ (τ ) = ±Cβ,µ;Â,B̂ (τ + β).
(4.26c)
(iv) If Â and B̂ are bosonic, then
β−
Z
1 X −i̟l τ
e
Cβ,µ;ÂB̂,i̟ ⇐⇒ Cβ,µ;ÂB̂,i̟ = dτ e+i̟l τ Cβ,µ;ÂB̂ (τ )
Cβ,µ;ÂB̂ (τ ) =
l
l
β l∈Z
0+
(4.26d)
with the bosonic Matsubara frequency ̟l = 2lπ/β. If Â and B̂ are fermionic, then
β−
Z
1 X −iωn τ
e
Cβ,µ;ÂB̂,iω ⇐⇒ Cβ,µ;ÂB̂,iω = dτ e+iωn τ Cβ,µ;ÂB̂ (τ )
Cβ,µ;ÂB̂ (τ ) =
n
n
β n∈Z
0+
(4.26e)
with the fermionic Matsubara frequency ωn = (2n + 1)π/β. The asymmetric convention for the normalization by β is the same as in Eq. (4.12).
36
Proof. Cyclicity of the trace with the definition (4.13) implies
h
i
TrF e−β Ĥµ e+(τ1 −τ2 )Ĥµ Âe−(τ1 −τ2 )Ĥµ B̂
Cβ,µ;Â,B̂ (τ1 , τ2 ) = − Θ(τ1 − τ2 )
TrF e−β Ĥµ
h
i
TrF e−β Ĥµ e+(τ2 −τ1 )Ĥµ B̂e−(τ2 −τ1 )Ĥµ Â
∓ Θ(τ2 − τ1 )
TrF e−β Ĥµ
(4.27)
from which (i) and (iii) follow. Insertion of a complete basis of eigenstates of Ĥµ in Eq. (4.27),
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. (4.26b).
The Fourier transforms (4.26d) and (4.26e) follow from the periodicity (iii).
The correlation function (4.25) cannot be evaluated exactly in practice. For a systematic
perturbation theory, a better suited representation of Eq. (4.25) is
"
#
TrF e−β Ĥ0,µ Tτ ÛI (β, 0)ÂI (τ1 )B̂I (τ2 )
Cβ,µ;Â,B̂ (τ1 − τ2 ) = −
h
TrF e−β Ĥ0,µ ÛI (β, 0)
i.
(4.28)
Proof. Equation (4.28) follows from Eq. (4.25) with the help of Eqs (4.24) and
ÂH (τ1 ) B̂H (τ2 ) = e+τ1 Ĥµ Â e−(τ1 −τ2 )Ĥµ B̂ e−τ2 Ĥµ
= e+τ1 Ĥµ Â ÛH (τ1 , τ2 ) B̂ e−τ2 Ĥµ
= e+τ1 Ĥµ e−τ1 Ĥ0,µ ÂI (τ1 ) e+τ1 Ĥ0,µ ÛH (τ1 , τ2 ) e−τ2 Ĥ0,µ B̂I (τ2 ) e+τ2 Ĥ0,µ e−τ2 Ĥµ
= ÛI (0, τ1 ) ÂI (τ1 ) ÛI (τ1 , τ2 ) B̂I (τ2 ) ÛI (τ2 , 0) .
(4.29)
Another useful tool to evaluate the correlation function (4.25) is the equation of motion
D
E
−∂τ1 Cβ,µ;Â,B̂ (τ1 − τ2 ) = δ (τ1 − τ2 ) ÂH (τ1 )B̂H (τ1 ) ∓ B̂H (τ1 )ÂH (τ1 )
β,µ
(4.30)
D nh
i
oE
+ Tτ Ĥµ , ÂH (τ1 ) B̂H (τ2 )
β,µ
that is obeyed for any unequal imaginary times τ1 and τ2 . In general, the commutator
h
i
Ĥµ , ÂH (τ1 ) is not proportional to ÂH (τ1 ) so that this equation does not close on its own.
In fact, a closed set of equations of motion is generically infinite.
37
The definition (4.25) readily generalizes to the 2n-point time-ordered correlation function
in imaginary time between operators Â1 , · · · , Ân and B̂1 , · · · , B̂n belonging to the Fock
space (4.1) on which the grand canonical partition function (4.4) is defined. It is
Cβ,µ;Â
1 ,··· ,Ân |B̂1 ,··· ,B̂n
(τ1 , · · · , τn |τ1′ , · · · , τn′ ) :=
D E
(−)n Tτ ÂH (τ1 , τ0 ) × · · · × ÂH (τn , τ0 ) × B̂H (τ1′ , τ0 ) × · · · × B̂H (τn′ , τ0 )
(4.31)
β,µ
.
Next, we are going to compute explicitly 2 and 4 points time-ordered Green functions for
the non-interacting jellium model, whereby we shall make the identifications
B̂ → ĉ†σ,k,
Â → ĉσ,k,
C.
ει → εσ,k ≡
~2 k 2
.
2m
(4.32)
Time-ordered Green functions in real time
Imaginary time τ and real time t are related by the analytical continuation
τ = it.
(4.33)
As before, let Â and B̂ be any pair of operator with the degrees deg(Â) and deg(B̂), respectively, from the Fock space (4.1) on which the grand canonical partition function (4.4)
is defined. The time-ordered correlation function in real time between Â and B̂ is the
expectation value
iCβ,µ;Â,B̂ (t1 , t2 ) = −
h
i
TrF e−β Ĥµ Tt ÂH (t1 , t0 )B̂H (t2 , t0 )
Tr e−β Ĥµ
D F
E
≡ − Tt ÂH (t1 , t0 )B̂H (t2 , t0 )
where
β,µ
ÂH (t, t0 ) := e+i(t−t0 )Ĥµ Â(t0 )e−i(t−t0 )Ĥµ
and
Tt
(4.34a)
(4.34b)
Â(t1 , t0 )B̂(t2 , t0 ) := Â(t1 , t0 )B̂(t2 , t0 )Θ (t1 − t2 )
≡
+ (−)deg(Â)deg(B̂) B̂(t2 , t0 )Â(t1 , t0 )Θ (t2 − t1 )



Â(t1 , t0 )B̂(t2 , t0 ),
when t1 > t2 ,





 (−)deg(Â)deg(B̂) B̂(t , t )Â(t , t ), when t > t .
2 0
1 0
2
1
(4.34c)
The sign on the right-hand side of Eq. (4.34a) is here convention, as is the imaginary factor
on the left-hand side of Eq. (4.34a).
38
D.
Time-ordered single-particle Green functions: non-interacting jellium model
The grand canonical partition function for the non-interacting jellium model is defined
in Eq. (3.1). We shall use the more compact notation
Ĥµ := Ĥ − µN̂ ,
1.
ξσ,k ≡ εσ,k − µ.
(4.35)
Momentum-space representation
The imaginary-time-ordered single-particle Green functions in momentum space is defined
by Eq. (4.25) with the identifications
B̂ → ĉ†σ2 ,k2 .
Â → ĉσ1 ,k1 ,
(4.36)
For any |τ1 − τ2 | < β, it is given by
E
D †
(τ
,
τ
)ĉ
(τ
,
τ
)
ĉ
(τ
−
τ
)
=
−
T
Cβ,µ;ĉ
†
1
2
τ
H σ1 ,k1 1 0 H σ2 ,k2 2 0
σ ,k ,ĉσ ,k
β,µ
2 2
1 1
i
h
TrF e−β Ĥµ e+(τ1 −τ2 )Ĥµ ĉσ1 ,k1 e+(τ2 −τ1 )Ĥµ ĉ†σ2 ,k2
= − Θ(τ1 − τ2 )
TrF e−β Ĥµ
i
h
TrF e−β Ĥµ e+(τ2 −τ1 )Ĥµ ĉ†σ2 ,k2 e+(τ1 −τ2 )Ĥµ ĉσ1 ,k1
+ Θ(τ2 − τ1 )
TrF e−β Ĥµ
≈ δσ1 ,σ2 δk1 ,k2 Gβ,µ (τ1 − τ2 , k1 )
(4.37a)
where
Gβ,µ (τ, k) = − Θ(+τ ) [1 − fF (ξk)] e−τ ξk − Θ(−τ )fF (ξk)e−τ ξk .
(4.37b)
It is extended to |τ1 − τ2 | > β by anti-periodicity. 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
∂Z(V, β, µ)
,
∂µ
(4.38)
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
the energy, Ne the electron number,
39
(4.39)
of Ĥµ between the creation and annihilation operators to go from the second equality to the
third equality of Eq. (4.37a). This brings the exponentials
”
“
(Ne +1)
(N )
−(τ1 −τ2 ) Eι,σ,k
−Eι e
e
≈ e−(τ1 −τ2 )ξk ,
(4.40)
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
”
“
(Ne −1)
(N )
+(τ1 −τ2 ) Eι,σ,k
−Eι e
e
≈ e−(τ1 −τ2 )ξk ,
(4.41)
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
lim [1 − fF (ξk)] = Θ(+ξk),
β→∞
lim fF (ξk) = Θ(−ξk),
β→∞
(4.42)
Eq. (4.37b) tells us that, at zero temperature, the imaginary-time-ordered 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→∞
(4.43)
Owing to the anti-periodic dependence (4.26e) for any fermionic Matsubara frequency ωn =
(2n + 1)π/β,
Gβ,µ (ωn , k) =
Zβ −
dτ e+iωn τ Gβ,µ (τ, k)
0+
=
Z0−
dτ e+iωn τ Gβ,µ (τ, k)
(4.44)
−β +
=
1
.
iωn − ξk
The real-time-ordered single-particle Green function in momentum space follows from
40
Eq. (4.37) with the analytical continuation (4.33), i.e., it is
D E
†
(t
−
t
)
=
−
T
ĉ
(t
,
t
)ĉ
(t
,
t
)
iCβ,µ;ĉ
†
1
2
t
H σ1 ,k1 1 0 H σ2 ,k2 2 0
σ1 ,k 1 ,ĉσ2 ,k 2
β,µ
i
h
TrF e−β Ĥµ e+(t1 −t2 )Ĥµ ĉσ1 ,k1 e+(t2 −t1 )Ĥµ ĉ†σ2 ,k2
= − Θ(t1 − t2 )
TrF e−β Ĥµ
i
h
TrF e−β Ĥµ e+(t2 −t1 )Ĥµ ĉ†σ2 ,k2 e+(t1 −t2 )Ĥµ ĉσ1 ,k1
+ Θ(t2 − t1 )
TrF e−β Ĥµ
≈ δσ1 ,σ2 δk1 ,k2 iGβ,µ (t1 − t2 , k1 )
(4.45a)
where
iGβ,µ (t, k) = − Θ(+t) [1 − fF (ξk)] e−itξk − Θ(−t)fF (ξk)e−itξk .
(4.45b)
At zero temperature,
lim iGβ,µ (t, k) = − Θ(+t)Θ(+ξk)e−itξk − Θ(−t)Θ(−ξk )e−itξk .
β→∞
(4.46)
In real-frequency space,
Gβ,µ (ω, k) :=
Z
R
=−
dt e+iωt Gβ,µ (t, k)
fF (ξk)
1 − fF (ξk)
+
ω − ξk + i0+ ω − ξk − i0+
(4.47)
with the zero-temperature limit
lim Gβ,µ (ω, k) =
β→∞
2.
−1
.
ω − ξk + i0+ sgn(ω)
(4.48)
Real-space representation
By combining Eqs. (4.11), (4.37), and (4.45), we obtain the real-space representation
iCβ,µ;ψ̂
†
σ1(r1 )ψ̂σ2(r2
(τ − τ2 ) =
) 1
1 X X −ik1 ·r1 +ik2 ·r2
(τ1 − τ2 )
e
iCβ,µ;ĉ
†
σ ,k ,ĉσ ,k
V k k
2 2
1 1
1
≈ δσ1 ,σ2
2
1 X −ik·(r1 −r2 )
iGβ,µ (τ1 − τ2 , k)
e
V k
≡ δσ1 ,σ2 iGβ,µ (τ1 − τ2 , r1 − r2 )
41
(4.49)
and
iCβ,µ;ψ̂
†
σ1(r1 )ψ̂σ2(r2 )
(t1 − t2 ) =
1 X X −ik1 ·r1 +ik2 ·r2
(t1 − t2 )
e
iCβ,µ;ĉ
†
σ1 ,k 1 ,ĉσ2 ,k 2
V k k
1
≈ δσ1 ,σ2
2
1 X −ik·(r1 −r2 )
iGβ,µ (t1 − t2 , k)
e
V k
(4.50)
≡ δσ1 ,σ2 iGβ,µ (t1 − t2 , r1 − r2 )
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
ν(ξ) :=
1 X
δ(ξk − ξ),
V k
(4.51)
in terms of which
iGβ,µ (τ, r = 0) = −
Z
iGβ,µ (t, r = 0) = −
Z
and
dξ ν(ξ) Θ(+τ ) [1 − fF (ξ)] e−τ ξ − Θ(−τ )fF (ξ)e−τ ξ
dξ ν(ξ) Θ(+t) [1 − fF (ξ)] e−itξ − Θ(−t)fF (ξ)e−itξ ,
respectively. For the parabolic spectrum of the jellium model,

p m


, for d = 1,

 2π2 ε
m
ε := ξ + µ,
ν(ε) =
,
for d = 2,
2π

√


ε
 m 2m
, for d = 3,
2π 2
(4.52)
(4.53)
(4.54)
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 imaginarytime single-particle Green function at equal points is, up to a proportionality constant, the
Laplace transform of the sign function, i.e.,
Z
ν
iGβ=∞,µ (τ, r = 0) ≈ −νF dξ Θ(+τ )Θ(+ξ)e−τ ξ − Θ(−τ )Θ(−ξ)e−τ ξ = − F .
τ
Analytical continuation to real time gives
Z
iGβ=∞,µ (t, r = 0) ≈ −νF dξ Θ(+t)Θ(+ξ)e−itξ − Θ(−t)Θ(−ξ)e−itξ = +
42
(4.55)
iνF
.
t − i0+ sgn(t)
(4.56)
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. (4.55) and 4.56) 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 right-hand sides
of Eqs. (4.55) and 4.56) 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, ν(ξ) ∼ |ξ|g with g > 0, the long-time
correlation probed by the single-particle Green function at equal points decay faster, e.g.,
Gβ=∞,µ (t, r = 0) ∼ +ie−iπ(1+g)/2
sgn(t)
.
|t|1+g
(4.57)
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.
3.
At equal times
We now combine Eqs. (4.50) and (4.45b) to study
1 X −ik·r
e
iGβ,µ (t, k)
V k
1 X −ik·r e
Θ(+t) [1 − fF (ξk)] e−itξk − Θ(−t)fF (ξk)e−itξk
=−
V k
iGβ,µ (t, r) :=
(4.58)
at equal times, i.e., in the limit t → −0+ (without loss of generality). In this limit, the equaltime single-particle Green function in real-space is the Fourier transform of the Fermi-Dirac
distribution,
iGβ,µ (t = −0+ , r) =
1 X −ik·r
e
fF (ξk).
V k
(4.59)
At zero temperature and in the thermodynamic limit, Eq. (4.59) reduces to the Fourier
43
transform over the Heaviside function
+
Z
dd k −ik·r
e
Θ(−ξk)
(2π)d
2 2
Z
dd k −ik·r
~ k F ~2 k 2
=
e
Θ
−
(2π)d
2m
2m
Z
d
d k −ik·r
e
Θ (kF − |k|) .
=
(2π)d
iGβ,µ (t = −0 , r) =
(4.60)
In d = 1,
iGβ,µ (t = −0+ , r) =
+kF
Z
dk −ikr
e
2π
(4.61)
−kF
=
2 sin kF r
.
r
For d > 1,
+
iGβ,µ (t = −0 , r) =
Z
dd k −ik·r
e
Θ (kF − |k|)
(2π)d
1
= d
|r|
Z
dΩd
(2π)d
1
= d
|r|
Z
b
dΩ
d−1
(2π)d
kF |r|
Z
dp pd−1 e−ip cos θ1
(4.62a)
0
2πδd,2 +(1−δd,2 )π
kF |r|
Z
dp p
d−1
Z
dθ1 sind−2 θ1 e−ip cos θ1
0
0
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 , (4.62b)
b
dΩ
d−1 :=
sind−3 θ2 dθ2 · · · sind−1−i θi dθi · · · sin θd−2 dθd−2 dθd−1 .
44
(4.62c)
Example d = 2,
kF |r|
Z
1/(2π)2
iGβ,µ (t = −0 , r) =
|r|2
+
Eq. (13.6.22) Ref. 1
0
dp p
X
Jn (p)
n∈Z
0
kF |r|
Z
dθ e−ip cos θ
0
kZF |r|
1/(2π)2
=
|r|2
1/(2π)
=
|r|2
8.472.3 from Ref. 2
dp p
Z2π
Z2π
dθ e−in( 2 −θ)
π
0
(4.63a)
dp p J0 (p)
0
kF |r|
Z
1/(2π)
=
|r|2
dp
0
d
p J1 (p)
dp
k
= F J1 (kF|r|).
2π|r|
Hence [see Eq. (4.4.5) Ref. 1],
k
lim iGβ,µ (t = −0 , r) ∼ − F
|r|→∞
2π|r|
+
s
π
sin kF |r| −
πkF |r|
4
2
(4.63b)
if d = 2. Example d = 3,
1/(2π)3
iGβ,µ (t = −0+ , r) =
|r|3
1/(2π)2
=
|r|3
kZF |r|
Z2π
kF |r|
Z+1
dp p2
0
e−iπ /(2π)2
=
|r|3
=
e
kF |r|
Z
dθ sin θ e−ip cos θ
0
dx e−ipx
dp p2
e−iπ/2 /(2π)2
=
|r|3
−iπ
0
0
Z
dϕ
Zπ
−1
dp p e+ip − e−ip
0
kZF |r|
dp
0
2
d p e+ip + e−ip − e+ip + e−ip
dp
/(2π) kF |r| e+ikF|r| + e−ikF |r| + i e+ikF|r| − e−ikF |r| .
3
|r|
(4.64a)
Hence [see Eq. (4.4.5) Ref. 1],
lim iGβ,µ (t = −0+ , r) ∼
|r|→∞
kF
e+ikF|r|−iπ + e−ikF |r|+iπ
2
2
(2π) |r|
45
(4.64b)
if d = 3. Examples (4.61), (4.63b), and (4.64b) illustrate the power-law decay of the equaltime 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 iGβ,µ (t = −0+ , r) ∼ c+
|r|→∞
e+bikF (+br)·r−iπ(d+1)/4
e−ikF(−br)·r+iπ(d+1)/4
+
c
+
|r|(d+1)/2
|r|(d+1)/2
(4.65)
where c± are numbers, rb ≡ r/|r|, and, given a coordinate system in momentum space with
the center of gravity of the Fermi surface as origin, kF (±b
r ) are the two Fermi points for
which the hyperplanes normal to rb are tangent in these points to the Fermi surface.
APPENDIX A: GRASSMANN COHERENT STATES
To simplify notation I will consider the two-dimensional fermion Fock space F spanned
by the fermion algebra of operators
{ĉ, ĉ† } = 1,
{ĉ, ĉ} = {ĉ† , ĉ† } = 0.
(A1)
In other words F is spanned by the two normalized and orthogonal vectors
|0i,
|1i := ĉ† |0i,
(A2)
where ĉ|0i = 0 defines the vacuum |0i. To be concrete, define the fermion harmonic oscillator
by the Hamiltonian
Ĥ := ĉ† ĉ.
(A3)
There are two eigenvalues, 0 and 1, and the partition function at inverse temperature β is
Zβ := tr|F e−β Ĥ = 1 + e−β .
(A4)
Is it possible to construct a path integral representation of this partition function as was
done for the boson harmonic oscillator in lecture 1? If the boson harmonic oscillator is
to be a guide, we need to construct coherent states defined by the conditions that they
are eigenstates of the annihilation operator ĉ, overcomplete, and provide a resolution of the
46
identity. The route to fermion coherent states goes through the introduction of a Grassmann
algebra.
A four dimensional Grassmann algebra is defined by considering all possible polynomials
with complex valued coefficients that can be built from monomials in the two independent
Grassmann numbers η ∗ and η that obey the following multiplication rules
{η, η ∗} = {η, η} = {η ∗ , η ∗ } = 0.
(A5)
In other words, η ∗ and η are anticommuting numbers and a generic element of the Grassmann
algebra G is written
a ≡ a1 + a2 η + a3 η ∗ + a4 η ∗ η = a1 + a2 η + a3 η ∗ − a4 ηη ∗ ,
a1,2,3,4 ∈ C.
(A6)
Observe that the Fock space F is a vector space spanned by |0i and |1i over the complex
numbers,
n
F := |a1 , a2 i |a1 , a2 i = a1 |0i + a2 |1i,
o
a1,2 ∈ C .
(A7)
By analogy, define the Grassmann Fock space FG to be the vector space spanned by |0i and
|1i over the Grassmann algebra (A5-A6),
n
o
FG := |a, bi|a, bi = a1 + a2 η + a3 η ∗ + a4 η ∗ η |0i + b1 + b2 η + b3 η ∗ + b4 η ∗ η |1i, ai , bi ∈ C , (A8)
whereby the consistency rule that Grassmann numbers η and η ∗ anticommute with fermion
annihilation ĉ and creation ĉ† operators,
{η, ĉ} = {η, ĉ† } = {η ∗ , ĉ} = {η ∗ , ĉ† } = 0
(A9a)
must be imposed.
Fermion coherent states |a2 ηi and |a∗3 η ∗ i of the Grassmann Fock space FG are defined by
−a2 η ĉ†
|a2 η i := e
∗ ∗ ĉ†
|a∗3 η ∗ i := e−a3 η
∞
X
−a2 η ĉ†
|0i ≡
n!
n=0
|0i ≡
∞
X
n=0
n
n
−a∗3 η ∗ ĉ†
n!
47
|0i = 1 − a2 η ĉ† |0i,
a2 ∈ C,
|0i = 1 − a∗3 η ∗ ĉ† |0i,
a∗3 ∈ C.
(A10a)
(A10b)
The corresponding adjoint coherent states ha2 η| and ha∗3 η ∗ | of the Grassmann Fock space FG
are defined by
ha2 η | := h0|e−ĉη
ha∗3 η ∗ | := h0|e−ĉη
∗ a∗
2
a3
≡ h0|
∞
X
(−ĉη ∗ a∗ )n
≡ h0|
∞
X
2
n!
n=0
n=0
= h0| (1 − ĉη ∗ a∗2 ) ,
a2 ∈ C,
(A11a)
(−ĉη a3 )n
= h0| (1 − ĉη a3 ) ,
n!
a∗3 ∈ C.
(A11b)
With these two definitions in hand, one verifies that |a2 ηi (|a∗3 η ∗ i) is a right eigenstate of ĉ
with Grassmann eigenvalue a2 η (a∗3 η ∗ ) and that ha2 η| (ha∗3 η ∗ |) is a left eigenstate of ĉ† with
Grassmann eigenvalue a∗2 η ∗ (a3 η),
ĉ|a2 η i = (−)2 a2 η |0i = a2 η |a2 η i,
ha2 η |ĉ† = h0|a∗2 η ∗ (−)2 = ha2 η |a∗2 η ∗ ,
(A12a)
ĉ|a∗3 η ∗ i = (−)2 a∗3 η ∗ |0i = a∗3 η ∗ |a∗3 η ∗ i,
ha∗3 η ∗ |ĉ† = h0|a3 η (−)2 = ha∗3 η ∗ |a3 η .
(A12b)
Fermion coherent states are neither normalized nor orthogonal,
hη |η i = h0| (1 − ĉη ∗ ) 1 − η ĉ† |0i = 1 + (−)4 η ∗ η
hη ∗ |η ∗ i = h0| (1 − ĉη ) 1 − η ∗ ĉ† |0i = 1 + (−)4 η η ∗
hη |η ∗ i = h0| (1 − ĉη ∗ ) 1 − η ∗ ĉ† |0i = 1 + (−)4 η ∗ η ∗
hη ∗ |η i = h0| (1 − ĉη ) 1 − η ĉ† |0i = 1 + (−)4 η η
∗
= e+η η ,
∗
(A13a)
= e−η η ,
(A13b)
= 1,
(A13c)
= 1.
(A13d)
The expectation value of any normal ordered operator Ĉ(ĉ† , ĉ) in the fermion coherent state
|ηi is
∗
hη|Ĉ(ĉ† , ĉ)|ηi = e+η η C(η ∗ , η).
(A14)
Here, the Grassmann valued function C(η ∗ , η) is obtained from the operator Ĉ(ĉ† , ĉ) by
replacing ĉ† by η ∗ and ĉ by η.
Fermion coherent states are merely a mathematical trick that allows a path integral
representation of partition functions for fermions.7 To this end, we still need a resolution
7
A word of caution here. Gauge potentials in classical electrodynamics were also thought to be mathematical curiosities before the advent of quantum mechanics.
48
of the identity, which, in turn, demands the notion of Grassmann integration. Grassmann
R
R
integrations dη and dη ∗ are multilinear mappings from the Grassmann algebra (A5-A6)
to the complex numbers which are defined by linear extension of the rules
Z
Z
Z
Z
∗
∗
dη 1,
1 = dη η ,
0 = dη 1,
0 = dη η = −η
Z
Z
Z
Z
∗
∗
∗
0 = dη 1,
0 = dη η = −η
dη 1,
1 = dη ∗ η ∗ ,
Z
Z
Z
Z
Z
∗
∗
dη
· · · = − dη
dη
· · · ≡ dη ∗ dη · · · .
dη
(A15a)
(A15b)
(A15c)
Thus,
Z
Z
dη (a1 + a2 η + a3 η ∗ + a4 η ∗ η) = a2 − a4 η ∗ ,
dη ∗ (a1 + a2 η + a3 η ∗ + a4 η ∗ η) = a3 + a4 η ,
Z
Z
∗
dη (a1 + a2 η + a3 η ∗ + a4 η ∗ η) = −a4 ,
dη
∀a1,2,3,4 ∈ C,
(A16a)
∀a1,2,3,4 ∈ C,
(A16b)
∀a1,2,3,4 ∈ C.
(A16c)
Grassmann integration over the Grassmann Fock space FG in Eq. (A9a-A9a) is the same as
Grassmann integration over the Grassmann algebra with the caveat that ĉ and ĉ† anticomR
R
mute with dη and dη ∗ .
We are now ready to establish a resolution of identity for fermion coherent states. Indeed,
with the help of the resolution of the identity
11F = |0ih0| + |1ih1|,
(A17)
in F , we have
Z
Z
Z
Z
−η∗ η
∗
∗
dη (1 − η ∗ η) 1 − ηĉ† |0ih0| (1 − ĉη ∗ )
dη e
|ηihη| =
dη
dη
Z Z
∗
∗
∗
∗
= dη dη |0ih0| − |0ih1|η − η|1ih0| + η|1ih1|η − η η|0ih0|
= |1ih1| + |0ih0|
= 11F .
(A18)
49
Moreover, we have the trace formula
Z
Z
Z
Z
∗
∗
−η∗ η
∗
∗
dη (1 − η η) h0| − η h1| Ĉ |0i + |1iη
dη e
h−η|ĉ| + ηi =
dη
dη
Z Z
=
dη ∗ dη h0|ĉ|0i + h0|Ĉ|1iη − η ∗ h1|Ĉ|0i − η ∗ h1|Ĉ|1iη
− η ∗ ηh0|Ĉ|0i
= h1|Ĉ|1i + h0|Ĉ|0i
= trF ĉ,
(A19)
for any linear operator Ĉ : F → F . The asymmetry in the sign of η in the fermion coherent
state trace formula should be contrasted with the bosonic coherent state trace formula. This
asymmetry will lead to the Fermi-Dirac distribution function.
APPENDIX B: PATH INTEGRAL REPRESENTATION FOR FERMIONS
With the help of the overlap, resolution of the identity, and trace formula in Eqs. (A13),
(A18), and (A19), respectively, it is possible to represent the partition function (A4) as the
Grassmann path integral,
!
!
M X
β
Zβ = lim
dηj∗ dηj exp −
ηj∗ ηj − ηj−1 + H(ηj∗ , ηj−1)
M →∞
M
j=1
j=0

 β
Z
Z
∗

≡
D[η , η] exp − dτ ηj∗ (τ )∂τ η(τ ) + H[η ∗ (τ ), η(τ )]  ,
(B1a)
M
−1 Z
Y
Z
0
where
ηM ≡ −η0 =⇒ η (τ + β) = −η (τ ),
(B1b)
∗
ηM
≡ −η0∗ =⇒ η ∗ (τ + β) = −η ∗ (τ ).
(B1c)
The manipulations that lead to Eq. (B1a) are identical to those made for the boson harmonic
oscillator in lecture 1. The only change comes about due to the asymmetry of the trace
formula (A19). It leads to the integration variables in the path integral representation
obeying antiperiodic boundary conditions in time. Equation (B1a) holds not only for Ĥ in
Eq. (A4) but for any normal ordered operator. However, since Eq. (A4) is quadratic, direct
50
evaluation of the first line in Eq. (B1a) can be performed,


β
1
0
0 ··· 0
0
0
1− M


β −1

1
0
·
·
·
0
0
0
0
M



β
 0

−
1
1
·
·
·
0
0
0
0


M






Zβ = lim det  .

.
.
.
.
.
.
.
M →∞
 ..
.. 
..
..
.. .. ..
..










β
 0

0
0
·
·
·
0
−
1
1
0
M


β
−1
1
0
0
0 ··· 0
0
M
#
"
MY
−1
β
β
M −1
(−) 1 −
1−
= lim 1 + (−)
M →∞
M j=1
M
"
#
M
β
= lim 1 + 1 −
M →∞
M
= 1 + e−β ,
(B2)
as it should be. It follows that the average energy level occupation number
fβ := −∂β ln Zβ
(B3)
is nothing but the Fermi-Dirac distribution function,
fβ := −∂β ln Zβ =
e−β
1
= +β
.
−β
1+e
e +1
(B4)
Had we imposed a boson algebra instead of the fermion algebra in Eq. (A1), the path
integral representation (B1a) would have to be modified in two ways. First, the integration
variables would be conventional complex numbers and the path integral would be an infinite
product of one dimensional Riemann integrals. Second, the integration variables would obey
periodic boundary conditions. These two changes would turn Eq. (B2) into
Zβ = 1 − e−β
−1
,
(B5)
from which follows that the average energy level occupation number
fβ := −∂β ln Zβ
(B6)
obeys the Bose-Einstein distribution
fβ := −∂β ln Zβ =
e−β
1
=
.
1 − e−β
e+β − 1
51
(B7)
APPENDIX C: JORDAN-WIGNER FERMIONS
1.
Introduction
To illustrate the physical relevance of spinless fermions, we are going to introduce JordanWigner fermions in this section. This will also allow us to demonstrate how spinless fermions
can emerge from a many-body Hamiltonian built out of hard-core bosonic operators.
Modern high-energy physics postulates the existence of two kinds of point-like elementary particle. There are bosons that mediate the strong (through the gluons), weak (through
the W ± and Z bosons), and electro-magnetic (through the photon) interactions. There are
fermions such as quarks or such as leptons. Quarks make up composite particles known as
baryons or mesons as a result of the strong interactions mediated by the gluons. Leptons interact with each others and with quarks through the weak and electro-magnetic interactions.
Finally, there is the Higgs boson that endows selected fermions with a mass.
All present scattering experiments probing length scales of the order of the weak interaction range can be understood within the standard model. The standard model is a
relativistic quantum field theory built out of quarks and leptons that interact by the exchange of gluons, W ± and Z bosons, and the photon through gauge invariant interactions.
In the standard model, quarks, leptons, gauge bosons, and the Higgs boson are all point-like
particles that are treated on equal footing. However, to this date, the Higgs bosons has yet
to be directly observed. Hence, it cannot be ruled out that it is a mere mathematical abstraction that quantifies a structure of the world below some characteristic length scale that
has yet to be observed by modern physics. Since the Higgs boson is the agent responsible
for the measured rest masses of leptons in the standard model, it could very well be that
leptons in the standard model are not elementary particle at sufficiently small length scale
but quasi-particles that emerge from a more fundamental organization principle than the
standard model.
Perhaps the most ambitious and radical organization principle for the physics beyond the
standard model is string theory. String theory is an attempt to deduce the standard model
with its elementary point-like particles and gravity from a more fundamental organization
principle based on elementary objects that have a one-dimensional extension in space, i.e.,
strings. In string theory, fermions and bosons have been replaced by more fundamental
52
objects: strings.
To this date, string theory has not been validated experimentally. It is a speculative
theory that has yet to be confronted with experiments. On the other hand, there are some
examples in condensed matter physics for which the interacting elementary constituents are
neither bosonic nor fermionic although the relevant low-energy degrees of freedom can be.
To illustrate this fact, we consider a condensed matter system made of point-like particles
that are (i) static (i.e., of infinite mass), but (ii) neither bosonic nor fermionic, and (iii)
interact with each others. These point-like particles are defined on the sites of some lattice Λ
(any countably set Λ of cardinality |Λ|), i.e., a regular arrangement of points in d-dimensional
space. We shall denote the sites of the lattice embedded in a space of dimension d larger
than one by boldfaced Latin letters, say i and j. To each lattice site i, we assign the
vector-valued operator
where

0
σix := 
1
1
0

,
 
σx
 i
 
σi := σiy  ,
 
σiz
(C1a)


+1 0
,
σiy := 
0 −1

0 −i
,
σiy := 
+i 0

(C1b)
are the usual Pauli matrices. These Pauli matrices act on the local two-dimensional Hilbert
space spanned by the eigenstates

 

1
Hi := span | ↑ii :=   ,

0
 
0 
| ↓ii :=  
1 
(C1c)
of σiz . Next, we assign to the lattice Λ the global Hilbert space
HΛ :=
O
i∈Λ
Hi
(C2a)
with the operator algebra
σia , σjb = 2δij iǫabc σic ,
(σia )2 = 11i ,
a, b = x, y, z,
i, j ∈ Λ.
(C2b)
Finally, given any choice of a |Λ| × |Λ| real-valued, symmetric, and positive matrix with the
matrix elements
∗
∗
Jij = Jij
= Jji
= Jji
53
(C3a)
where i, j ∈ Λ together with the real-valued number 0 ≤ λ < ∞, we define the Hamiltonian
Ĥ xxz :=
1 X
Ji,j σix σjx + σiy σjy + λσiz σjz .
2
(C3b)
i,j∈Λ
When λ = 0, we have defined the so-called quantum xy model. When λ = 1, we have defined
the antiferromagnetic Heisenberg model. When λ = ∞, we have defined the classical Ising
model. All three cases are of relevance to certain classes of materials in condensed matter
physics.
For any value of 0 ≤ λ < ∞, the local degrees of freedom (C1) are neither bosonic
nor fermionic according to the global Hilbert space and algebra (C2). The fact that they
commute when localized on different sites is reminiscent of bosons. The fact that the local
Hilbert space is finite is reminiscent of hard-core bosons or, alternatively, of fermions.
In fact, it is always possible to reformulate the problem solely in terms of spinless fermions
defined by the non-local (Jordan-Wigner) transformation
ĉ†iι
Y
1 x
y
σiz ′ ,
σiι + iσiι
:=
ι
2
′
ι <ι
Y
1 x
y
ĉiι :=
σiz ′
σiι − iσiι
ι
2
′
(C4a)
ι <ι
where we have chosen the ordering of the lattice
Λ = i1 , i2 , · · · , iι , · · · , i|Λ| ,
(C4b)
for it can be shown (see below) that
n
ĉi , ĉ†i ′
ι
ι
o
= διι′ ,
n
o n
o
0 = ĉi , ĉi ′ = ĉ†i , ĉ†i ′ .
ι
ι
ι
ι
(C4c)
Observe however that the transformation (C4a) does not generically simplify the Hamiltonian (C3b) since the quantum xy limit of Hamiltonian (C3b) is generically non-local when
expressed in terms of the Jordan-Wigner fermions (C4a). A remarkable exception to this
rule was discovered by Jordan and Wigner. It occurs when the lattice is one-dimensional
and the exchange couplings (C3a) are only non-vanishing for pairs of nearest-neighbor sites
(see below).
Before we consider the case of the nearest-neighbor quantum xy model in one dimension,
we should remember that the nearest-neighbor quantum Heisenberg antiferromagnet on a
cubic lattice breaks spontaneously the SU(2) spin rotation symmetry down to its U(1)
subgroup so that, by the Goldstone theorem, the low-lying excitations are bosonic modes
54
called magnons. Anticipating the results of the following section, we thus conclude that the
low energy excitations of xxz Hamiltonian (C3b) can either be bosons or fermions depending
on the dimensionality of space, the lattice structure, and the sign, range, and symmetries of
the exchange couplings.
2.
Nearest-neighbor quantum xy limit in one-dimensional space
When the lattice Λ is a one-dimensional ring made of |Λ| = N sites and the exchange
couplings are only vanishing between ordered pairs i and i + 1 (i = 1, · · · , N with i + N ≡ i)
of nearest-neighbor sites, the quantum xy Hamiltonian reduces to
Ĥ
xx
N
Jx X
+ −
− +
:=
[1 − δiN (1 − cos φ)] Ŝi Ŝi+1 + Ŝi Ŝi+1 ,
2 i=1
(C5a)
where
1
1
1 x
Ŝi ± iŜiy ≡ (σix ± iσiy ) ,
(C5b)
Ŝiz := σiz ,
2
4
2
acting on the Hilbert space (C2a) with the ring-parameter 0 ≤ φ < 2π (i + N ≡ i). We are
Ŝi± :=
going to show that excitations above the ground state form a continuum. Remarkably, this
continuum is not separated by an energy gap from the ground sate energy. Correspondingly,
spin-spin correlation functions decay algebraically and not exponentially with separation at
zero temperature. The quantum xy limit at zero temperature defines a quantum critical
point in that all correlation functions between local spin operators are algebraic functions
of the space arguments when sufficiently far apart. At a quantum critical point there is
no characteristic intrinsic length scale, scale invariance rules. The key step in deriving this
result is the Jordan-Wigner transformation, a remarkable identity that relates the raising
and lowering spin-1/2 operators to spinless fermions through a non-local transformation,
but preserves the locality of the xy Hamiltonian when expressed in terms of these spinless
fermions.
For any i ∈ Λ, define the operators
fî† := K̂i Ŝi+ ,
fî := Ŝi− K̂i† ,
(C6a)
!
i−1 X
1
Ŝjz +
K̂i := exp iπ
2
j=1
(C6b)
where the unitary operator
55
has been introduced.
For any i ∈ Λ, the non-local operator
i(i−1)π/2
K̂i = e
i−1
Y
z
eiπŜj
j=1
= ii−1
i−1 Y
π
π cos σ̂j0 + i sin σ̂jz
2
2
j=1
(C7a)
z
= (−)i−1 σ̂1z · · · σ̂i−1
rotates by the angle π around the z axis in spin space all the spins left to site i,
K̂i Ŝjx K̂i† = −Θ(i − j)Ŝjy + Θ(j − i)Ŝjx ,
K̂i Ŝjy K̂i† = −Θ(i − j)Ŝjx + Θ(j − i)Ŝjy ,
(C7b)
K̂i Ŝjz K̂i† = Ŝjz ,
where i, j = 1, · · · , N and Θ(x) is the Heaviside function. It is thus the non-local nature
of the operator K̂i that allows it to either anticommute or commute with the raising or
lowering operators Ŝj± depending on whether j < i or j ≥ i, respectively. The choice of
the phase factor exp i(i − 1)π/2 insures that its eigenvalues are ±1. Finally, since K̂i is
unitary by construction and has only real eigenvalues, it must satisfy
K̂i K̂i† = K̂i† K̂i = 1,
K̂i = K̂i† ,
i = 1, · · · , N.
(C7c)
Furthermore, we observe that K̂i is built exclusively from Ŝjz with j = 1, · · · , i − 1. Thus,
it obeys the commutation relations
i
h
K̂i , K̂j = 0,
and
h
i
K̂i , Ŝjz = 0,
i, j = 1, · · · , N,
(C7d)
1 ≤ i ≤ j ≤ N.
(C7e)
Hence, we can rewrite Eq. (C6) as
fî† = Ŝi+ K̂i = K̂i Ŝi+ ,
fî = Ŝi− K̂i = K̂i Ŝi− ,
We are now in position to prove that
n
o
n
o n
o
fî , fˆj† = δij ,
fî† , fˆj† = fî , fˆj = 0,
i = 1, · · · , N.
(C8)
i, j = 1, · · · , N,
(C9)
i.e., that we have constructed out of the spin-1/2 operators fermionic operators called Jordan-
Wigner fermions.
56
Proof. Let i, j ∈ Λ. When i < j, we have,
fî fˆj† + fˆj† fî = Ŝi− K̂i Ŝj+ K̂j + Ŝj+ K̂j Ŝi− K̂i
= Ŝi− Ŝj+ K̂i K̂j + Ŝj+ K̂j Ŝi− K̂j K̂i K̂j
= Ŝi− Ŝj+ K̂i K̂j − Ŝj+ Ŝi− K̂i K̂j
h
i
= Ŝi− , Ŝj+ K̂i K̂j
(C10)
=0
and
fî fˆj + fˆj fî = Ŝi− K̂i Ŝj− K̂j + Ŝj− K̂j Ŝi− K̂i
= Ŝi− Ŝj− K̂i K̂j + Ŝj− K̂j Ŝi− K̂j K̂i K̂j
= Ŝi− Ŝj− K̂i K̂j − Ŝj− Ŝi− K̂i K̂j
h
i
= Ŝi− , Ŝj− K̂i K̂j
(C11)
= 0.
When i = j, we have,
fî fî† + fî† fî = Ŝi− K̂i Ŝi+ K̂i + Ŝi+ K̂i Ŝi− K̂i
n
o
= Ŝi− , Ŝi+ K̂i K̂i
(C12)
fî fî + fî fî = Ŝi− K̂i Ŝi− K̂i + Ŝi− K̂i Ŝi− K̂i
n
o
−
−
= Ŝi , Ŝi K̂i K̂i
(C13)
=1
and
= 0.
2
Observe that only Ŝi− = 0 is needed to reach the last line. The case of j > i follows by
interchanging i and j in the proof for the case of i < j.
The interpretation of the fermion creation operator fî is that it creates a defect, a kink,
in an ordered state. To see this, consider the states
N
O
1
√ (| ↑ii + | ↓ii )
|Fi :=
2
i=1
57
(C14)
and
N
O
1
√ (| ↑ii − | ↓ii )
|F̄i :=
2
i=1
(C15)
which are eigenstates of Ŝix with eigenvalues +1/2 and −1/2, respectively, for all sites i ∈ Λ.
For any j ∈ Λ, the state
fj† |Fi
=
j−1
O
1
√ (| ↑ii − | ↓ii )
2
i=1
!
1
√ | ↑ij
2
N
O
1
√ (| ↑ii + | ↓ii )
2
i=j+1
!
(C16)
is an eigenstate of Ŝix with eigenvalue −1/2 for all sites i = 1, · · · , j − 1, an eigenstate
of Ŝjz with eigenvalue +1/2, and an eigenstate of Ŝix with eigenvalue +1/2 for all sites
i = j + 1, · · · , N. The state (C16) interpolates between |Fi and F̄i with a spin up at the
boundary j ∈ Λ.
Having established that the definition (C6) yields operators obeying the fermion algebra,
we are going to express the spin operators in terms of these fermions. To this end, we assume
the fermion algebra (C9). One then verifies that the spin operators defined by
Ŝi+ := K̂i fî† ,
where
K̂i := exp iπ
i−1
X
fˆj† fˆj
j=1
satisfy the su(2) Lie algebra
h
i
Ŝi+ , Ŝj− = δij 2Ŝ z ,
1
Ŝiz := fî† fî − ,
2
Ŝi− := fî K̂i† ,
!
i−1 Y
=
1 − 2fˆj† fˆj
(C17a)
(C17b)
j=1
h
i
Ŝi− , Ŝjz = δij Ŝi− ,
h
i
Ŝiz , Ŝj+ = δij Ŝi+ ,
(C17c)
for all i, j = 1, · · · , N.
The fermion representation (C17b) of the kink operator (C6b) is useful to establish the
identities
ˆ† ˆ
K̂i K̂i+1 = K̂i+1 K̂i = eiπfi fi = 1 − 2fî† fî ,
fî† K̂i K̂i+1 = fî† K̂i+1 K̂i = fî† 1 − 2fî† fî = +fî† ,
†ˆ
ˆ
ˆ
ˆ
ˆ
fi K̂i K̂i+1 = fi K̂i+1 K̂i = fi 1 − 2fi fi = −fî ,
−
Ŝi+ Ŝi+1
= fî† K̂i K̂i+1 fî+1 = +fî† fî+1 ,
†
†
+
Ŝi− Ŝi+1
= fî K̂i K̂i+1 fî+1
= −fî fî+1
,
† ˆ
−
+
Ŝi+ Ŝi+1
+ Ŝi− Ŝi+1
= fî† fî+1 + fî+1
fi ,
58
(C18)
for i = 1, · · · , N − 1. The term
+ −
− +
+ −
− +
ŜN
ŜN +1 + ŜN
ŜN +1 ≡ ŜN
Ŝ1 + ŜN
Ŝ1
= K̂N fˆN† fˆ1 K̂1 + fˆN K̂N K̂1 fˆ1†
that encodes a ring topology must be treated with care. Since
h
i
K̂1 = 1,
fˆN , K̂N = 0,
(C19)
(C20)
we can make the simplification
+ −
− +
ŜN
Ŝ1 + ŜN
Ŝ1 = K̂N fˆN† fˆ1 + K̂N fˆN fˆ1† .
(C21)
Next, we observe that
K̂N = exp iπ
N
−1
X
fˆj† fˆj
j=1
= exp iπ
= K̂
where we have introduced
N̂f :=
N
X
fˆj† fˆj
j=1
1 − 2fˆN† fˆN
N
X
!
fˆj† fˆj ,
j=1
!
† ˆ
ˆ
exp −iπ fN fN
(C22a)
K̂ := exp iπ N̂f .
(C22b)
K̂N fˆN = +K̂ fˆN ,
(C23)
Now,
K̂N fN† = −K̂fN† ,
so that the ring topology is encoded by the relation
+ −
− +
+ −
− +
ŜN
ŜN +1 + ŜN
ŜN +1 ≡ ŜN
Ŝ1 + ŜN
Ŝ1
= − K̂ fˆN† fˆ1 + fˆ1† fˆN .
(C24)
We conclude that in the sector of the Hilbert space (C2a) with the given number Nf of
fermions, the ring topology is achieved by the condition
†
fî+N
= (−)Nf +1 fî† ,
fî+N = (−)Nf +1 fî ,
i ∈ Λ.
(C25)
Since, the fermion number operator is related to the total spin
Ŝ :=
N
X
i=1
59
Ŝi
(C26)
by
N
N̂f = Ŝ + ,
2
z
z
Ŝ :=
N
X
Ŝiz ,
(C27)
i=1
the sector of the Hilbert space with vanishing total spin Ŝ z = 0 is equivalent to demanding
that the number of fermion Nf + N/2 is half the number of lattice sites, i.e., the half-filled
condition for spinless fermions.
With the help of the Jordan-Wigner transformation (C17), the xy Hamiltonian (C5a)
has been fermionized,
Ĥ
xx
N
Jx X
† ˆ
[1 − δiN (1 − cos φ)] fî† fî+1 + fî+1
fi
:=
2 i=1
(C28a)
where the spinless fermion obey the boundary conditions
†
fî+N
= (−)Nf +1 fî† ,
fî+N = (−)Nf +1 fî ,
i∈Λ
(C28b)
in the subspace of the Hilbert space with the fermion number operator constrained to
N̂f = Ŝ z +
N
= Nf ,
2
Nf = 1, · · · , N.
(C28c)
When the ring-parameter φ = 0, the energy spectrum is solved using the Bloch states
1 X −iki ˆ†
e
fk ,
fî† = √
N k∈Λ∗
1 X +iki ˆ
fî = √
e
fk ,
N k∈Λ∗
with the reciprocal spaces
2πn N
N
N
N
∗
Λ = k=π+
n=−
,−
+ 1, · · · , +
− 1, +
,
N
2
2
2
2
when periodic boundary condition hold and the reciprocal spaces
πn N
N
N
N
∗
Λ = k=π+
,−
+ 1, · · · , +
− 1, +
n=−
N
2
2
2
2
(C29a)
(C29b)
(C29c)
when antiperiodic boundary conditions hold,
Ĥ xx :=
X
εk fˆk† fˆk ,
k∈Λ∗
εk = −J x cos k.
(C29d)
When N is even, the ground state can be shown rigorously to be a singlet. The ground
state can be represented by the Fermi sea of Jordan-Wigner fermions at half-filling as shown
in Fig. 6(a). The first excited states of the quantum xy chain have total spin-1 quantum
60
ε
ε
k
−π
−π/2
+π/2
+π
(a)
FIG. 6:
(b)
π
2π
q
(a) Single-particle eigenvalues for Jordan-Wigner fermions and the Fermi sea at half-
filling. (b) Jordan-Wigner particle-hole continuum in the nearest-neighbor quantum xy chain with
q = kp − kh where kp is the particle and kh the hole wave number.
number and can be represented by particle-hole excitations of Jordan-Wigner fermions. Consistency implies that one associates Jordan-Wigner fermions with spin-1/2 representations
of SU(2).
Reinstating a small J z /J x ≡ λ does not violate the locality of the representation of the
xy Hamiltonian in terms of the Jordan-Wigner fermions,
Ĥ
xxz
N
Jx X
†ˆ
† ˆ
ˆ
ˆ
[1 − δiN (1 − cos φ)] fi fi+1 + fi+1 fi
=
2 i=1
N
X
1
1
†ˆ
† ˆ
z
ˆ
ˆ
+J
[1 − δiN (1 − cos φ)] fi fi −
.
fi+1 fi+1 −
2
2
i=1
(C30)
The quartic interaction for the Jordan-Wigner fermions cannot be treated by perturbation
theory. Nonperturbative techniques are available and go under the name of Bethe Ansatz
or bosonization. The qualitative nature of the non-interacting xy limit survives until the
Heisenberg point J z /J x = 1 is reached, i.e., the line 0 ≤ J z /J x ≤ 1 realizes a line of critical
points with all local spin-correlation functions decaying like power laws with exponents
determined by the ratio J z /J x . Beyond the Heisenberg point J z /J x = 1, an energy gap
opens between the ground state and the low lying excitations as does the onset of long-range
order in the ground state. The quantum transition at the Heisenberg point J z /J x = 1 is,
mathematically, closely related to the (classical) Kosterlitz-Thouless transition studied in
lecture 5.
61
APPENDIX D:
THE GROUND STATE ENERGY AND THE SINGLE-
PARTICLE TIME-ORDERED GREEN FUNCTION
We are going to perturb the non-interacting jellium model with a Coulomb interaction.
This is to say that the index ι that labels the fermionic generators of the Fock space (4.1)
becomes the quantized momenta
k=
2π
n,
L
n ∈ Z3 ,
(D1a)
in a cubic box of volume L3 upon imposition of periodic boundary conditions and the spin1/2 index
σ =↑, ↓ .
(D1b)
Let 0 ≤ λ ≤ 1 be a dimensionless coupling that allows us to treat the Coulomb interaction
adiabatically, i.e., we define
Ĥµ (λ) := Ĥ0,µ + λĤ1 ,
X k2
Ĥ0,µ :=
− µ ĉ†σ,kĉσ,k,
2m
σ,k
Ĥcb :=
(D2)
1 X 2πe2 X X †
ĉ
ĉ† ′ ′ ĉ ′ ′ ĉ .
V q6=0 q 2 σ,σ′ k,k′ σ,k+q σ ,k −q σ ,k σ,k
We shall use the notation
Ĥµ ≡ Ĥµ (λ = 1)
(D3)
so that Ĥµ (λ) interpolates between Ĥ0,µ and Ĥµ as λ varies between 0 and 1. The restriction
to non-vanising transfer momenta
q 6= 0
(D4)
implements the charge neutrality condition. We work in the grand canonical ensemble with
the grand canonical partition function
Z(β, µ, λ) := TrF e−β Ĥµ (λ) .
(D5)
The free energy in the grand canonical ensemble is defined by
1
1
F (β, µ; λ) := U(β, µ; λ) − T S(β, µ; λ) ≡ − ln TrF e−β Ĥµ (λ) ≡ − ln Z(β, µ; λ).
β
β
62
(D6)
The thermal expectation value of the Coulomb interaction is
h
i
−β Ĥµ (λ)
E
D
Ĥcb
∂F (β, µ; λ) TrF e
=
≡ Ĥcb
.
∂λ
β,µ;λ
TrF e−β Ĥµ (λ)
(D7)
The change in the free energy induced by switching on the Coulomb interaction adiabatically
is
F (β, µ, 1) − F (β, µ, 0) =
Z1
∂F (β, µ; λ)
=
dλ
∂λ
Z1
0
0
E
dλ D
λĤcb
.
λ
β,µ;λ
Our goal is now to relate the grand canonical expectation value
E
D
λĤcb
(D8)
(D9)
β,µ;λ
to the so-called single-particle Green function. We first define time-ordered Green functions
for the Coulomb gas in the grand canonical ensemble.
Define the single-particle time-ordered Green function
E
D Gσ,k(τ ; λ) := − Tτ ĉHσ,k(τ ; λ) ĉ†Hσ,k(0; λ)
(D10)
β,µ;λ
where the λ dependence comes from the use of Eq. (D2) to define the grand canonical
partition function and the Heisenberg picture. The terminology single-particle stems from
the fact that the ĉ’s create particle-like excitations in the non-interacting limit. According
to Eq. (4.30), the single-particle Green function (D10) obeys the equation of motion
i
oE
D nh
†
Ĥµ (λ), ĉHσ,k(τ ; λ) ĉHσ,k(0; λ)
−∂τ Gσ,k(τ ; λ) = δ(τ ) + Tτ
.
(D11)
β,µ;λ
We take advantage of the fact that Ĥµ (λ) is conserved, i.e.,
Ĥµ (λ) = ĤHµ (τ ; λ)
(D12)
for any imaginary time τ . We can bring the equations of motion (D11) to the form
D nh
i
oE
†
−∂τ Gσ,k(τ ; λ) = δ(τ ) + Tτ
ĤHµ (τ ; λ), ĉHσ,k(τ ; λ) ĉHσ,k(0; λ)
(D13)
β,µ;λ
in which the commutator on the right-hand side now only involves operators at equal imaginary time. When evaluated at an infinitesimal time τ = 0− before τ = 0, Eq. (D13) can be
brought to the form
h
ioE
X
XD n †
Tτ ĉHσ,k(0; λ) ĤHµ (0− ; λ), ĉHσ,k(0− ; λ)
∂τ Gσ,k (0− ; λ) + δ(0− ) =
σ,k
σ,k
63
β,µ;λ
.
(D14)
Evaluation of the commutator on the right-hand side of Eq. (D14) yields
E
D
X
X
−
−
−
∂τ Gσ,k (0 ; λ) + δ(0 ) = −
ξµkGσ,k(0 ; λ) − 2 × λ Ĥcb
σ,k
σ,k
β,µ;λ
,
(D15a)
with
ξµk :=
k2
− µ,
2m
(D15b)
as we now show.
Proof. With the help of the identity
h
i
n
o n
o
ÂB̂, Ĉ = ÂB̂ Ĉ − Ĉ ÂB̂ = ÂB̂ Ĉ + ÂĈ B̂ − ÂĈ B̂ − Ĉ ÂB̂ = Â B̂, Ĉ − Â, Ĉ B̂ (D16)
valid for any triplet of operators in Hµ , we deduce, for λ = 1 without loss of generality,
!
#
"
h
i
X
X
X
X
1
ĉ†σ′ ,k′ +q ĉ†σ′′ ,k′′ −q ĉσ′′ ,k′′ ĉσ′ ,k′ , ĉσ,k
Vcb q
ξµk′ ĉ†σ′ ,k′ ĉσ′ ,k′ +
Ĥµ , ĉσ,k =
2V q6=0
σ′ ,σ′′ k′ ,k′′
σ′ ,k′
X X
1 X
= − ξµkĉσ,k +
Vcb q
2V q6=0
σ′ ,σ′′ k′ ,k′′
†
†
× ĉσ′ ,k′+q ĉσ′′ ,k′′ ĉσ′ ,k′ δσ′′ ,σ δk′′ −q,k − ĉσ′′ ,k′′−q ĉσ′′ ,k′′ ĉσ′ ,k′ δσ′ ,σ δk′ +q,k
1 X
= − ξµkĉσ,k +
2V ′ ′′
k ,k
!
X †
X †
× Vcb k′′ −k6=0
ĉσ′′ ,k′+k′′ −kĉσ′′ ,k′′ ĉσ,k′
ĉσ′ ,k′ +k′′ −kĉσ,k′′ ĉσ′ ,k′ − Vcb k−k′6=0
σ′′
σ′
1 X
= − ξµkĉσ,k −
2V ′ ′′
k ,k
×
Vcb k′′ −k6=0
X
σ′
ĉ†σ′ ,k′ +k′′ −kĉσ′ ,k′ ĉσ,k′′ + Vcb k−k′ 6=0
1 X
= − ξµkĉσ,k −
2V ′ ′′
X
σ′′
ĉ†σ′′ ,k′ +k′′ −kĉσ′′ ,k′′ ĉσ,k′
k ,k
×
Vcb k′′ −k6=0
X
σ′
ĉ†σ′ ,k′ +k′′ −kĉσ′ ,k′ ĉσ,k′′ + Vcb k−k′′ 6=0
X
σ′
ĉ†σ′ ,k′ +k′′ −kĉσ′ ,k′ ĉσ,k′′
!
!
(D17)
where we have introduced the short-hand notation
ξµk :=
k2
− µ.
2m
64
(D18)
The Coulomb potential is an even function of momentum. It can thus be factorized from
Eq. (D17),
h
i
X †
1 X
Ĥµ , ĉσ,k = − ξµkĉσ,k −
ĉσ′ ,k′+k′′ −kĉσ′ ,k′ ĉσ,k′′ .
Vcb k′′ −k6=0
V k′ ,k′′
′
σ
Observe that
XX
σ
ĉ†σ,k
k
(D19)
h
i
XX
ξµkĉ†σ,kĉσ,k
Ĥµ , ĉσ,k = −
σ
k
X † †
1 X
−
ĉσ,kĉσ′ ,k′ +k′′ −kĉσ′ ,k′ ĉσ,k′′ .
Vcb k′′ −k6=0
V k,k′,k′′
σ,σ′
XX
=−
ξµkĉ†σ,kĉσ,k
σ
−2×
(D20)
k
XX † †
1 X
Vcb q6=0
ĉσ,kĉσ′ ,k′ +q ĉσ′ ,k′ ĉσ,k+q
2V q
′
′
k,k σ,σ
= − Ĥ0,µ − 2 × Ĥcb .
Equation (D15a) follows from inspection of Eqs. (D10), (D14), and (D20).
Equation (D15a) allows to express the expectation value of the Coulomb interaction in
terms of single-particle properties as
E
D
X
X
∂τ Gσ,k (0− ; λ) + δ(0− ) −
ξµkGσ,k(0− ; λ)
2 × λ Ĥcb
=−
β,µ;λ
(D21)
σ,k
σ,k
If we use the Fourier conventions [compare with Eq. (4.26e)]
Gσ,k(τ ; λ) =
1 X −iωn τ
e
Gσ,k,iωn (λ),
β n∈Z
1 X −iωn τ
δ(τ ) =
e
,
β n∈Z
we arrive at
E
D
λ Ĥcb
β,µ;λ
1 X X −iωn 0− e
−iωn + ξµk Gσ,k,iωn (λ) + 1
2β n∈Z σ,k
i
1 X X −iωn 0− h
=+
e
iωn − ξµk − G−1 σ,k,iω (λ) Gσ,k,iωn (λ).
n
2β n∈Z σ,k
(D22)
=−
(D23)
Here, the inverse operator G−1 to the Green function G is defined by the condition
G−1 G = GG−1 = 1
65
(D24)
that holds as an operator equation on the space of propagators that evolve the many-body
wave functions from the Fock space Hµ in imaginary time following the dynamics set by Ĥµ .
Equation (D23) suggests that we introduce the non-interacting Green function through the
matrix elements
G−1
0µ
and the selfenergy
Σσ,k,iωn (λ) := G−1
0µ
σ,k,iωn
− G−1
σ,k,iωn
σ,k,iωn
:= iωn − ξµk
(λ) ⇐⇒ G−1
σ,k,iωn
(D25a)
(λ) = iωn − ξµk − Σσ,k,iωn (λ)
(D25b)
that measures the difference induced by the Coulomb interaction between the non-interacting
−1
G−1
0µ and the exact Gµ . With these definitions, Eqs. (D23) and (D8) take the final forms
D
λ Ĥcb
E
β,µ;λ
=+
1 X X −iωn 0−
e
Σσ,k,iωn (λ)Gσ,k,iωn (λ)
2β n∈Z σ,k
(D26)
and
1
Z
1 X X dλ −iωn 0−
F (β, µ; 1) − F (β, µ; 0) = +
e
Σσ,k,iωn (λ)Gσ,k,iωn (λ),
2β n∈Z σ,k
λ
(D27)
0
respectively, where the chemical potential µ is fixed by the condition that there is an average
number of Ne electrons in the cubic box of volume L3 . The change in the ground state energy
E0µ of the Fermi sea induced by switching on adiabatically the Coulomb potential is
1
Z
1 X X dλ −iωn 0−
e
Σσ,k,iωn (λ)Gσ,k,iωn (λ)
E(µ; 1) − E(µ; 0) = lim
β→∞ 2β
λ
n∈Z σ,k
0
1
β→∞ β
= lim
1
XXZ
n∈Z
k
dλ −iωn 0−
e
Σσ,k,iωn (λ)Gσ,k,iωn (λ),
λ
0
(D28)
σ =↑, ↓ .
APPENDIX E: LINEAR RESPONSE
1.
Introduction
To characterize experimentally the different states or phases in condensed matter physics,
a macroscopic sample is subjected to weak external perturbations. A sample can thus be
probed in various different ways. Mechanical forces are exerted on it. Leads are attached
66
to it to drive an electric or thermal current with varying applied magnetic fields or applied
pressures. Beams of light, electrons, neutrons, or muons are aimed to it with detectors
recording the outcome of these scattering experiments. As long as these external perturbations preserve the integrity of the sample under study on the relevant length and time
scales, one hopes that intrinsic properties of the sample can be deduced at these length and
time scales. The mathematical description of such experiments is the following.
Let Ĥ0 be the Hamiltonian describing the macroscopic sample to be investigated under
the assumption that it is well isolated from its environment. Thus, Ĥ0 is conserved and acts
on the Hilbert space H0 . In thermodynamic equilibrium at the inverse temperature β the
sample is described by the partition function
Z0 := TrH0 e−β Ĥ0 ≡
X
e−βE0 ι .
(E1a)
e−βE0 ι |ιi hι|
(E1b)
ι
If the density operator is defined by
ρ̂0 β := e−β Ĥ0 ≡
X
ι
then the thermal average of any operator Â acting on H0 is
D E
Â
0β
:=
TrH0 e−β Ĥ0 Â
TrH0 e−β Ĥ0
≡ Z0−1 TrH0
ρ̂0 β Â .
(E1c)
Let the Hilbert space of the environment be denoted by H′ . The Hilbert space of the
“universe” is then
H := H0 ⊗ H′ .
(E2)
We now imagine that at time t0 a weak time-dependent coupling to the environment is
switched on. This physics is modeled by the time-dependent Hamiltonian
Ĥ(t) := Ĥ0 + Θ(t − t0 )Ĥ ′(t)
(E3)
acting on the Hilbert space H. Here, Θ is the Heaviside function. The question we then
pose is the following. What is the change in the expectation value (E1c) induced by the
time-dependent coupling (E3) to first order in perturbation theory? The answer to this
question is the Kubo formula that we now derive.
67
^ ^
H+H’(t)
0
H^0
t
t0
FIG. 7: A material is described by the conserved Hamiltonian Ĥ0 at times t < t0 . At time t0 a
time-dependent interaction Ĥ ′ (t) to the environment is switched on.
2.
The Kubo formula
Assume that we have solved the time-dependent Schrödinger equation
i~∂t |ι(t)i = Ĥ(t) |ι(t)i
(E4a)
|ι(t0 )i = |ιi
(E4b)
with the initial condition that
is the eigenstate of Ĥ0 with eigenvalue E0 ι . We then define the time-dependent density
matrix by
ρ̂0 β (t, t0 ) :=
X
ι
e−βE0 ι |ι(t)i hι(t)| ,
ρ̂0 β (t0 , t0 ) = ρ̂0 β .
(E5a)
The time-dependent density matrix allows to define the out-of-thermodynamic-equilibrium
expectation value
D
E
Â(t, t0 )
:= Z0−1 TrH ρ̂0 β (t, t0 )Â ,
0β
D
Â(t0 , t0 )
E
0β
D E
= Â
0β
.
(E5b)
The Kubo formula states that, to first order in time-dependent perturbation theory,
D
Â(t, t0 )
E
0β
D E
= Â
0β
+
Z∞
dt′ C0Rβ Â,Ĥ ′ (t, t′ )
(E6a)
t0
where the retarded Green function
Dh
iE
i
C0Rβ Â,Ĥ ′ (t, t′ ) := − Θ(t − t′ ) ÂI (t, t0 ), ĤI′ (t′ , t0 )
~
0β
(E6b)
has been introduced and
ÂI (t, t0 ) = e+iĤ0 (t−t0 )/~ Â e−iĤ0 (t−t0 )/~,
ĤI′ (t, t0 ) = e+iĤ0 (t−t0 )/~ Ĥ ′ (t) e−iĤ0 (t−t0 )/~. (E6c)
68
Proof. Step 1: Make the Ansatz
|ι(t)i = e−iĤ0 (t−t0 )/~ ÛI (t, t0 ) |ι(t0 )i
(E7)
that defines the unitary time-evolution operator ÛI (t, t0 ). Insertion of Eq. (E7) into the
left-hand side of Schrödinger equation (E4a) gives
h
i
i~∂t |ι(t)i = e−iĤ0 (t−t0 )/~ Ĥ0 ÛI (t, t0 ) + i~ ∂t ÛI (t, t0 ) |ι(t0 )i
(E8a)
on the one hand. Insertion of Eq. (E7) into the right-hand side of Schrödinger equation
(E4a) gives
i~∂t |ι(t)i = Ĥ0 + Θ(t − t0 )Ĥ ′ (t) e−iĤ0 (t−t0 )/~ÛI (t, t0 ) |ι(t0 )i
(E8b)
on the other hand. In other words, the unitary operator ÛI (t, t0 ) obeys the differential
equation
i~ ∂t ÛI (t, t0 ) = e+iĤ0 (t−t0 )/~ Θ(t − t0 )Ĥ ′ (t) e−iĤ0 (t−t0 )/~ ÛI (t, t0 )
≡ Θ(t −
t0 )ĤI′ (t, t0 ) ÛI
(E9a)
(t, t0 )
with the initial condition
ÛI (t0 , t0 ) = 1,
i.e.,

 i
ÛI (t, t0 ) = Tt exp −
~
We recall that
Zt
t0
(E9b)


dt′ ĤI′ (t′ , t0 ) .
B̂I (t, t0 ) = e+iĤ0 (t−t0 )/~ B̂ e−iĤ0 (t−t0 )/~.
(E9c)
(E10)
Step 2: We insert into
D
E
X
Â(t, t0 )
= Z0−1
e−βE0 ι hι(t)| Â |ι(t)i
0β
ι
= Z0−1
X
ι
= Z0−1
X
ι
e−βE0 ι hι(t0 )| ÛI† (t, t0 ) e+iĤ0 (t−t0 )/~ Â e−iĤ0 (t−t0 )/~ ÛI (t, t0 ) |ι(t0 )i
e−βE0 ι hι(t0 )| ÛI† (t, t0 ) ÂI (t, t0 ) ÛI (t, t0 ) |ι(t0 )i
(E11)
69
the expansion
i
ÛI (t, t0 ) = 1 −
~
Zt
dt′ ĤI′ (t′ )
(E12)
t0
to first order in Ĥ ′ . We find
D
E
E
D
X
Â(t, t0 )
= Z0−1
e−βE0 ι ι(t0 ) ÂI (t, t0 ) ι(t0 )
0β
ι
i
−
~
Zt
′
dt
Z0−1
X
−βE0 ι
e
ι
t0
which we rewrite as
D
E
D E
X
−1
−βE0 ι
Â(t, t0 )
= Z0
e
ι Â ι
0β
h
i
E
D
ι(t0 ) ÂI (t, t0 ), ĤI′(t′ , t0 ) ι(t0 ) ,
(E13)
ι
−
i
~
Z∞
dt′ Θ(t − t′ )Z0−1
t0
X
ι
D h
i E
e−βE0 ι ι ÂI (t, t0 ), ĤI′ (t′ , t0 ) ι .
(E14)
To first order in perturbation theory, we have thus expressed the time evolution out of thermodynamic equilibrium solely in terms of expectation values in thermodynamic equilibrium,
D
Â(t, t0 )
E
0β
D E
= Â
0β
+
Z∞
t0
Dh
iE i
′
′ ′
dt − Θ(t − t ) ÂI (t, t0 ), ĤI (t , t0 )
.
~
0β
′
(E15)
The Kubo formula in the frequency domain applies to the case when
Ĥ ′ (t) = f (t) B̂
(E16a)
with f any complex-valued function and B̂ any operator acting on H. Indeed, Eq. (E6)
becomes
D
E
Â(t, t0 )
0β
D E
= Â
0β
+
Z∞
t0
dt′ C0Rβ Â,B̂ (t − t′ ) f (t′ )
(E16b)
where
C0Rβ Â,B̂ (t
Dh
iE
i
′
′
Â
B̂
Θ(t
−
t
)
(t,
t
),
(t
,
t
)
.
− t ) := −
I
0
I
0
~
0β
′
70
(E16c)
Invariance under time translation of Eq. (E16c) follows from the fact that both Â and B̂ are
time-independent. If the Fourier transform of f exists on the open line R and if one is not
interested in transients, then the convolution in time
Z
D
E
D
E
D E
Â(t)
:= lim Â(t, t0 )
= Â
+ dt′ C0Rβ Â,B̂ (t − t′ ) f (t′ )
t0 →−∞
0β
0β
0β
(E17a)
R
is represented in frequency space by the product
D
Â(ω)
E
0β
D E
= 2π Â
0β
δ(ω) + C0Rβ Â,B̂ (ω) f (ω)
where we are using the Fourier convention
Z
Z
dω −iωt
g(t) =
e
g(ω) ⇐⇒ g(ω) = dt e+iωt g(t)
2π
R
(E17b)
(E18)
R
for all three functions in Eq. (E17).
By construction, the Kubo formula is linear in the perturbation so that
Z
D
E
D E
XZ
d
Â(t)
= Â
+
d r
dt′ C0Rβ Â,B̂a (r) (t − t′ ) f a(r, t′ )
0β
and
hold when
D
Â(ω)
0β
E
0β
a
D E
= 2π Â
′
(E19a)
R
0β
Ĥ (t) =
δ(ω) +
XZ
dd r C0Rβ Â,B̂a (r) (ω) f a(r, ω)
(E19b)
a
XZ
dd r f a (r, t) B̂ a (r).
(E19c)
a
3.
Kubo formula for the conductivity
We start from the time-dependent Hamiltonian
2
Z
XZ
~
e
1
d
†
∂ − A(r, t) ψ̂σ (r) + e dd r n̂(r) ϕ(r, t)
d r ψ̂σ (r)
Ĥ(t) =
2m
i
c
σ=↑,↓
Z
Z
1 X X
dd r dd r ′ ψ̂σ† (r)ψ̂σ† ′ (r ′ )ψ̂σ′ (r ′ )ψ̂σ (r) Wint (r − r ′ )
+
2
′
(E20a)
σ=↑,↓ σ =↑,↓
where
n̂(r) :=
X
σ=↑,↓
71
ψ̂σ† (r)ψ̂σ (r).
(E20b)
The time-dependence originates from the classical gauge fields in terms of which the classical
electromagnetic fields read
E(r, t) = − (∂ϕ) (r, t) −
1
∂ A (r, t)
c t
(E21a)
and
B(r, t) = (∂ ∧ A) (r, t).
(E21b)
The corresponding classical Lagrangian
Z
Z
XZ
d
∗
dt d r ψσ i~∂t ψσ − dt H
L :=
(E22a)
σ=↑,↓
is invariant under the local gauge transformation
ψ ∗ → ψ ∗ e−ieχ/(~c) ,
ψ → eieχ/(~c) ψ,
A → A + ∂χ,
1
ϕ → ϕ − ∂t χ.
c
(E22b)
At the operator level, this implies that the continuity equation
1
∂ n̂ + ∂ · ĵH = 0
c t H
(E23a)
is obeyed in the Heisenberg picture. Here, the gauge-invariant number-density and currentdensity operators are
n̂H =
X
ψ̂H† σ ψ̂H σ
(E23b)
σ=↑,↓
and
~
e
1 X
†
ψ̂
∂ − A ψ̂H σ + H.c. ,
ĵH =
2m σ=↑,↓ H σ i
c
(E23c)
respectively.
We choose a gauge in which
ϕ(r, t) = 0
(E24)
A(r, t) = A0 (r) + Θ(t − t0 )A′ (r, t)
(E25)
and make the linear-response Ansatz
where the vector potential A0 (r) is static as a consequence of the thermodynamic equilibrium
prior to switching on the coupling A′ (r, t) to the environment at time t0 . Correspondingly,
72
we do the decomposition
Ĥ(t) = Ĥ0 + Θ(t − t0 )Ĥ ′ (t) + · · · ,
2
XZ
1
~
e
d
†
d r ψ̂σ (r)
Ĥ0 =
∂ − A0 (r) ψ̂σ (r)
2m i
c
σ=↑,↓
Z
Z
1 X X
dd r dd r ′ ψ̂σ† (r)ψ̂σ† ′ (r ′ )ψ̂σ′ (r ′ )ψ̂σ (r) Wint (r − r ′ ) ,
+
2 σ=↑,↓ ′
σ =↑,↓
Z
e
Ĥ ′(t) = −
dd r A′ (r, t) · ĵ0 (r).
c
The gauge-invariant current-density operator in thermodynamic equilibrium
1 X
~
e
†
ψ̂
ĵ0 (r) =
∂ − A0 ψ̂σ + H.c.
2m σ=↑,↓ σ i
c
(E26)
(E27a)
is sometimes broken up into
ĵ0 (r) = ĵ0 d (r) + ĵ0 p (r)
(E27b)
with the so-called diamagnetic contribution
!
ψ̂σ† ∂ ψ̂σ − ∂ ψ̂σ† ψ̂σ (r)
~ X
ĵ0 d (r) =
2mi σ=↑,↓
(E27c)
and with the paramagnetic contribution
ĵ0 p (r) = −
e
A (r)n̂(r).
mc 0
(E27d)
Observe that ĵ0 (r) = ĵ0 d (r) when A0 = 0. Terms omitted in · · · are total derivatives and
the second-order term
Z
1 e ′ 2
A (r) .
d r n̂(r)
2m c
d
(E28)
First, we seek the out-of-thermodynamic-equilibrium current density
D
E
D
E
ĵ0 (r, ω)
= ĵ0 d (r, ω) + ĵ0 p (r, ω)
0β
0β
(E29)
to linear order in A′ . To this end, we observe that Ĥ ′ (t) is of the form (E19c) with the
identifications
e
f b (r, t) → − A′b (r, t),
c
B̂ b (r) → ĵ0b (r).
(E30)
We can then apply Eq. (E19b) to the computation of the out-of-thermodynamic-equilibrium
expectation value of ĵ0 (r, t) provided we make the identification
Â → ĵ0a (r).
73
(E31)
We thus find
D
E
a
ĵ0 (r, ω)
0β
= 2π
D
ĵ0a (r)
E
eX
δ(ω) −
c b
0β
Z
dd r ′ CĵRa (r),ĵ b (r′ ) (ω) A′b (r ′ , ω)
0
0
(E32)
in the frequency domain and to linear order in A′ . Furthermore, with our choice of gauge
iω
A′b (r ′ , ω) = E ′b (r ′ , ω),
(E33)
c
so that
D
ĵ0a (r, ω)
E
0β
= 2π
D
ĵ0a (r)
E
Z
e X
dd r ′ CĵRa (r),ĵ b (r′ ) (ω) E ′b(r ′ , ω)
δ(ω) −
0
0
iω b
0β
(E34)
in the frequency domain and to linear order in E ′ .
Second, we seek the out-of-thermodynamic-equilibrium current density
D
E
D
E
e
ĵ(r, ω)
:= ĵ0 (r, ω)
−
hA′ (r, ω) n̂(r)i0 β
mc
0β
0β
(E35)
to first order in A′ . However
hA′ (r, ω) n̂(r)i0 β = A′ (r, ω) hn̂(r)i0 β =
c ′
E (r, ω) hn̂(r)i0 β
iω
(E36)
to first order in A′ , while the average local number of electrons per unit volume
n0 β (r) := hn̂(r)i0 β
is time-independent in thermodynamic equilibrium, i.e., the current density
D
E
ĵ0 (r)
=0
0β
(E37)
(E38)
vanishes in thermodynamic equilibrium by the continuity equation. [Indeed, (div j0 )(r) = 0
for all r with the proper boundary conditions implies Eq. (E38). The continuity equation
breaks down when the global gauge symmetry responsible for the continuity equation is
spontaneously broken as is the case in a superconducting phase of matter. In a superconducting phase, the magnetic field B0 = rot A0 is screened by the diamagnetic current
rot B0 =
4π
j
c 0d
conclude that
D
E
a
ĵ0 (r, ω)
where j0 d = −
0β
e2 n0 β sc
A0
m
and n0 β sc is the so-called superfluid density.] We
Z
e X
e
=−
dd r ′ CĵRa (r),ĵ b (r′ ) (ω) E ′b(r ′ , ω) −
n0 β (r)E ′a (r, ω)
0
0
iω b
iωm
Z
i
h
X
e
e
n0 β (r)δ(r − r ′ )δ ab E ′b (r ′ , ω).
dd r ′ − CĵRa (r),ĵ b (r′ ) (ω) −
=
0
iω 0
iωm
b
(E39)
74
If we define the charge-current density out of thermodynamic equilibrium
D
E
J0 β (r, ω) := e ĵ0 (r, ω)
0β
and the nonlocal conductivity tensor
Z
XZ
′
a
dd r ′ σ0abβ (r, t′ ; r ′ , t′ )E ′b (r ′ , t′ )
dt
J0 β (r, t) =:
b
⇐⇒
R
J0aβ (r, ω)
=:
XZ
d ′
d r
(E40)
(E41)
σ0abβ (r, r ′ , ω)E ′b (r ′ , ω)
b
we arrive at the Kubo formula for the conductivity
ie2
1
ab
′
R ab
′
′ ab
σ0 β (r, r , ω) =
Π0 β (r, r , ω) + n0 β (r)δ(r − r )δ
ω
m
(E42)
of a non-relativistic interacting electron gas. Here, we have introduced the notation
Z
Dh
iE
R ab
′
R
+iωt Θ(t)
a
b
′
Π0 β (r, r , ω) ≡ Cĵ a (r),ĵ b (r′ ) (ω) = dt e
ĵ0I (r, t), ĵ0I (r , 0)
.
(E43)
0
0
i~
0β
R
As it should be, the Kubo conductivity (E42) is gauge invariant.
4.
Kubo formula for the dc conductance
The conductivity (E42) relates linearly the charge-current density to an applied electric
field. In an infinite system (or far away from the boundaries of a macroscopic sample), the
conductivity (E42) is an intrinsic property of the system (macroscopic sample). On the
other hand, Ohm’s law,
2
e
charge charge
,
×
=
I = GU =⇒ [G] =
time
energy
~
(E44)
relates linearly the current I through a macroscopic sample to the voltage U applied to
it, thereby defining the conductance G. The conductance depends on the shape of the
material and is thus not an intrinsic property. For three-dimensional materials with a local
conductivity, the conductance and conductivity are related by the aspect ratio
G=
W
σ
L
where L is the sample length and W the area of its cross-section, as
2
e
charge
charge×length
× length2−d .
=
j = σE =⇒ [σ] =
×
d−1
energy
~
length ×time
75
(E45)
(E46)
For materials for which the local approximation to the conductivity tensor is not good, a
Kubo Formula for the conductance is needed. Mesoscopic conductors are characterized by
a size smaller than the typical thermalization length so that a local approximation for the
conductivity is inadequate. We shall assume that the wave length of the external perturbation is much longer than the sample, in which case the perturbing frequency is usually very
small and can be set to zero. This is the static or dc approximation. We want to compute
the time evolution out of thermodynamic equilibrium of the current
Z
I0 β dc := da ξ̂(r) · Re J0 β dc (r),
J0 β dc (r) ≡ J0 β (r, ω = 0),
(E47)
where the integration is over a (d−1)-dimensional cross-section of the d-dimensional volume.
Current conservation gives us the freedom to choose the cross-section as we wish. The
perturbing dc electric field
′
Edc
(r) ≡ E ′ (r, ω = 0)
(E48)
defines equipotential cross-sections by the condition
′
â(r) · Êdc
(r) = 0,
â2 (r) = 1.
(E49)
We thus choose the coordinate system
a(r)â(r) + ξ(r)ξ̂(r),
ξ̂(r) :=
′
Edc
(r)
′
|Edc (r)|
(E50)
to parametrize the oriented cross-section with infinitesimal volume element da ξ̂(r). In this
coordinate system, Eq. (E47) becomes
Z
I0 β dc =
da ξ̂(r) · Re J0 β dc (r)
Z
Z
Z
X
′
′
′
Re σ0abβ dc (r, r ′) ξâ (r)ξˆb(r ′ )
=
dξ(r ) |Edc (r )| da da′
=
Z
dξ(r
′
′
) |Edc
(r ′ )|
Z
da
Z
ab
′
da
X
a
ie2 R ab
′
lim Re Π0 β (r, r , ω) ξâ (r)ξˆb (r ′ ).
ω→0
ω
By definition
Z
Z
X
ab
′
â ˆb ′
da da′
ΠR
0 β dc (r, r , ω)ξ (r)ξ (r ) =
ab
Z
R
Θ(t)
dt
i~
Z
a
da ξâ (r)ĵ0I
(r, t),
76
(E51)
Z
b
da ξˆb (r ′ )ĵ0I
(r ′ , 0)
′
(E52)
.
0β
Current conservation implies that
Z
a
da ξ̂ a(r)ĵ0I
(r, t) = Iˆ0I (t),
Z
b
da′ ξˆb (r ′ )ĵ0I
(r ′ , 0) = Iˆ0I (t = 0).
(E53)
Hence,
I0 β dc
Z
ie2 R
′
= lim Re
C
(ω)
dξ(r ′) |Edc
(r ′ )|
ω→0
ω 0 β I,ˆ Iˆ
2
ie R
C
(ω) U.
= lim Re
ω→0
ω 0 β I,ˆ Iˆ
(E54)
The dc conductance is thus given by
ie2 R
G = lim Re
C
(ω) .
ω→0
ω 0 β I,ˆ Iˆ
5.
(E55)
Kubo formula for the dielectric function
What if we had used a time-dependent scalar potential as coupling to the environment
instead of a time-dependent vector gauge field in Sec. E 3? In other words, we start from
the time-dependent Hamiltonian
Ĥ(t) =
XZ
d
d
r ψ̂σ† (r)
σ=↑,↓
1
2m
~
∂
i
2
ψ̂σ (r) + e
Z
dd r n̂(r) ϕ(r, t)
Z
Z
1 X X
d
d r dd r ′ ψ̂σ† (r)ψ̂σ† ′ (r ′ )ψ̂σ′ (r ′ )ψ̂σ (r) Wint (r − r ′ )
+
2 σ=↑,↓ ′
(E56a)
σ =↑,↓
where
n̂(r) :=
X
ψ̂σ† (r)ψ̂σ (r).
(E56b)
σ=↑,↓
We make the linear-response Ansatz
ϕ(r, t) = ϕ0 (r) + Θ(t − t0 )ϕ′ (r, t)
(E57)
where the scalar potential ϕ0 (r) is static as a consequence of the thermodynamic equilibrium
prior to switching on the coupling ϕ′ (r, t) to the environment at time t0 . Correspondingly,
77
we do the decomposition
Ĥ(t) = Ĥ0 + Θ(t − t0 )Ĥ ′ (t),
2
Z
XZ
1
~
d
†
Ĥ0 =
d r ψ̂σ (r)
∂ ψ̂σ (r) + e dd r ϕ0 (r) n̂(r)
2m i
σ=↑,↓
Z
Z
1 X X
d
d r dd r ′ ψ̂σ† (r)ψ̂σ† ′ (r ′ )ψ̂σ′ (r ′ )ψ̂σ (r) Wint (r − r ′ ) ,
+
2 σ=↑,↓ σ′ =↑,↓
Z
′
Ĥ (t) = e dd r ϕ′ (r, t) n̂(r).
(E58)
We observe that Ĥ ′ (t) is of the form (E19c) with the identifications
f b (r, t) → eϕ′ (r, t),
B̂ b (r) → n̂(r).
(E59)
We can then apply Eq. (E19b) to the computation of the out-of-thermodynamic-equilibrium
expectation value of n̂(r, t) provided we make the identification
Â → n̂(r).
(E60)
Thus, to linear order in ϕ′ , we find
hn̂(r, ω)i0 β = 2π hn̂(r)i0 β δ(ω) + e
Z
dd r ′ C0Rβ n̂(r),n̂(r′ ) (ω) ϕ′(r ′ , ω)
(E61)
dt′ C0Rβ n̂(r),n̂(r′ ) (t − t′ ) ϕ′ (r ′ , t′ )
(E62)
in the frequency domain or
hn̂(r, t)i0 β = hn̂(r)i0 β + e
Z
d ′
d r
Z
R
in the time domain.
It is customary to denote the retarded density-density correlation function by
′ ′
R
′
χR
0 β (r, t; r , t ) ≡ C0 β n̂(r),n̂(r′ ) (t − t ).
(E63)
This correlation function is related to the dielectric constant as follows. Define the classical
scalar gauge potential induced in linear response by the solution to Laplace equation
∆ϕ0 β tot (r, t) = −4πρ0 β tot (r, t)
where
′
ρ0 β tot (r, t) = e n (r, t) + hn̂(r, t)i0 β − hn̂(r)i0 β
78
(E64a)
(E64b)
and
∆ϕ′ (r, t) = −4πen′ (r, t).
(E64c)
∆Wcb (r) = −4πeδ(r).
(E65)
Let Wcb be the solution to
It then follows that
Z
dd r ′′ Wcb (r − r ′′ ) hn̂(r ′′ , t)i0 β − hn̂(r ′′ )i0 β
Z
Z
Z
′′
′ ′
′ ′ ′
′
d ′′
′′
d ′
dt′ χR
= ϕ (r, t) + e d r Wcb (r − r ) d r
0 β (r , t; r , t )ϕ (r , t ) (E66a)
Z
Z
′ ′
′ ′ ′
d ′
dt′ ε−1
= d r
0 β (r, t; r , t ) ϕ (r , t )
′
ϕ0 β tot (r, t) = ϕ (r, t) + e
where we have introduced the dielectric kernel through its inverse
Z
−1
′ ′
′
′
′′
′ ′
ε0 β (r, t; r , t ) := δ(r − r )δ(t − t ) + e dd r ′′ Wcb (r − r ′′ )χR
0 β (r , t; r , t ).
(E66b)
This result should be compared with the discussion of the dielectric function in lecture 7.
6.
Fluctuation-dissipation theorem
The Kubo formula is an example of the fluctuation-dissipation theorem that relates dissipative processes in a system out of thermodynamic equilibrium due to weak couplings to
the environment (the dissipations) to its fluctuations in thermodynamic equilibrium when
isolated from the environment. To emphasize the notion of fluctuations in the Kubo formula,
we are going to relate the retarded correlation function to a spectral density function.
We consider the closed system defined in Eq. (E1). Let Â and B̂ be any pair of operators
acting on H0 . It is standard practice to call
Z+∞
D
E
J0 β Â,B̂ (ω) :=
dt e+iωt ÂH (t)B̂H (0)
0β
−∞
Z+∞
D
E
=
dt e+iωt ÂH (0)B̂H (−t)
0β
−∞
Z+∞
≡
dt e+iωt J0 β Â,B̂ (t)
−∞
79
(E67a)
the spectral density function associated with the time correlation
−β
Ĥ
+i
Ĥ
t
−i
Ĥ
t
0
0
0
D
E
D
E
Tr e
e
Âe
B̂
= ÂH (0)B̂H (−t)
J0 β Â,B̂ (t) ≡ ÂH (t)B̂H (0)
:=
.
0β
0β
Tr e−β Ĥ0
(E67b)
The second line in Eq. (E67a) or the last equality in Eq. (E67b) follows from translation
invariance in time, i.e., from energy conservation as the system is assumed closed. By
construction the spectral density function depends on both the energy transfer ~ω and the
inverse temperature β aside from its dependence on the operators Â and B̂. In the limit
β = ∞ corresponding to zero temperature, only one term survives in the trace (E67b),
namely the ground state expectation value of the product ÂH (t)B̂H (0).
The desired link between the physics of fluctuations encoded by the spectral density
function and the physics of dissipation is made by relating the spectral density function to
the retarded Green function for operators Â and B̂
Dh
iE
ÂH (t)B̂H (0) − B̂H (0)ÂH (t)
0β
h
i
≡ −iΘ(t) J0 β Â,B̂ (+t) − J0 β B̂,Â (−t) .
C0Rβ Â,B̂ (t) := −iΘ(t)
(E68)
We recall that the prefactor −i is convention and that the function Θ is the Heaviside
function, i.e., the step function taking the value 0 when t < 0 and 1 otherwise. The meaning
of the terminology retarded is made transparent after time Fourier transformation
C0Rβ Â,B̂ (ω)
Z+∞
dt e+iωt C0Rβ Â,B̂ (t)
:=
−∞
Z+∞
dt e+iωt C0Rβ Â,B̂ (t)
=
0
Z+∞
Dh
iE
= −i
dt e+iωt ÂH (t)B̂H (0) − B̂H (0)ÂH (t)
= −i
0
Z+∞
0
h
i
dt e+iωt J0 β Â,B̂ (+t) − J0 β B̂,Â (−t)
to which only C0Rβ Â,B̂ (t) with t > 0 contributes.
80
0β
(E69)
The retarded Green function of operators Â and B̂ as a function of t has the integral
representation
Dh
iE
ÂH (t)B̂H (0) − B̂H (0)ÂH (t)
0β
h
i
= −iΘ(t) J0 β Â,B̂ (+t) − J0 β B̂,Â (−t)
C0Rβ Â,B̂ (t) = −iΘ(t)
= −iΘ(t)
Z+∞
−∞
i
dω ′ −iω′ t h
e
J0 β Â,B̂ (+ω ′) − J0 β B̂,Â (−ω ′ )
2π
(E70)
in terms of the spectral density function of operators Â and B̂. Now, J0 β Â,B̂ (+ω ′ ) and
J0 β B̂,Â (−ω ′ ) are related. To see this denote by {|µi} the exact basis of eigenstates of Ĥ0
with eigenvalues {E0 µ }. From the definitions (E67a) and (E67b),
Z+∞
J0 β Â,B̂ (+ω) =
dt e+iωt hÂH (t)B̂H (0)i0 β
−∞
+∞
E
ED D XZ
+i(ω+E0 µ −E0 ν )t −βE0 µ
Â
(0)
ν
ν
B̂
(0)
µ
dt
e
e
= Z0−1
µ
H
H
β
µ,ν −∞
= 2πZ0−1
β
X
µ,ν
and
E
ED D e−βE0 µ µ ÂH (0) ν ν B̂H (0) µ δ(ω + E0 µ − E0 ν )(E71)
Z+∞
J0 β B̂,Â (−ω) =
dt e+iωt hB̂H (0)ÂH (t)i0 β
−∞
= Z0−1
β
+∞
XZ
µ,ν −∞
= 2πZ0−1
β
X
µ,ν
= 2πZ0−1
β
X
µ,ν
E
ED D dt e+i(ω+E0 ν −E0 µ )t e−βE0 µ µ B̂H (0) ν ν ÂH (0) µ
E
ED D e−βE0 µ µ B̂H (0) ν ν ÂH (0) µ δ(ω + E0 ν − E0 µ )
ED E
D e−βE0 ν µ ÂH (0) ν ν B̂H (0) µ δ(ω + E0 µ − E0 ν ),(E72)
where the canonical partition function is given by
X
Z0 β :=
e−βE0 µ .
(E73)
µ
Making use of the constraint of energy conservation in the so-called Lehmann expansions
(E71) and (E72), we infer that
J0 β B̂,Â (−ω) = e−βω J0 β Â,B̂ (+ω).
81
(E74)
Insertion of Eq. (E74) into Eq. (E70) gives
C0Rβ Â,B̂ (t)
= −iΘ(t)
Z+∞
i
dω ′ −iω′ t h
′
′
e
J0 β Â,B̂ (+ω ) − J0 β B̂,Â (−ω )
2π
Z+∞
dω ′ −iω′ t
′
−βω ′
e
J0 β Â,B̂ (+ω ) 1 − e
.
2π
−∞
= −iΘ(t)
−∞
(E75)
Time-Fourier transformation of the retarded Green function demands a regularization at
long positive time which is implemented by the addition of an infinitesimal convergence
factor |η| (P1/x denotes the principal value of 1/x):
Z+∞
C0Rβ Â,B̂ (ω) = lim
dt e+iωt−|η|t C0Rβ Â,B̂ (t)
η→0
−∞
= −i
Z+∞
−∞
= lim
Z+∞
η→0
−∞
=
=
dω ′
′
J0 β Â,B̂ (ω ′) 1 − e−βω lim
η→0
2π
Z+∞
′
dt e+i(ω−ω +i|η|)t
0
dω ′
1
′
J0 β Â,B̂ (ω ′) 1 − e−βω
2π
ω − ω ′ + i|η|
Z+∞
−∞
h
i
dω ′
′
P 1 − iπδ(ω − ω ′ )
J0 β Â,B̂ (ω ′ ) 1 − e−βω
ω−ω ′
2π
Z+∞
−∞
dω ′
i
′
J0 β Â,B̂ (ω ′ ) 1 − e−βω P 1 − J0 β Â,B̂ (ω) 1 − e−βω .
′
ω−ω
2π
2
(E76)
Whenever J0 β Â,B̂ (ω) is real valued for all ω, we conclude that
J0 β Â,B̂ (ω) = −
2
Im C0Rβ Â,B̂ (ω),
1 − e−βω
∀ω 6= 0.
(E77)
This is one form of the fluctuation-dissipation theorem. The condition for J0 β Â,B̂ (ω) to be
real is that
Â = B̂ † .
[1] H. W. Wyld, Mathematical methods for physics, Benjamin/Cummings (1976)
82
(E78)
[2] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, fifth edition Academic Press, London (1980).
[3] H. Bruus and K. Flensberg, Many-body quantum theory in condensed matter physics, Oxford
University press (2004).
83

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