Lecture 8: Jellium model for electrons in a solid: Part II
Christopher Mudry∗
Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland.
(Dated: April 26, 2010)
Abstract
A diagrammatic interpretation to the RPA approximation for the jellium model is given. The
ground state energy in the RPA approximation is calculated. The dependence on momenta and
frequencies of the RPA polarization function for the jellium model is studied. A qualitative argument is given for the existence of a particle-hole continuum and for a branch of sharp excitations
called plasmons. The quasi-static and dynamic limits of the polarization function are studied. The
quasi-static limit is characterized by screening, Kohn effect, and Friedel oscillations. The dynamic
limit is characterized by plasmons and Landau damping. The physical content of the RPA approximation for a repulsive short-hand interaction is derived. The physics of zero-sound is discussed.
The feedback effect of phonons on the RPA effective interaction between electrons is sketched.
∗
Electronic address: [email protected]; URL: http://people.web.psi.ch/mudry
1
Contents
I. Diagrammatic interpretation of the RPA
3
7
II. Ground state energy in the RPA
8
III. Lindhard response function
A. Long-wave lengths and quasi-static limit at zero temperature
16
1. The physics of screening
16
2. Thomas-Fermi approximation
16
3. Kohn effect
17
4. Friedel or Ruderman-Kittel-Kasuya-Yosida oscillations
20
B. Long-wave lengths and dymanic limit at zero temperature
23
1. The physics of plasmons
24
2. Landau damping
25
IV. RPA for a short-range repulsive interaction
V. Feedback effect on and by phonons
25
28
30
References
2
I.
DIAGRAMMATIC INTERPRETATION OF THE RPA
We derived in lecture 3 the random phase approximation (RPA) to the jellium model.
The RPA amounts to expanding the logarithm of the fermionic determinant up to quadratic
order in the electron charge in the effective action for the collective field ϕ . The collective
field ϕ was introduced through a Hubbard-Stratonovich transformation. It couples to local
electronic charge fluctuations as the scalar potential does in electrodymanics. Hence, ϕ can
be thought of as an effective scalar potential. The RPA thus trades a fermionic partition
function for an effective bosonic partition function,
RPA
Zβ,µ ≈ Zβ,µ
,
RPA
Zβ,µ
∝
RPA
Sβ,µ
Z
(1.1a)
RPA
D[ϕ] exp −Sβ,µ
,
2
X 1
q
2 RPA
=
ϕ−q ,
ϕ
− e Πq
2 +q 4π
(1.1b)
ϕq=0,̟l = 0,
q=(q,̟l )
ΠRPA
=+
q
G0k =
L
q ∈ Z3 ,
2π
β
̟l = l ∈ Z,
2π
(1.1c)
2 X
G G
,
βV k 0k 0(k+q)
1
iωn −
k2
2m
+µ
≡
L
k ∈ Z3 ,
2π
1
,
iωn − ξk
ωn =
π
(2n + 1), n ∈ Z.
β
(1.1d)
The polarization function ΠRPA
encodes the effects of the Coulomb interaction within RPA.
q
The limit e → 0 tells us that if we insert in the non-interacting Fermi gas two static
(infinitely heavy) unit point charges at r and r ′ , respectively, then they interact through
the (instantaneous) bare Coulomb potential δ(τ − τ ′ )/|r − r ′ |. The bare Coulomb potential
is renormalized by the response of the Fermi sea to switching on e.
The RPA has a straightforward interpretation in terms of diagrams. The Euclidean
propagator for the scalar potential in the RPA is, by definition and up to a sign, the inverse
of the kernel
q2
4π
− e2 ΠRPA
in Eq (1.1d),
q
DqRPA := − q2
4π
1
− e2 ΠRPA
q
3
,
q ≡ (q, ̟l ).
(1.2)
The sign is convention. In the absence of Coulomb interaction, e = 0, the Euclidean propagator D0q is instantaneous as it is independent of the Matsubara frequency ̟l ,
D0q := −
4π
.
q2
(1.3)
This is nothing but the bare Coulomb potential in Fourier space. We thus have
1
(D0q )−1 + e2 ΠRPA
q
1
= D0q
2
1 + e D0q ΠRPA
q
∞
X
n
= D0q
(−)n e2 D0q ΠRPA
q
DqRPA =
n=0
= D0q
∞
X
(ie)2 ΠRPA
D0q
q
n=0
n
.
(1.4)
Equation (1.4) is the approximate solution to Dyson’s equation,
Dq = D0q + D0q Σq Dq ,
where Dq is the exact propagator, i.e., (the sign is convention)
R
′′
D[ϕ] ϕ+q ϕ−q exp −Sβ,µ
,
Dq := − R
′′
D[ϕ]
exp −Sβ,µ
Z β Z
∆
′′
3 1
Sβ,µ =
dτ
d r (−ϕ∆ϕ)(r, τ ) − 2Tr ln ∂τ −
− µ + ieϕ ,
8π
2m
0
V
(1.5a)
(1.5b)
(1.5c)
and the right-hand side defines the so-called self-energy Σq of the collective field ϕ. The
RPA replaces the self-energy Σq of ϕ by the RPA polarization function,
Σq → (ie)2 ΠRPA
.
q
(1.6)
The diagrammatic or perturbative definition of the self-energy of ϕ goes as follows.
• Draw a dotted thin line for the unperturbed propagator D0q [Fig. 1(a)].
• Draw a dotted thick line for the exact propagator Dq [Fig. 1(b)].
• Draw a thin line for the unperturbed propagator G0q [Fig. 1(c)].
• The rule for connecting dotted and non-dotted lines is that a dotted (thin or thick)
line can only be connected to two distinct non-dotted lines. The connection point is
4
a vertex. It is to be associated with the factor ie [Fig. 1(d)]. Momenta and energies
on the three lines meeting at a vertex must obey momentum and energy conservation.
The electronic spin index is a by-stander.
• Each perturbative contribution to the exact propagator is related to a connected diagram that has been built from dotted and non-dotted lines according to the preceding
rules with no more and no less than two open ended lines. These two lines are dotted
thin lines that are called external legs. External legs carry the momentum and energy
q = (q, ̟l ). All other lines are called internal lines. Momenta and energies on the
internal lines that differ from q = (q, ̟l ) are said to be virtual and are to be summed
over. One must also sum over the spin degrees of freedom of the fermionic (non-dotted)
internal lines. This gives an extra degeneracy factor of 2 for all diagrams.
• An irreducible diagram contributing to the exact propagator is a connected diagram
that cannot be divided into two sub-diagrams joined solely by a single dotted line.
• An irreducible self-energy diagram is an irreducible diagram with the two external legs
removed (amputated).
• The irreducible self-energy Σq is the sum of all irreducible self-energy diagrams.
The diagrammatic counterparts to Eqs. (1.5a) and (1.4) are given in Figs. 1(e) and 1(g),
respectively.
In terms of the original electrons, Dq is closely related to the density-density correlation
function
hρ+q ρ−q iSβ,µ
Sβ,µ =
X
k,σ
ρ+q :=
X
R
D[ψ ∗ ]D[ψ] ρ+q ρ−q exp (−Sβ,µ )
R
:=
,
D[ψ ∗ ]D[ψ]
exp (−Sβ,µ )
1 X 2πe2
k2
∗
− µ ψk,σ +
ρ+q ρ−q ,
ψk,σ −iωn +
2m
βV q6=0,̟ q 2
(1.7a)
l
∗
ψk,σ
ψk+q,σ ,
L
k ∈ Z3 ,
2π
k = (k, ωn ),
k,σ
ωn =
(1.7b)
π
(2n + 1), n ∈ Z.
β
(1.7c)
To see this, note that the saddle-point equation
0=
δS ′ β,µ
δϕq
5
(1.8)
applied to the exact partition function
+1/2 Z
Z
∆
Zβ,µ = Det −
× D[ϕ] D[ψ ∗ ]D[ψ] exp [−S ′ β,µ ] ,
4π
S ′ β,µ =
Z
β
dτ
0
X
Z
V
#
X ∆
1
(−ϕ∆ϕ) +
ψσ∗ ∂τ −
− µ + ieϕ ψσ (r, τ )
d3 r
8π
2m
σ
"
(1.9a)
q2
ϕ−q,−̟l
8π
q,l
)
2
X
k
ie
∗
+
ψk+q,ω
−iωn +
ϕ
ψ
− µ δq,0 δ̟l ,0 + √
n +̟l ,σ
2m
βV +q,+̟l k,ωn,σ
k,n,σ
=
ϕ+q,+̟l
(1.9b)
yields
δ ln Zβ,µ
δϕq
Z
Z
1
′
δS ′ β,µ −Sβ,µ
=
D[ϕ] D[ψ ∗ ]D[ψ]
e
Zβ,µ
δϕq
2
ie
q
ϕ−q + √ ρ−q
=
.
4π
βV
S ′ β,µ
0 = −
Equation (1.10) suggests the identification
2
(ie)2
4π
hρ+q ρ−q iSβ,µ .
Dq −→
q2
βV
(1.10)
(1.11)
More generally, any m-point correlation function for the scalar potential ϕ corresponds to
a n = 2m-point correlation function for electrons. The converse is not true. Not all n-point
fermionic correlation functions can be written as m-point correlation functions for the scalar
field ϕ. For example, the two-point fermionic correlation function (electron propagator)
R
D[ψ ∗ ]D[ψ] ψk ψk∗ exp (−Sβ,µ )
(1.12)
Gk = R
D[ψ ∗ ]D[ψ]
exp (−Sβ,µ )
has no simple expression in terms of correlation functions for the scalar field ϕ. The electronic
density-density correlation function can be measured by inelastic x-ray scattering. The
electronic 2-point function can be measured by angular resolved photoemission scattering
(ARPES). At zero temperature and as a function of the Matsubara frequency analytically
continued to the imaginary axis, poles of the propagator Dq are interpreted as collective
excitations of the underlying jellium model. Similarly, poles of the 2-point function fermion
Green function Gk are called single-particle excitations.
6
(a)
D0q =
(b)
Dq =
(c)
G0q =
q
q
q
k+q
(d)
(e)
(f )
(g)
q
ie =
q
@
@ k
@
@
=
q
=
− e2 ΠRPA
q
q
=
RPA
q
q
q
k+q
+
Σq
≡
+
RPA
k
q
q
RPA
RPA
FIG. 1: (a − d) Rules to construct (Feynmann) diagrams. (e) Dyson’s equation. (f ) RPA electronhole bubble. (e) RPA propagator follows from (e) with the substitution of the self-energy Σq by
.
the electron-hole bubble −e2 ΠRPA
q
II.
GROUND STATE ENERGY IN THE RPA
The ground state energy follows from the partition function by taking the zero temperature limit β → ∞
1
lim − ln Zβ,µ =: EGS .
β→∞
β
(2.1)
Remember that I chose to define Ĥµ to be normal ordered from the outset in lecture 3, i.e.,
that
N X 2πe2
h0| ĤN − Ĥµ |0i = −
.
V q6=0 q 2
7
(2.2)
I need to account for the shift in the energy due this choice. The RPA provides an upper
bound to the exact ground state energy,
#−1/2
2
− e2 ΠRPA
q,̟l
2 /(8π)
q
q6=0,̟l
#
" 2
q
e2 RPA
2
X
X
X
−
Π
2πe
N
1
q,̟l
= −
+ lim
ln 8π 2 2
2
β→∞
V q6=0 q
2β q6=0 ̟
q /(8π)
l
Z
+∞
X
4πe2 RPA
d̟
N 2πe2
+
ln 1 − 2 Πq (̟) .
−
=
2
V
q
4π
q
−∞
q6=0
N X 2πe2
RPA −
E
=
−
+ lim (−)β −1 ln
GS e=0
e6=0
β→∞
V q6=0 q 2
E RPA GS
In the limit
as :=
RPA
EGS
=
N
3 V
4π N
1/3
≪ aB :=
"
Y
q2
8π
~2
,
me2
(2.3)
(2.4)
0.916
2.21
−
+ 0.062 ln(as /aB ) − 0.096 + O (as /aB ) ln(as /aB ) Ry,
(as /aB )2 (as /aB )
(2.5)
where
Ry :=
~2
e2
=
.
2ma2B
2aB
(2.6)
The first term is called the Hartree term. The first two terms are the Hartree-Fock terms
(see chapter 17 of Ref. [1]). In conventional metals as /aB range from 2 to 6 which indicates
that electronic interactions need to be accounted for to calculate the energy of a metal with
any hope of precision. The RPA gives the next two leading corrections in an expansion
in powers of as /aB [2]. Evidently, it is doubtful that such an expansion is of relevance to
metals in a computational sense. The RPA is, however, instructive conceptually and was,
historically, the first attempt to calculate systematically the effects of electron interactions
in a metal. Our next task is to identify the excitation spectrum above the RPA ground
state.
III.
LINDHARD RESPONSE FUNCTION
It is time to evaluate the polarization function
1
1 X1X
,
ΠRPA
:=
2
q,̟l
V k β ω (iωn − ξk) (iωn + i̟l − ξk+q)
n
8
q 6= 0.
(3.1)
Im z
b
6
b
b
iωn
b
Γk
b
d
6
? b
∂Uξk
b
6 t ?
-
Re z
∂Uξk+q −i̟l
b
b
b
FIG. 2: Poles of Euclidean polarization function on and off imaginary z-axis in the representation
of Eq. (3.5) arising from an arbitrarily chosen k contribution. Hole- (particle-) like poles are
off the imaginary axis and denoted by an empty (filled) circle. Poles on the imaginary axis at
the Matsubara frequencies ωn are denoted by smaller filled circles. Closed integration paths Γk,
∂Uξk+q −i̟l and ∂Uξk are also drawned.
The first step consists in performing the summation over fermionic Matsubara frequencies
ωn = π(2n + 1)/β, n ∈ Z for any given k = 2πl/L, l ∈ Z3 . As an intermediary step, observe
that the Fermi-Dirac distribution function
fFD (z) =
eβz
1
,
+1
z ∈ C,
(3.2)
has equidistant first order poles at
zn = iωn ,
n ∈ Z,
(3.3)
with residues
1
Res fFD (z)iωn = − .
β
9
(3.4)
For any given k, let Γk be the path running anti-parallel to the imaginary axis infinitesimally
close to its left and parallel to the imaginary axis infinitesimally close to its right, i.e., it
goes around the imaginary axis in a counterclockwise fashion. By the residue theorem,
Z
1 X
dz
fFD (z)
RPA
Πq,̟l = −2
.
(3.5)
V k Γk 2πi (z − ξk) (z + i̟l − ξk+q )
Since q 6= 0, the integrand with k fixed has, asides from first order poles along the imaginary
axis, two isolated first order poles at
zk := ξk,
zk+q,̟l := ξk+q − i̟l ,
(3.6)
with residues
fFD (ξk)
1
2πi ξk − ξk+q + i̟l
(3.7)
1
fFD (ξk+q )
1 fFD (ξk+q − i̟l )
=−
,
2πi ξk+q − ξk − i̟l
2πi ξk − ξk+q + i̟l
(3.8)
and
respectively. These two first order poles merge into a second order pole when ̟l = 0 and
q → 0. By Cauchy theorem, the contour of integration can be deformed into two small
circles ∂Uzk and ∂Uzk+q,̟l encircling zk and zk+q,̟l , respectively, in a clockwise fashion (see
Fig. 2),
Γk → ∂Uzk ∪ ∂Uzk+q,̟l .
(3.9)
A second application of the residue theorem gives [be aware of the extra (−)]
X f (ξk) − f (ξk+q)
FD
FD
2 1
ΠRPA
q,̟l = (−) 2
V k
ξk − ξk+q + i̟l
= +2
1 X fFD (ξk− 2q ) − fFD (ξk+ q2 )
.
V k
ξk− q2 − ξk+ q2 + i̟l
(3.10)
It is customary to define the (Euclidean) dielectric constant εq to be the proportionality constant between the bare, Eq. (1.3), and renormalized propagators in Dyson’s equation (1.5a),
D0q = 1 − D0q Σq Dq =: εq Dq .
(3.11)
The RPA for the (Euclidean) dielectric constant is obtained with the substitution (1.6),
4πe2 RPA
Π
q 2 q,̟l
4πe2 1 X fFD (ξk− q2 ) − fFD (ξk+ q2 )
= 1−2 2
.
q V k
ξk− q2 − ξk+ q2 + i̟l
εRPA
q,̟l = 1 −
10
(3.12)
2 4πe
q2
2
1
V
P fFD (ξk )−fFD (ξk+q )
̟−
e (ξk+q −ξk −i0+ )
k
t= ̟
e qmin
'$
6k
&%
.......................
k+q
q fixed with |q| > 2kF
6
1
...................
........... 0
.......
......
.....
....
...
...
...
...
..
..
..
..
.
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...
......
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..
..
..
k+q
..
.......
.
.
.
.
..
..
...
...
..
.........
..
..
..
.
.
.
..
...........
..
..
...
...
...
...............
..
..
.............................
..
..
..
.............
.
.
.
.
.
.
.
t ...
.. d
..
..
..
..
..
..
..
...
..
...
...
..
..
̟
e
̟
e
..
..
Pq
.
.
..
..
..
..
..
...
..
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..
..
..
..
..
.
..
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.
..
..
..
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..
d= ̟
..
..
..
..
e qmax ..
..
...
...
...
..
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..
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.
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..
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..
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..
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..
..
.
..
..
...
..
..
..
..
.
.
.
-
1/L
&%
e at fixed
FIG. 3: Qualitative plot of (4πe2 /q 2 )ΠRPA
+ as a function of the real time frequency ̟
q,−i̟+0
e
momentum q, |q| > 2kF . The polarization function decays like 1/̟
e when |̟|
e ≫ |̟
e qmin |, |̟
e qmax |.
e qmin ≤ ̟
e ≤
The number of intercepts between (4πe2 /q 2 )ΠRPA
+ and the constant line at 1 for ̟
q,−i̟+0
e
̟
e qmax scales like the inverse of the level spacing 1/L, i.e., like L = V 1/3 . There can be one more
intercept between (4πe2 /q 2 )ΠRPA
e qmax < ̟.
e This intercept
+ and the constant line at 1 for ̟
q,−i̟+0
e
takes place at the plasma frequency ̟
e Pq .
Equation (3.12) is known as the Lindhard dielectric constant. Equation (3.12) was first
derived in the static limit ̟l = 0.[3] The static limit ̟l → 0 of the (Euclidean) polarization
function is called the Lindhard function.
A useful property of the Euclidean dielectric function at zero temperature is that an
excitation in the jellium model with momentum q and real time frequency ̟
e q shows up
as a zero of the analytic continuation of the Euclidean dielectric function to the negative
11
imaginary axis1
lim
̟→−i̟+0
e +
εq,̟ = 0.
(3.14)
Indeed, the physical interpretation of Eqs. (3.14) and (3.11) is that a harmonic perturbation
with arbitrarily small amplitude induces a finite response of the Fermi sea in the form of a
finite renormalization of the Coulomb potential.2 The jellium model supports free modes of
oscillations with the dispersion ̟
e q since these oscillations need not be forced by an external
probe to the electronic system. The excitation spectrum within the RPA is obtained from
solving
0 =
lim
̟→−i̟+0
e +
2
= 1−2
εRPA
q,̟
4πe 1 X fFD (ξk) − fFD (ξk+q )
.
q2 V k ̟
e − (ξk+q − ξk − i0+ )
(3.15)
Figure 3 displays a graphical solution to Eq. (3.15). One distinguishes two types of excitations. There is a continuum of particle-hole excitations when, for ̟ ≥ 0 and zero
temperature say,
̟
e qmin ≤ ̟
e ≤̟
e qmax ,
2
kF |q|
|q|
−
,
|q| > 2kF
̟
e qmin :=
2m m
|q|2
kF |q|
= inf
,
+ cos θ
0≤θ<2π
2m
m
(ξk+q − ξk) ,
=
inf
|k|<kF,|k+q|>kF
̟
e qmax
|q| > 2kF
|q| > 2kF ,
|q|2 kF |q|
+
:=
2m m
|q|2
kF |q|
= sup
+ cos θ
2m
m
0≤θ<2π
=
sup
(ξk+q − ξk) .
(3.16)
|k|<kF,|k+q|>kF
1
2
Remember that real time t is related to imaginary time τ by τ = it. A Matsubara frequency ̟l that
enters as ̟l τ in the imaginary-time Fourier expansion of fields is related to the real time frequency ̟
fl by
̟l τ = (+i̟l )(−iτ ) ≡ ̟
fl t,
̟
fl := +i̟l ,
t := −iτ.
(3.13)
The harmonic perturbation can be imposed, for example, by forcing a charge fluctuation in the electron
gas that varies periodically in space and time with wave vector q and real time frequency ̟
e q , respectively.
The infinitesimal frequency 0+ in Eq. (3.14) ensures that the perturbation on the jellium model is switched
on adiabatically slowly.
12
̟
e qmax
̟
e qmin
.
.
.
..
...
6
..
...
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.
.
.
̟P ............................................................
....
....
....
....
.
.
.
.
.
.
...
...
....
....
....
....
.
.
.
.
.
.
...
...
....
....
....
....
.
.
.
.
.
.
.
.
...
...
....
....
......
.....
.
.
.
.
.
.
.
.
.
.
.....
.....
.....
.....
......
.....
.
.
.
.
.
.
.
.
.
.
.
.......
.......
........
........
.........
.........
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
..
............
............
.................
.................
̟
e
0
FIG. 4:
2kF
|q|
Qualitative excitation spectrum for the jellium model within the RPA approximation.
The hashed marked region represents the particle-hole continuum. The line emanating from (|q| =
0, ̟P ) is the plasmon branch of collective excitations.
There is another branch of excitations called plasmons that merges into the continuum for
sufficiently large momentum transfer. Figure 4 sketches the excitation spectrum for the
jellium model within the RPA.
References have been made to the Fermi sea and Fermi wave vector kF . At zero temperature, the Fermi-Dirac distribution becomes the Heavyside step function,

 0, if ξ > 0,
lim fFD (ξ) = Θ(−ξ) =
β→∞
 1, otherwise ,
(3.17)
i
h
q
q
lim fFD (ξk− 2 ) − fFD (ξk+ 2 ) 6= 0 ⇐⇒ ξk− q2 × ξk+ q2 < 0.
(3.18)
and
β→∞
At low temperatures, the polarization function is thus controlled by the geometrical properties of the Fermi sea, the unperturbed ground state of the Fermi gas. I will need some
13
characteristic scales of the Fermi sea. The Fermi wave vector kF is defined by filling up all
available single-particle energy levels,
XX
N
Θ(−ξk)
= V −1
V
σ=↑,↓ k
X k2
k2
F
−1
Θ
=: 2V
−
2m 2m
k
1 4π
(kF )3
(2π)3 3
(kF )3
=
.
3π 2
= 2×
The Fermi velocity and Fermi energy are
1/3
N
kF
∝
,
vF :=
m
V
(kF )2
∝
εF :=
2m
(3.19)
N
V
2/3
,
(3.20)
respectively. The Fermi velocity appears naturally when expanding the numerator in powers
of |q|/kF,
∂fFD (ξk)
= −δ(ξk)
∂ξk
2
k
kF2
= −δ
−
2m 2m
1
= − δ(|k| − kF ).
vF
(3.21)
At temperatures much smaller than the Fermi energy, only those single-particle electronic
states within a distance β −1 of the Fermi surface contribute to the polarization function. At
zero temperature and in the infinite-volume limit, the Lindhard function can be calculated
explicitly,
lim
̟l →0
ΠRPA
q,̟l
mkF
= −
~2 π 2
1 1 − x2 1 + x ,
+
ln 2
4x
1 − x
x=
|q|
.
2kF
(3.22)
Note the presence of logarithmic singularities when the magnitude of the momentum transfer
|q| is twice the Fermi wave vector. These logarithmic singularities are responsible for the
so-called Friedel or Ruderman-Kittel-Kasuya-Yosida oscillations. Note also that
mkF
RPA
× 1.
lim lim Πq,̟l = −
q→0 ̟l →0
~2 π 2
(3.23)
In fact the full dependence of the polarization function on transfer momentum q and transfer
energy ̟l can be expressed in terms of elementary functions (see chapter 12 of [5]). I will
restrict myself to the derivation of some limiting cases below.
14
From now on, both the zero temperature and infinite volume limits are understood,
Z
Z
Z
1 X
d3 k
d3 q
1 X1X
d̟
−→
−→
,
.
(3.24)
3
3
V k
V q β ̟
R3 (2π)
R3 (2π)
R 2π
Furthermore, the long wave-length limit
|q| ≪ kF
(3.25)
will also be assumed for some given transfer momentum q. Choose a spherical coordinate
system in k-space with the angle between q and k the polar angle θ:
k · q = |k||q| cos θ ≡ |k||q|ν,
Z
Z +∞
Z 2π
Z
3
2
d k=
d|k||k|
dφ
R3
0
0
0
π
dθ sin θ ≡
Insertion of
Z
+∞
d|k||k|
0
2
Z
2π
dφ
0
k·q
,
m
∂fFD (ξk) k · q
+ O |q|3
fFD (ξk+ q2 ) − fFD (ξk− q2 ) =
∂ξk
m
1
k·q
= − δ(|k| − kF )
+ O |q|3 ,
vF
m
Z
+1
dν. (3.26)
−1
ξk+ q2 − ξk− q2 =
(3.27)
into the polarization function (3.10) yields
1 X fFD (ξk− q2 ) − fFD (ξk+ q2 )
ΠRPA
=
+2
q,̟
V k
ξk− q2 − ξk+ q2 + i̟
Z ∞
Z 2π
Z +1
1
δ(|k| − kF )|k||q|ν + O (|q|3)
d|k|
kF
2
= +2
|k|
dφ
dν
(2π)3
ν
i̟ − |k||q|
0
0
−1
m
Z +1
vF |q|ν + O (|q|3 /(vF )3 )
mkF
dν
= +2
(2π)2 −1
i̟ − vF |q|ν
" #
3
|q|
4mkF
̟
vF |q|
+O
.
(3.28)
Eqs. 1.622 and2.112 from Ref. 4 = −
1−
arctan
(2π)2
vF |q|
̟
vF
Next, two limits of Eq. (3.28) will be considered:
• The quasi-static limit,
|̟| ≪ vF |q|,
|q| ≪ mvF .
(3.29)
In this limit, the RPA encodes the physics of screening.
• The dynamic limit,
vF |q| ≪ |̟|,
|q| ≪ mvF .
In this limit, the RPA encodes the physics of plasma oscillations.
15
(3.30)
A.
Long-wave lengths and quasi-static limit at zero temperature
The regime
|̟| ≪ vF |q|,
|q| ≪ mvF ,
(3.31)
suggests using the expansion
arctan z =
π 1
1
− + 3 +··· ,
2 z 3z
|z| > 1,
z→
vF |q|
,
̟
in Eq. (3.28). To leading order in this expansion,
" 2 #
2
π
|q|
4mk
̟
̟
F
+O
.
1−
ΠRPA
,
q,̟ = −
(2π)2
2 vF |q|
vF
vF |q|
(3.32)
(3.33)
In turn, the RPA propagator in Eq. (1.2) is approximated by
RPA
Dq,̟
= − q2
4π
= −
1.
1
− e2 ΠRPA
q,̟
4π
F
|q|2 + 8πe2 2mk
1−
2
(2π)
π ̟
2 vF |q|
+O
"
|q|
vF
2 2 #
̟
.
,
vF |q|
(3.34)
The physics of screening
Analytical continuation of Eq. (3.34) onto the negative imaginary axis yields
" 2 #
2
|q|
̟
4π
RPA
+O
. (3.35)
,
lim + Dq,̟ = −
2mkF
e
π ̟
̟→−i̟+0
e
vF
vF |q|
2
2
|q| + 8πe (2π)2 1 + i 2 vF |q|
The imaginary part of the denominator indicates that the “lifetime” of the field ϕ is finite
in the quasi-static limit. The bare Coulomb interaction is thus profoundly modified by the
Fermi sea. The Fermi sea is characterized by a continuum of particle-hole excitations causing
a finite life time of ϕ at finite frequencies and screening in the static limit. As we shall see,
screening is non-perturbative in powers of e.
2.
Thomas-Fermi approximation
In the static limit, ̟
e = 0, the field ϕ acquires an infinite lifetime,
" 2 #
2
4π
̟
|q|
RPA
lim Dq,̟
= − 2
,
,
+O
̟→0
vF
vF |q|
|q| + (λTF )−2
16
(3.36)
where I have introduced the Thomas-Fermi screening length
−1/2
2 2mkF
λTF := 8πe
(2π)2
−1/2
2 mkF
= 4e
.
π
(3.37)
The dependence on e of the Thomas-Fermi screening length is non-analytic in the vicinity
of e = 0. A real space Fourier transformation of the right-hand side yields the Yukawa
potential
− λ|r|
e
TF
|r|
.
(3.38)
However, this Fourier transform extends the range of validity of Eq. (3.36) beyond the long
wave-length limit. Fourier transform to real space of the Lindhard function amounts to the
replacement
− λ|r|
e
TF
|r|
−→ |r|−3 cos(2kF |r|).
(3.39)
This oscillatory behavior is known as a Friedel oscillation. I will rederive this result by
“elementary” means below.
3.
Kohn effect
I would like to revisit the Thomas-Fermi approximation to the RPA propagator DqRPA
and the condition under which it breaks down. We have seen seen that in the static limit,
ξk+q − ξk = + (q · ∇kξk) + O(q 2 ),
∂fFD
(q · ∇kξk) + O(q 2 ),
fFD (ξk+q ) − fFD (ξk) =
∂ξ
2 Z
3
4πe
∂fFD
dk
RPA
2
εq,̟=0 = 1 + 2 2
+ O(q )
−
q
(2π)3
∂ξ
(k )2
= 1 + TF2 + O(q 0 ),
q
(3.40a)
(3.40b)
(3.40c)
where the Thomas-Fermi wave vector kTF is proportional to the density of states at the
Fermi energy. What happens for larger |q|’s? We can use the Lindhard function (3.22)
1 4(kF )2 − q 2 2kF + |q| 4πe2 mkF
,
(3.41)
+
ln εq,̟=0 = 1 + 2
q
~2 π 2
2
8kF|q|
2kF − |q| 17
....................................................................................
........
.....
.....
....
.....
....
....
.
.
....
.
...
...
....
.....
...
...
.
.
...
...
....
....
...
...
q
..
..
...
- .. II .....
... I
...
...
...
.
.
...
...
..
..
..
..
...
...
.
.
.
.
.
....
.
....
...
...
.....
.....
....
....
.......
.
............. ................................ ...................
................
................
|q| < 2kF
...........................................
...........................................
......
........
......
........
.....
.....
....
....
.
.
.
.
.
.
...
.
... ...
.
.
...
... ..
..
...
......
....
.
...
.
.... q
....
..
.
I
II
....
...
..
...
......
..
.
.
...
.
. .
...
...
... ..
....
...
... .......
.
.
.
.
.
.....
.
......
.....
........
......
........
.........................................
........................................
|q| = 2kF
.................................
...........
.......
......
.....
.
.
.
....
....
.
.
...
....
...
...
..
....
...
I
.
...
...
...
.
.
...
.
.
.
...
....
....
.....
....
.
.
.
.
........
.
.........................................
q
...............................
......
............
.....
......
.
.
.
....
..
...
.....
...
...
...
...
...
...
- II
..
..
...
...
.
...
..
...
..
....
...
.
.....
.
.
...
.......
.................. .........................
..
|q| > 2kF
FIG. 5: Two Fermi spheres are drawned to represent pictorially the Kohn effect at zero temperature. The
center of the Fermi spheres are shifted in reciprocal space by the transfer momentum −q. Contributions
to the dielectric constant are only possible when fFD (ξk ) = 1 and fFD (ξk−q ) = 0 or fFD (ξk ) = 0 and
fFD (ξk−q ) = 1. The condition fFD (ξk ) = 1 defines the interior of the Fermi sphere centered at the origin,
the condition fFD (ξk−q ) = 0 defines the outside of the Fermi sphere centered at +q, combining those two
conditions yields region I. The condition fFD (ξk ) = 0 defines the outside of the Fermi sphere centered at the
origin, the condition fFD (ξk−q ) = 0 defines the inside of the Fermi sphere centered at +q, combining those
two conditions yields region II. The union of regions I and II is the Fermi surface ξk = 0 to a very good
approximation for very small momentum transfer q. As the momentum transfer increases in magnitude so
does the volume of the union of region I and II. The volume of the union of region I and II saturates to
twice the Fermi volume at and beyond the value |q| = 2kF .
to infer that the effective screening length increases with the momentum transfer |q|. It
is becoming more and more difficult to make electrons screen out potentials on shorter
wave lengths. Moreover, when |q| = 2kF , the dielectric constant becomes singular. This
singularity comes about from the fact that the summand in the polarization function is
18
proportional to
fFD (ξk+q) − fFD (ξk).
(3.42)
Only those single-particle states with momenta k and k + q contribute to the sum in the
polarization function provided either of one is occupied but not both simultaneously. For
small values of |q| the pairs of single-particle states k and k + q contributing to
fFD (ξk+q ) − fFD (ξk)
(3.43)
belong to two regions I and II that are essentially equal to the surface of the Fermi sea (see
Fig. 5). As |q| is increased, regions I and II increase in size and converge smoothly to the
P
Fermi sea. There thus exists a functional change of k [fFD (ξk+q ) − fFD (ξk)] upon a small
variation δq of q,
|q| < 2kF =⇒
δ X
[fFD (ξk+q) − fFD (ξk)] 6= 0.
δq k
(3.44)
However, as soon as q equals in magnitude twice the Fermi wave vector and beyond, there
P
is no functional change of k [fFD (ξk+q ) − fFD (ξk)] anymore upon a small variation δq of
q,
|q| ≥ 2kF =⇒
δ X
[fFD (ξk+q ) − fFD (ξk)] = 0.
δq k
(3.45)
Transfer momenta obeying |q| = 2kF must be singular points.
This argument does not depend on the shape of the Fermi surface. It also has consequences for the ability of electrons to screen out the electrostatic potential set up by collective
modes propagating through the solids. For example, a phonon of wave vector K sets up an
external potential due to the coherent motion of ions. The electrons respond by screening
the electric field induced by the phonon. Evidently, screening of the ions by the much more
mobile electrons changes the effective interaction between the ions in a nearly instantaneous
way. This last change should thus be encoded by the electronic dielectric constant in the
static limit. Moreover, any singularity in the electronic dielectric constant should show up
in the phonon spectrum thereby opening the possibility to measure the Fermi wave vector
by inspection of the phonon spectrum. This phenomenon is called the Kohn effect.
19
4.
Friedel or Ruderman-Kittel-Kasuya-Yosida oscillations
The dependence on position r of the RPA propagator in the static limit is given by
2
−1
Z
q
RPA
3
−iq·r
2
lim D̟ (r) = − d q e
+ e χq
,
(3.46)
̟→0
4π
where χq is, up to a sign, the static limit of the polarization function, i.e., (~ = 1)
|q|
1 1 − x2 1 + x mkF
,
x=
.
+
ln χq = 2 ×
2
2π
2
4x
1−x
2kF
(3.47)
I have explicitly factorized a factor of 2 arising from the two-fold spin degeneracy. As
already noted in Eq. (3.22), the logarithmic singularity of χq when |q| = 2kF shows up as
an oscillatory behavior at long distances. This oscillatory behavior is known as a Friedel
oscillation in the context of the jellium model. It is also known as the Ruderman-KittelKasuya-Yosida (RKKY) oscillation of the static spin susceptibility induced by a magnetic
impurity in a free electron gas.
I am going to sketch an alternative derivation of the Friedel oscillations. In this derivation,
the emphasis is on the response of the electron gas to a s-wave static charge impurity.
Consider the Schrödinger equation
p2
+ V (|r|) Ψ
i∂t Ψ =
2m
(3.48)
for a (spinless) particle subjected to a spherically symmetric potential V (|r|) that decays
faster than 1/|r| for large |r|. The boundary conditions
Ψ(r = 0, t) finite,
Ψ ∼ outgoing plane wave for large r,
(3.49)
are imposed. Stationary states have an energy spectrum {ε|k|,l} that depends on the angular
momentum quantum number l and on the magnitude of the momentum k of the outgoing
wave. Stationary states behave at large r as
1
π
ψk,l (r) ∼
sin |k||r| − l + ηl Pl (cos θ),
|r|
2
l ∈ N,
(3.50)
where θ is the angle between the momentum k and r and Pl is a Legendre polynomials. The
phase shifts ηl , which are implicit functions of k, encode all informations on the impurity
potential V (|r|). For V = 0, ηl = 0 and stationary states behave at large r as
π 1
(0)
sin |k||r| − l Pl (cos θ),
l ∈ N.
ψk,l (r) ∼
|r|
2
20
(3.51)
l given, ηl = 0
s
-
(0)
(0)
|k1 |
π/R
l given, ηl 6= 0
s
|k1 |
-
s
-
|k2 |
|kn+1,l |
|kn,l |
s
|k2 |
|kn,l |
|k|
|k|
FIG. 6: By switching on the spherical impurity potential V , eigenvalues are shifted along the momentum quantization axis that characterizes the large r asymptotic behavior of energy eigenstates.
This shift of the spectrum can cause a net change in the number of eigenvalues in the fixed interval
(0)
|k1 | ≤ |k| ≤ |k2 |. The shift induced by the spherical impurity potential between |kn,l | and |kn,l | is
(0)
ηl (|kn,l |)/R.
The energy spectrum is discrete if we impose the hard wall boundary condition
lim ψk,l (r) = 0
(3.52)
|r|→R
where R is the radius of a large sphere centered about the origin, i.e., about the impurity,
since
π
1
nπ + l − ηl ,
|k| =
R
2
n ∈ Z,
l∈N
(3.53)
must then hold. Notice that in the absence of the impurity, i.e., when ηl = 0 ∀l ∈ N, the
quantization condition
|k| =
1
π nπ + l ,
R
2
n ∈ Z,
l∈N
(3.54)
yields the same number of energy eigenstates below the Fermi energy εF as if we had chosen
to impose periodic boundary conditions in a box of volume 4πR3 /3 instead. I will denote
(0)
solutions to Eq. (3.53) by kn,l and solutions to Eq. (3.54) by kn,l .
Consider the momentum range |k1 | ≤ |k| ≤ |k2 | as is depicted in Fig. 6. In the absence
of the impurity at the origin we can enumerate all eigenfunctions that decay like
π 1
(0)
sin |kn,l ||r| − l Pl (cos θ)
|r|
2
21
(3.55)
by the integers l and n allowed by the hard wall boundary conditions on the very large
sphere of radius R and for which
|k1 | ≤
(0)
|kn,l |
l π
= n+
≤ |k2 |
2 R
(3.56)
must hold. If we fix l, the spacing in momentum space between neighboring eigenvalues
is π/R. Under the terminology of adiabatic switching of the s-wave impurity potential
one understands the hypothesis that there exists a one to one corespondence between the
eigenfunctions (3.55) with the quantization condition (3.56) and all eigenfunctions that decay
like
π
1
sin |kn,l ||r| − l + ηl Pl (cos θ)
|r|
2
(3.57)
with the quantization condition
|k1 | ≤ |kn,l | =
l
ηl
n+ −
2 π
π
≤ |k2 |,
R
(3.58)
up to few states with wavevectors in the vicinity of k1 and k2 . In the spirit of adiabatic
(0)
switching, the phase shift ηl should be thought of as a function of |kn,l |. Moreover, it is
worthwile to keep in mind that the shift
(0)
|kn,l | − |kn,l | = −
ηl
R
(3.59)
vanishes in the thermodynamic limit R → ∞.
In the thermodynamic limit R → ∞, the change in the number of energy eigenvalues
with fixed l in the range |k1 | ≤ |k| ≤ |k2 | before and after adiabatically switching the s-wave
impurity potential V (|r|) is
1
[ηl (|k2 |) − ηl (|k1 |)] .
π
(3.60a)
If k1 and k2 are chosen to be infinitesimaly far apart, i.e., k1 → k and k2 → k + dk, then
the number (3.60a) takes the differential form
1 dηl
.
π d|k|
(3.60b)
Let us further assume that:
1. First,
lim ηl (k) = 0.
|k|→0
22
(3.61)
2. Second, the Fermi momentum kF or, more generally, the volume of the Fermi sea, is
left unchanged by switching on V (|r|).
We can then integrate Eq. (3.60b) to obtain the total number
∞
1X
2×
(2l + 1)ηl (kF )
π l=0
(3.62)
of new electrons required to fill up all single-particle energy levels up to the Fermi energy
after switching on the s−wave impurity potential V (|r|). (The factor of 2 accounts for the
two-fold spin degeneracy.) If we further require that the electric charge of a s-wave impurity
must be neutralized by an excess of electrons within a finite distance R, then the valence
difference Z between the impurity and the solvent metal is given by
∞
1X
Z =2×
(2l + 1)ηl (kF ).
π l=0
(3.63)
Equation (3.63) is known as Friedel sum rule.
Associated to the phase shifts ηl there are changes in the local electronic density. In the
thermodynamic limit and at large distances from the s-wave impurity, the excess charge is
given by
∞
X
Z
kF
dk |ψk,l;ηl6=0 (r)|2 − |ψk,l;ηl=0 (r)|2
R→∞
π/R
0
l=0
Z
∞
X
1 kF h 2 π i
π
(2l + 1) 2
dk sin kr − l + ηl − sin2 kr − l
∝ e
r 0
2
2
l=0
δρ(r) ∝ lim 2 × e
∝ e
(2l + 1)
∞
X
(2l + 1)(−)l sin ηl cos(2kF r + ηl )
l=0
r3
(3.64)
in the static limit.3 The Yukawa decay predicted by the Thomas-Fermi approximation is
replaced by the slower algebraic decay with superimposed periodic oscillations (quantum
interferences) with periodicity of twice the Fermi wave vector.
B.
Long-wave lengths and dymanic limit at zero temperature
The regime
vF |q| ≪ |̟|,
3
|q| ≪ mvF ,
(3.65)
The normalization of a wave function decaying like r−1 sin(kr) in a sphere of radius R is proportional to
R−1/2 .
23
suggests using the expansion
arctan z = z −
z3 z5
+
−··· ,
3
5
|z| < 1,
z→
vF |q|
,
̟
in Eq. (3.28). To leading order in this expansion,
" 2
4 #
2
v
|q|
4mk
|q|
v
|q|
F
F
F
.
ΠRPA
+O
,
q,̟ = −
3(2π)2
̟
vF
̟
(3.66)
(3.67)
In turn, the RPA propagator in Eq. (1.2) is approximated by
RPA
Dq,̟
= − q2
4π
= −
= −
1
− e2 ΠRPA
q,̟
4π
|q|2
+
4mkF
4πe2 3(2π)
2
4π
h
|q|2 1 +
̟P 2
̟
vF |q|
̟
i +O
2 + O
"
|q|
vF
"
|q|
vF
2 4 #
vF |q|
,
̟
2 4 #
vF |q|
,
,
̟
(3.68)
whereby the so-called plasma frequency is
(̟P )2 := 4πe2
2
4
4 2 N e2
N e2
4mkF
2
3e
(v
)
=
(k
)
=
3π
=
4π
.
F
F
3(2π)2
3π
m
3π
V m
V m
(3.69)
Observe that the factorization of q 2 in the denominator of Eq. (3.68) is special to the
Coulomb interaction.
1.
The physics of plasmons
Analytical continuation of Eq. (3.68) onto the negative imaginary axis yields
" 4 #
2
4π
|q|
vF |q|
RPA
h
lim + Dq,̟ =
.
,
2 i + O
̟→−i̟+0
e
vF
̟
e
q 2 1 − ̟̟eP
(3.70)
After this analytical continuation, we find poles whenever
̟
e q = ̟P ,
∀q.
(3.71)
Of course the independence on the momentum transfer q is an artifact of truncating the
gradient expansion to leading order. Including higher order contributions in the gradient
expansion gives, up to some numerical constant #, the so-called plasmon branch of excitations
̟
e q = ̟P
"
vF
1+#
q
̟P
provided the momentum transfer is not too large.
24
2
#
+··· ,
(3.72)
2.
Landau damping
Once the dispersion curve of plasmons enters in the particle-hole continuum, plasmons
become unstable to decay into an electron-hole pair. This phenomenon is signaled by
lim
̟→−i̟+0
e +
RPA
Dq,̟
−1
acquiring an imaginary part, [use (x − i0+ )−1 = P1/x + iπδ(x)]
Z
d3 k
k·q RPA −1
∝
Im
lim + Dq,̟
q · ∇kfFD (ξk) .
δ ̟
eq −
3
̟→−i̟+0
e
(2π)
m
(3.73)
(3.74)
The factor
k·q
δ ̟
eq −
m
(3.75)
select electrons whose velocities |k|/m are close to the phase velocity ̟
e q /|q| of the plasmon
density wave in that k · q/m = ̟
e q . There is thus a small range of electron velocities for
which the electrons are able to surf the plasmon wave. Electrons moving initially slightly
more slowly than the plasmon wave will pump energy from the plasmon wave as they are
accelerated up to the wave speed by the wave leading edge. Conversely, electrons moving
initially faster than the plasmon wave will give up energy to the plasmon wave as they are
decelerated up to the wave speed by the wave trailing edge. Because the velocity distribution
of electrons
q · ∇kfFD (ξk)
(3.76)
is skewed in favor of low energy electrons, the net effect is to damp the wave. This damping
is called Landau damping.
IV.
RPA FOR A SHORT-RANGE REPULSIVE INTERACTION
So far, we have been dealing exclusively with the two-body repulsive potential
Vcb (r1 − r2 ) = +
e2
.
|r1 − r2 |
(4.1)
(The coupling constant e2 has units of energy × length.) What if we work instead with a
short-range repulsive potential, say
Vλ (r1 − r2 ) = +λδ(r1 − r2 )?
25
(4.2)
(The coupling constant λ has units of energy × volume.) This type of modeling of a two-
body interaction is made, for example, to describe the interaction between 3 He atoms in
liquid 3 He. Since
δ(r) =
1 X +iq·r
e
V q
(4.3)
in a box of volume V with the imposition of periodic boundary conditions, we have the
Fourier transforms
4πe2
,
|q|2
= +λ,
Vcb q =
(4.4a)
Vλ;q
(4.4b)
for the Coulomb and contact repulsive interactions, respectively. RPA fluctuations of the
order parameter ϕ around the mean field ϕmf = 0 is now encoded by the effective action
X 1
1
RPA
RPA
Sλ =
ϕ−q
(4.5a)
ϕ
− Πq
2 +q λ
q=(q,̟l )
instead of [compare with Eq. (1.1d) and note that the convention for the (engineering)
dimension of ϕ has been changed]
RPA
Scb
2
X 1
q
RPA
ϕ
− Πq
ϕ−q .
=
2 +q 4πe2
(4.5b)
q=(q,̟l )
The locations of the poles of
−1
DqRPA := (Vq )−1 − ΠRPA
q
(4.6)
depend dramatically on the short distance behavior of Vq in the dynamic limit |̟l | ≫ vF |q|,
|q| ≪ kF . Indeed, the plasma dispersion (3.72) becomes gapless for our naive modeling of
3
He as
lim
̟→−i̟+0
e +
RPA
Dλ;q,̟
= 1−
1
c|q|
̟
e
2 + O
"
|q|
vF
where
λ
× (̟P )2
2
4πe
N λ
.
=
V m
2 4 #
vF |q|
,
,
̟
e
(4.7)
c2 :=
Eq. (3.69)
26
(4.8)
These excitations are just above the particle-hole continuum and are called zero-sound.
Collective modes whose energies go to zero at large wavelength are the general rule. A finite
energy mode at large wavelengths such as the plasmon is the exception as it is associated with
infinite range forces. Infinite range forces are very special. In the context of phase transition
they cause the breakdown of Goldstone theorem, i.e., of the existence of excitations with
arbitrary small energies when a continuous symmetry is spontaneously broken.
Coming back to zero-sound, one can show that zero-sound is a coherent superposition
of particle-hole excitations near the Fermi surface tantamount to some q-resolved periodic
oscillation of the local (in space) Fermi surface (see chapter 5.4 in Ref. 6). Zero-sound is thus
completely different from thermodynamic sound in a Fermi gas. Thermodynamic sound is a
classical phenomenon that can only be observed on time scales much larger than the typical
time scale τ (smallest between microscopic time scale and inverse temperature) for particles
to interact since the system needs to relax into thermodynamic equilibrium. Conventional
sound results from a time-dependent perturbation whose characteristic time 1/ω is much
larger than τ :
ωτ ≪ 1.
(4.9)
On such time scales, quasiparticle and collective modes have already decayed at finite temperature and thus are unrelated to thermodynamic sound. From a geometrical point of view,
thermodynamic sound can be viewed as an isotropic pulsating local (in space) Fermi sphere
(see chapter 5.4 in Ref. 6). Zero-sound is the opposite extreme to thermodynamic sound.
Zero-sound is built out of quasiparticles. At zero temperature, the coherent superposition
of quasiparticles responsible for zero-sound acquires an infinite lifetime. Hence, zero-sound
can propagate at finite frequencies. At finite temperatures, a necessary condition for the
observation of zero-sound is that the frequency of the external perturbing frequency ω be
large enough for the characteristic observation time 1/ω to be smaller than the lifetime of
quasiparticles,
ωτ ≫ 1.
27
(4.10)
V.
FEEDBACK EFFECT ON AND BY PHONONS
We now consider a jellium model for ions. Ions are point charges of mass M immersed
in an (initially) uniform electron gas of density ρ0 = N/V . The ionic charge is denoted Ze.
The averaged number of ions per unit volume is denoted ρion = Nion /V . Charge neutrality
reads
ZNion = N,
Zρion = ρ0 .
(5.1)
In the absence of any electronic motion but allowing the ionic density to fluctuate in space
and time according to
M v̇ion = (Ze)E,
(5.2a)
∇ · E = 4πe(Z ρion − ρ0 ),
(5.2b)
0 = ρ̇ion + ∇ · Jion ,
(5.2c)
Jion := ρion vion ,
we find, after linearization, a plasma oscillation with frequency [compare with Eq. (3.69)]
(ΩP )2 = 4π
Nion (Ze)2
.
V
M
(5.3)
The ionic plasma frequency ΩP is much lower then the electronic one since
ΩP
̟P
2
2
=
4π NVion (Ze)
M
4π N
V
(−e)2
m
.
=
Zm
≪ 1.
M
(5.4)
Ions move much more slowly that electrons. Electrons can thus adapt to the motion
of ions. In particular, any (infinitesimal) local excess of ionic charge δρion induced by a
collective motion of the ions that solves Eq. (5.2) is screened by the electrons. To account
for this physics, we set up the following model for the coupled system of ions and electrons:
Mδ r̈ion = −(Ze)∇φ,
(5.5a)
−∆φ + (k0 )2 φ = 4π(Ze) δρion + 4πeφext ,
Nion
(Ze)∇ · δrion .
(Ze)δρion = −
V
(5.5b)
(5.5c)
Here, (k0 )2 = 4e2 mkF /π is the squared Thomas-Fermi screening length. We have assumed
that the characteristic frequency ΩP that enters the electronic polarization function is, for all
28
intent and purposes, so small that we can use the Thomas-Fermi approximation to account
for the screening by the electrons. Space and time Fourier transformation of
− ∆φ̈ + (k0 )2 φ̈ = 4π
Nion (Ze)2
∆φ + 4πeφ̈ext
V
M
(5.6)
gives
q 2 (−̟ 2 )φq,̟ + (k0 )2 (−̟ 2 )φq,̟ = (Ωp )2 (−q 2 )φq,̟ + 4πe(−̟ 2 )φext q,̟ ,
(5.7)
i.e.,
1 1
φext q,̟ ,
q 2 εRPA
q,̟
4πe
̟2
=
,
1 + (k0 )2 /q 2 ̟ 2 − (̟q )2
φq,̟ =
1
εRPA
q,̟
(̟q )2 =
(ΩP )2
.
1 + (k0 )2 /q 2
(5.8)
We conclude that:
• The response to an external test charge diverges when ̟ 2 = (̟q )2 . A longitudinal
density fluctuation can thus propagate at this frequency. For small |q|, ̟ ≈ c|q| where
the sound velocity is given by
c2 = (ΩP /k0 )2
1 m
= Z (vF )2 .
3 M
(5.9)
This approximate relation between the speed of sound c and the Fermi velocity vF is
called the Bohm-Staver relation.
• The effective Coulomb propagator mediating the interaction between electrons is modified by the slow motion of the ions. It becomes
gq,̟ =
1
̟2
4πe
.
q 2 1 + (k0 )2 /q 2 ̟ 2 − (̟q )2
(5.10)
This propagator is frequency dependent, i.e., the force between two electrons is not
instantaneous anymore. More importantly, whenever
̟ 2 < (̟q )2 ,
29
(5.11)
the force between electrons has effectively changed signed and become attractive. This
arises because the passage of an electron nearby an ion draws the ion to the electron.
However, in view of the difference in the characteristic energy scales ΩP /̟P ≪ 1, the
ion relaxes to its equilibrium position on time scales much larger than the time needed
for the electron to be far away. In the mean time, another electron can take advantage
of the gain in potential energy caused by moving in the wake of the positive charge
induced by the displaced ion. As both ΩP , or, more generally, the Debye energy, are
small compared to the Fermi energy εF , only electrons near the Fermi surface can take
advantage of the gain in potential energy induced by the ionic motion.
[1] N. W. Ashcroft and N. D. Mermin, Solid state physics, (Holt-Saunders, Philadelphia, 1976)
[2] M. Gell-Mann and K. Brueckner, Phys. Rev. 106, 364 (1957).
[3] J. Lindhard, Kgl. Danske Videnskab. Selskab Mat.-Fys. Medd. 28, No. 8 (1954).
[4] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academuc Press,
San Diego, 1994).
[5] A. L. Fetter and J. D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, NewYork, 1971).
[6] J. W. Negele and H. Orland, Quantum many-particle systems, (Addison-Wesley, Redwood City,
1988).
30