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= +


⎝ A
p’
Ep’

16 Interms of diagrams the Di¬uson is given by the impurity ladder diagrams; see Exercise 7.8
and chapter 8 of reference [1].
198 7. Quantum kinetic equations


R R
p+


E+ p+ E+ p+


+ · · ·⎟ .
+ (7.105)



A A
p’
Ep’ Ep’



Show that for ql, ω„ 1, E F, the Di¬uson exhibits the di¬usion pole
1
D(q, ω) ≡ „ u’2 DE (q, ω) = , (7.106)
’iω + D0 q 2
2
where D0 = vF „ /d is the di¬usion constant in d dimensions.


7.5 Quasi-classical Green™s function technique
When particles interact there can be strong dependence of the self-energy on the
energy variable E, as in the case of electron“phonon interaction in strong coupling
materials, say as in a metal such as lead, which is the type of system we for exam-
ple shall have in mind in this section. The employed quasi-particle approximation
Eq. (7.47) is not valid and the structure in the spectral weight, Eq. (7.45), must be
respected, leaving no chance of simplicity by integrating over the energy variable E,
i.e. of obtaining equations involving only equal-time Green™s functions.
There exists a consistent and self-contained approximation scheme for a degen-
erate Fermi system, valid for a wide range of phenomena, that does not employ the
restrictive quasi-particle approximation. It is called the quasi-classical approxima-
tion.17 The electron“phonon interaction can lead to an important structure in the
self-energy, i.e. in eΣ and “, as a function of the variable E. In contrast, as noted
by Migdal, the momentum dependence is very weak as a consequence of the phonon
energy being small compared with the Fermi energy [25]. The spectral weight func-
tion thus becomes a peaked function of the momentum, and we shall exploit this
peaked character.
The left“right subtraction trick dismissed the strong linear E-dependence in the
inverse propagator G’1 , and similarly its strong momentum dependence, its ξp -
0
dependence, ξp = p ’ μ. It therefore allows, when there is only weak momentum
dependence of the self-energy, i.e. short-range e¬ective interaction, which is typi-
cally the case for electronic interactions, integration over the variable ξp , so-called
ξ-integration. The peaked character of the spectral weight in the variable ξ will, in
conjunction with multiplying other quantities, restrict their momentum dependence
to the Fermi surface. We shall therefore consider the ξ-integrated Green™s function
17 This scheme was ¬rst applied by Prange and Kadano¬ in their treatment of transport phenomena
in the electron“phonon system [23]. It was later extended to describe transport in super¬‚uid systems
by Eilenberger [24], the topic of the next chapter.
7.5. Quasi-classical Green™s function technique 199


or quasi-classical Green™s function18
i
ˆ
g(R, p, t1 , t1 ) = dξ G(R, p, t1 , t1 ) . (7.107)
π
We note that care should be exercised with respect to ξ-integration, since the in-
tegrand is not well behaved for large values of ξ, falling o¬ only as 1/ξ. The ξ-
integration should be understood in the following sense of deforming the integration
contour as depicted in Figure 7.1: the ξ-integration is split into a low- and high-
energy contribution, and only the low-energy contribution is important in the kinetic
equation since high-energy contributions do not contribute.




1
+1
= 2 2



Figure 7.1 Splitting in high- and low-energy contributions.



The semicircles are speci¬ed by a cut-o¬ energy Ec , which is chosen much larger than
the Fermi energy. The remaining high-energy contribution to the Green™s function
does not depend on the non-equilibrium state, i.e. it is a constant, and therefore
drops out of the left“right subtracted Dyson equation. We immediately return to
this point again when expressing physical quantities, such as average currents and
densities in terms of the quasi-classical Green™s function, and later in Section 8.3 to
provide a less formal and more physical understanding of the quasi-classical Green™s
function.
Let us ¬rst determine measurable quantities in terms of the quasi-classical Green™s
function, say density and currents, in the presence of an electromagnetic ¬eld (A, •).
The charge density becomes, in terms of the quasi-classical Green™s function,
1 dˆ
p
ρ(R, T ) = ’ eN0 dE g K (E, p, R, T ) ’ 2e2 N0 •(R, T ) ,
ˆ (7.108)
2 4π
where N0 is the density of states at the Fermi energy, and the current density is given
by
1 dˆ
p
j(R, T ) = ’ eN0 dE vF g K (E, p, R, T ) .
ˆ (7.109)
2 4π
18 Here and in the following, we assume for simplicity a spherical Fermi surface. For a general
Fermi surface one decomposes according to
dξp dsF
dp ˆ
= ,
3 vF (2π)3
(2π)
where dξp /vF is the length of the momentum increment measured from the Fermi surface in the
ˆ
directions ±ˆ , and dsF is the corresponding Fermi surface area element.
p
200 7. Quantum kinetic equations


The ξ-integration does not respect the proper order of integrations, momentum in-
tegration being last because of the convergence property of the Green™s function.
We thus encounter the high energy contribution to the density, the second term on
the right in Eq. (7.108), whereas in the current density the high-energy contribution
cancels the term proportional to the vector potential, the so-called diamagnetic term.
We observe, as discussed in Section 6.2, that as far as the high-energy contributions
are concerned, the analysis and their calculation is equivalent to their appearance
in linear response expressions.19 The high-energy contributions also follow from the
gauge transformation properties of the Green™s function.

7.5.1 Electron“phonon interaction
Here we apply the quasi-classical technique to the case of strong electron“phonon
interaction, thereby obtaining the kinetic equation for the electrons that includes the
renormalization e¬ects.
Let us ¬rst make sure that the electron“phonon self-energy is susceptible to ξ-
integration, i.e. it can be expressed solely in terms of the ξ-integrated Green™s func-
tion or quasi-classical Green™s function. Migdal™s theorem states that the electron“
phonon self-energy diagrams for the electron Green™s function where phonon lines
cross are small in the parameter ωD / F , where ωD is the typical phonon energy, i.e.
vertex corrections are negligible [25].20 In this approximation, which indeed is a good
one in metals, with an accuracy of order 1%, the electronic self-energy is represented
by a single skeleton diagram as depicted in Figure 7.2.




1 1
Figure 7.2 Electron“phonon self-energy.



or analytically for the lowest order in ωD / contribution to the electron self-energy
F

(e’ph)
(1, 1 ) = ig 2 γii Gi j (1, 1 ) Dkk (1, 1 ) γj j .
k
˜k
Σij (7.110)

In the mixed coordinates with respect to the spatial coordinates we get (suppressing
19 A detailed discussion of this is given in chapters 7 and 8 of reference [1].
20 The demonstration of Migdal™s theorem is quite analogous to that of crossing impurity diagrams
being small. Crossing lines result in propagators having restriction on the momentum range for which
they provide a large contribution. Contributions from such diagrams thus become small owing to
phase space restrictions. In the case of electron“phonon interaction, the range is set by the typical
phonon energy. For details on diagram estimation see chapter 3 of reference [1].
7.5. Quasi-classical Green™s function technique 201


the arguments on the left (R, p, t1 , t1 ))21
dp
(e’ph)
Gi j (R, p , t1 , t1 ) Dkk (R, p ’ p , t1 , t1 ) γj j . (7.111)
= ig 2 γii
k
˜k
Σij
(2π)3
The momentum integration can be split into integrations over angular (or in general
Fermi surface) and length of the momentum measured from the Fermi surface
dp dˆ dˆ
p p
= dξ N (ξ) = N0 dξ (7.112)
(2π)3 4π 4π
and the last equality is valid when particle“hole symmetry applies.22 Using the fact
that the Debye energy is small compared with the Fermi energy,23 the various electron
Green™s function are tied to the Fermi surface, and we obtain the electron“phonon
matrix self-energy expressed in terms of the quasi-classical matrix electron Green™s
function
»k
(e’ph)
(R, p, t1 , t1 ) = γii dˆ gi j (R, p , t1 , t1 ) Dkk (R, pF (ˆ ’ p ), t1 , t1 )˜j j ,
γk
ˆ ˆ pˆ
σij p
4
(7.113)
where » = g 2 N0 is the dimensionless electron“phonon coupling constant. The matrix
components of the matrix self-energy are therefore
»
R(A)
g K (R, p , t1 , t1 ) DR(A) (R, pF (ˆ ’ p ), t1 , t1 )
ˆ ˆ pˆ
σe’ph (R, p, t1 , t1 ) = dˆ
p
8

g R(A) (R, p , t1 , t1 ) DK (R, pF (ˆ ’ p ), t1 , t1 )
ˆ pˆ
+ (7.114)

and
»
dˆ (g R (R, p , t1 , t1 ) DR (R, pF (ˆ ’ p ), t1 , t1 )
K
ˆ ˆ pˆ
σe’ph (R, p, t1 , t1 ) = p
8

+ g A (R, p , t1 , t1 ) DA (R, pF (ˆ ’ p ), t1 , t1 )
ˆ pˆ

+ g K (R, p , t1 , t1 ) DK (R, pF (ˆ ’ p ), t1 , t1 ))
ˆ pˆ (7.115)
or equivalently
»
dˆ (g R ’ g A )(DR ’ DA ) + g K DK
K
σe’ph = (7.116)
p
8
21 Inthe case of impurity scattering, the self-energy is expressed in terms of the quasi-classical
Green™s function according to
p
i dˆ
σimp (E, R, T ) = ’ g(ˆ , E, R, T ) ,
p
2„ 4π
where the high-energy cut-o¬ is provided by the momentum dependence of the impurity potential,
providing necessary convergence.
22 Or rather, owing to this step the quasi-classical approximation is unable to account for e¬ects

due to particle“hole asymmetry.
23 Or equivalently, the sound velocity is small compared with the Fermi velocity.
202 7. Quantum kinetic equations


since

g R (t1 , t1 ) DA (t1 , t1 ) = 0 = g A (t1 , t1 ) DR (t1 , t1 ) . (7.117)

Utilizing the peaked character of the electron spectral weight function in the ξ-
variable, the momentum dependence of the self-energy can be neglected, and the
left“right subtracted Dyson equations, Eq. (7.1), can be integrated with respect to
ξ, giving the quantum kinetic equation
’1
g0 + i eσ —¦ g K = 2iσ K ’ i (σ R ’ σ A ) —¦ g K
, , , (7.118)
’ +

where
’1 ’1
g0 (R, p, t1 , t1 ) = g0 (R, p, t1 ) δ(t1 ’ t1 )
ˆ ˆ (7.119)
and
e2 2
’1
= ‚t1 + vF · (∇x1 ’ ieA(R, t1 )) + ieφ(R, t1 ) ’
ˆ
g0 (R, p, t1 ) A (R, t1 ). (7.120)
2m
Here vF = pF /m, the Fermi velocity, speci¬es the Fermi surface direction, and —¦
implies matrix multiplication in the time variable. We have considered the case
where, say, the electrons in a metal are subject to electromagnetic ¬elds.
From the spectral representation

dE A(E , p, R, T )
R(A)
G (E, p, R, T ) = (7.121)
’∞ 2π E ’ E (’) i0
+


it follows that ξ-integrating eG gives a state independent constant and the last
term on the left-hand side in Eq. (7.3) vanishes upon ξ-integration.
’1
The form of g0 follows from the following observation where for de¬niteness we
focus on the scalar potential term. First transform to the mixed spatial coordinates

•(x1 , t1 ) G(x1 , t1 , x1 , t1 ) = •(R + r/2, t1 ) G(R, r, t1 , t1 ) . (7.122)

Since the Green™s function G(R, r) is a wildly oscillating function in the relative
’1
coordinate r, the function is essentially zero when r kF , and since we shall
assume that the scalar potential is slowly varying on the atomic length scale we have

•(x1 , , t1 ) G(x1 , t1 , x1 , t1 ) •(R) G(R, r, t1 , t1 ). (7.123)

It can be instructive to perform the equivalent argument on the Fourier-transformed
product giving (being irrelevant for the manipulations, the time variables are sup-
pressed)

dkdPdp ei(R·P+r·p) •(k) G(P ’ k, p ’ k/2) , (7.124)
•(R + r/2) G(R, r) =

where the shifts of variables, P + k ’ P and P + k/2 ’ p, have been performed.
The quasi-classical approximation consists of the weak assumption that the external
perturbation only has Fourier components for wave vectors small compared with the
7.5. Quasi-classical Green™s function technique 203


Fermi wave vector, k kF , so that G(P’k, p’k/2) G(P’k, p), again leading to
the stated result, Eq. (7.123). In the quasi-classical approximation the e¬ect of the
Lorentz force is lost, and for a perturbing electric ¬eld we might as well transform
to a gauge where the vector potential is absent. This is the price paid for the quasi-
classical approximation, which is less severe in the superconducting state, and we will
return to e¬ects of the Lorentz force in the normal state in Section 7.6. However, we
note that the in¬‚uence on the phase of the Green™s function is fully incorpotated in
the quasi-classical approximation, a fact we shall exploit when considering the weak
localization e¬ect in Chapter 11.
A simpli¬cation which arises in the normal state, and should be contrasted with
the more complicated situation in the superconducting state to be discussed in Sec-
tion 8.2.3, is the lack of structure in the ξ-integrated retarded and advanced Green™s
functions

δ(t1 ’ t1 ) ,
g R(A) (R, p, t1 , t1 ) = +
g R(A) (R, p, E, T ) = +
ˆ ˆ 1 (7.125)
(’) (’)

and they thus contain no information since particle“hole asymmetry e¬ects are ne-
glected, i.e. the variation of the density of states through the Fermi surface is ne-
glected. This fact leaves the quantum kinetic equation, Eq. (7.118), together with
the self-energy expressions a closed set of equations for g K .
We emphasize again that in obtaining the quasi-classical equation of motion only
the degeneracy of the Fermi system is used, restricting the characteristic frequency
and wave vectors to modestly obey the restrictions

q kF , ω . (7.126)
F

These criteria are well satis¬ed for transport phenomena in degenerate Fermi systems.
In contrast to the performed approximation for the convolution in space due to
the degeneracy of the Fermi system, there is in general no simple approximation
for the convolution in the time variables. Two di¬erent approximation schemes are
immediately available: one consists of linearization with respect to a perturbation
such as an electric ¬eld, allowing frequencies restricted only by the Fermi energy,
ω < F , to be considered, but of course restricted to weak ¬elds. The other assumes
perturbations to be su¬ciently slowly varying in time that a lowest-order expansion
in the time derivative is valid

[A —¦ B]’ ‚E A ‚T B ’ ‚T A ‚E B .
A B A B
, (7.127)

For de¬niteness, we shall employ the second scheme here. In order to reduce the
general quantum kinetic equation, Eq. (7.118), to a simpler looking transport equa-
tion, we introduce the mixed coordinates with respect to the temporal coordinates
and perform the gradient expansion in these variables giving

((1 ’ ‚e eσ)‚T + ‚T eσ ‚E + vF · ∇R + e‚T • ‚E ) g K = Ie’ph , (7.128)

where the collision integral is

Ie’ph = 2iσ K ’ γg K , (7.129)
204 7. Quantum kinetic equations


where

γ = i(σ R ’ σ A ) . (7.130)

The two terms in the collision integral constitute the scattering-in and scattering-out
terms, respectively. According to Eq. (7.114) and Eq. (7.116) they are determined
by (space and time variables suppressed)

E ’E

p
γ(E, p) = ’π dE μ(pF (ˆ ’ p ), E ’ E ) coth ’ h(E , p )
ˆ pˆ ˆ
4π 2T

(7.131)

and
E ’E

p
iσe’ph (E, p) = ’π dE μ(pF (ˆ ’ p ), E ’ E ) h(E , p ) coth ’1 ,
K
ˆ pˆ ˆ
4π 2T
(7.132)

where we have introduced the distribution function

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