⎜

⎜

⎝ 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

dˆ

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

dˆ

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