functions and self-energies.5 In the following we assume the non-equilibrium state is

induced by an applied potential, V (R, T ), which is a slowly varying function of its

variables compared to the characteristic scales of equilibrium Green™s functions and

self-energies.6 We shall, for example, have a degenerate Fermi system in mind, say

conduction electrons in a metal, where the characteristic scales are the Fermi energy

and momentum. This allows for the approximation where only lowest-order terms

in the variation is kept, the so-called gradient approximation. In this approximation

we thus have

[A — B]+ ’ 2A(X, p) B(X, p)

, (7.34)

and

’i[A — B]’ ’ [A, B]p ,

, (7.35)

where

= ‚X A ‚p B ’ ‚p A ‚X B

A B A B

[A, B]p

‚E ‚T ’ ‚T ‚E ’ ∇A · ∇B + ∇A · ∇B A(X, p) B(X, p) ,

AB AB

= p R R p

(7.36)

5 If

one is interested only in the linear response, such an assumption is not needed, but the gradient

approximation allows, in principle, inclusion of all the nonlinear e¬ects of a slightly inhomogeneous

perturbation.

6 The coupling to a vector potential will be handled in Section 7.6.

7.3. Gradient approximation 185

and the subscript p on the bracket signi¬es its resemblance to the Poisson bracket of

classical mechanics.

In the gradient approximation, the quantum kinetic equation, Eq. (7.3), becomes

[G’1 ’ eΣ, GK ]p ’ [ΣK — eG]p = iΣK A ’ i“ GK .

, (7.37)

0

The ¬rst term on the left-hand side becomes, in the gradient approximation,

[G’1 — GK ]’ ’ [G’1 — GK ]p

, ,

0 0

‚T GK (E, p, R, T ) + ‚E GK (E, p, R, T ) ‚T V (R, T )

=

∇R GK (E, p, R, T ) · ∇p ξp ’ ∇p GK (E, p, R, T ) · ∇R V (R, T ).

+

(7.38)

In fact, the ¬rst term is always exact, and so is the third term for the case of quadratic

dispersion.7 We note that they are identical in form to the driving terms in the

Boltzmann equation, whereas the last term on the right, which also appears in the

Boltzmann equation, here is valid only in the gradient approximation, i.e. the mag-

nitude of the characteristic wave vector of the potential, q, is small compared with

the characteristic wave vector of the system, which in the case of degenerate fermions

is the Fermi wave vector, q < kF (usually no restriction at all for transport situations

in degenerate Fermi systems). The second term on the right looks strange in the

Boltzmann context, but we shall soon integrate the equation over E, upon which

this term disappears.

Since in equilibrium a Poisson bracket vanishes, the kinetic equation reduces to

0 = ΣK (E, p) A(E, p) ’ “(E, p) GK (E, p) (7.39)

and this identity can be interpreted as the statement of determining the equilibrium

distribution function as the one for which the right-hand side, the collision integral,

vanishes.

7.3.1 Spectral weight function

To make further progress we study the spectral weight function. The equation of

motion for the spectral weight function is obtained by subtracting the diagonal com-

ponents of Eq. (7.1), giving

[G’1 ’ eΣ — A]’ ’ [“ —

, , eG]’ = 0 . (7.40)

0

In the gradient approximation, the non-equilibrium spectral function satis¬es

(according to Eq. (7.40)) the equation

[E ’ ξp ’ V (R, T ) ’ eΣR , A]p + [ eGR , “]p = 0 . (7.41)

7 The ¬rst term is not dependent on the gradient approximation, but as usual is exact, simply

owing to the equation being ¬rst order in time, and similarly for the second term for the case of

quadratic dispersion.

186 7. Quantum kinetic equations

We note that

“(E, p, R, T )

A(E, p, R, T ) = (7.42)

2 2

“(E,p,R,T )

E ’ ξp ’ V (R, T ) ’ eΣR (E, p, R, T ) + 2

solves Eq. (7.41) since, because [A, B]p = ’[B, A]p , and noting that

’1

= E ’ ξp ’ V (R, T ) ’

e GR (E, p, R, T ) eΣR (E, p, R, T ) , (7.43)

the left-hand side of equation Eq. (7.41) can then be rewritten in the form

’1

i i

R ’1 R ’1

’i ’ “, ’“

eG eG

2 2

p

’1

i i

R ’1 R ’1

+i eG + “, eG +“ , (7.44)

2 2

p

which vanishes, since for any function F , we have [A, F (A)]p = 0. In the far past,

where the system is assumed undisturbed, i.e. V vanishes, the presented solution,

Eq. (7.42), reduces to the equilibrium spectral function

“(E, p)

A(E, p) = , (7.45)

2

2

E ’ ξp ’ eΣ(E, p) + (“(E, p)/2)

which in this case can be obtained directly from Eq. (7.40). The solution Eq. (7.42)

is therefore the sought solution since it satis¬es the correct initial condition.

Adding the left and right Dyson equations for the retarded non-equilibrium Green™s

function, and performing the expansion within the gradient approximation, Eq. (7.34),

we similarly obtain the result

1

GR (E, p, R, T ) =

G’1 (E, p, R, T ) ’ ΣR (E, p, R, T )

0

1

= , (7.46)

E ’ ξp ’ V (R, T ) ’ ΣR (E, p, R, T )

and similarly for the advanced Green™s function.

7.3.2 Quasi-particle approximation

If the interaction is weak the self-energies are small, and the spectral weight function

is a peaked function in the variable E, in fact in the absence of interactions according

to Eq. (7.42)

A(E, p, R, T ) = 2π δ(E ’ ξp ’ V (R, T )) (7.47)

7.3. Gradient approximation 187

and therefore is GK also a peaked function in the variable E. We ¬rst consider this

so-called quasi-particle approximation.8 In Section 7.5 we will consider the case of

strong electron“phonon interaction and the spectral weight can not be approximated

by a delta function, and a di¬erent approach to obtaining a kinetic equation must

be developed.

The reason for subtracting the left and right Dyson equations is that the term

linear in E in G’1 then disappears, thereby, in view of Eq. (7.47), allowing the

0

equation, Eq. (7.37), to be integrated with respect to this variable giving

∇p ξp · ∇R ’ ∇R V (R, T ) · ∇p ) h(p, R, T )

(‚T +

ΣK (E = ξp + V (R, T ), p, R, T )

=

’ “(E = ξp + V (R, T ), p, R, T ) h(p, R, T ) , (7.48)

where we have introduced the distribution function

∞

dE K

h(p, R, T ) = ’ G (E, p, R, T ) . (7.49)

’∞ 2πi

The two self-energy terms on the left in Eq. (7.37) must be neglected in this ap-

proximation since they are by assumption small and in addition multiplied by the

characteristic frequency, ω0 , of the external potential which is small compared with

the characteristic frequency of the system, which in the case of degenerate fermions

is the Fermi energy, ω0 F . In the event that the left“right subtracted Dyson equa-

tion allows for integrating over E, equal time quantities appear, and the distribution

function is of the Wigner type, and is related similarly to densities and currents.9

In equilibrium the distribution function is for fermions given by

ξp

h0 (p) = tanh (7.50)

2kT

in which case the sum of the two terms on the right in Eq. (7.48) vanish. We shall

now focus on the terms on the right-hand side of equation Eq. (7.48), and realize

they describe collisions and dissipative e¬ects.

Since the equation for the Green™s function is not closed we will eventually have

to make an approximation that cuts o¬ the hierarchy of correlations. For states

not too far from equilibrium, this can be done at the level of self-energies if, for

example, vertex corrections can be shown to be small in some parameter, viz. the one

characterizing the equilibrium approximation. To this end we recall the usefulness

of the diagrammatic estimation technique.

8 This is of course a most unfortunate choice of labeling used in the literature. The physical

implication of the approximation simply being that in between collisions, the particle motion is that

of a free particle.

9 For a discussion of the Wigner function see chapter 4 of reference [1].

188 7. Quantum kinetic equations

7.4 Impurity scattering

We now start to consider interactions of relevance, and begin with the simplest

case; that of impurity scattering. In the clean limit where impurity scattering say of

electrons in a metal or semiconductor is weak, so that any tendency to localization in

,10 diagrams with crossing

a three-dimensional sample can be neglected, i.e. F „

of impurity lines can be neglected, and the impurity self-energy is11

≡ pE pE

Σ(E, p, R, T ) (7.51)

p ERT

corresponding to the analytical expression for the real-time matrix self-energy

dp

|Vimp (p ’ p )|2 G(p , E, R, T ) .

Σ(p, E, R, T ) = ni (7.52)

(2π )3

For the kinetic component of the self-energy we have

dp

|Vimp (p ’ p )|2 GK (p , E, R, T )

ΣK (p, E, R, T ) = ni (7.53)

3

(2π)

and

“(p, E, R, T ) = i(ΣR (p, E, R, T ) ’ ΣA (p, E, R, T ))

dp

|Vimp (p ’ p )|2 A(p , E, R, T ) .

= ni (7.54)

(2π)3

Since we work to lowest order in the impurity concentration, ni , the spectral weight

should be replaced by the delta function expression, and we obtain

(‚T + ∇p ξp · ∇R ’ ∇R V (R, T ) · ∇p ) h(p, R, T ) = I (1) [f ] (7.55)

where the right side, the electron-impurity collision integral, is

dp

I (1) [f ] = ’2πni |Vimp (p’p )|2 δ(ξp ’ξp )(h(p, R, T )’h(p , R, T )) . (7.56)

3

(2π)

We have arrived at the classical kinetic equation describing the motion of a particle

in a weakly disordered system, the Boltzmann equation for a particle in a random

10 In a strictly one-dimensional sample localization is typically dominant and in a two-dimensional

sample it is important at low enough temperatures. The ¬rst quantum correction to this classical

limit, the weak localization e¬ect, is discussed in Chapter 11.

11 For a detailed description of the standard impurity average Green™s function technique and

diagrammatic estimation, we refer the reader to reference [1], where also inclusion of multiple

impurity scattering is shown to be equivalent to the considered Born approximation by inclusion of

the t-matrix.

7.4. Impurity scattering 189

potential. The derived equation is called a kinetic equation because the collision

integral is not a functional in time (or space), i.e. local in both the space and time

variables, and a functional only with respect to the momentum variable. The only

di¬erence signaling we are considering the degenerate electron gas is the quantum

statistics, which dictates the distribution function to respect the Pauli principle, i.e

the equilibrium distribution is speci¬ed by Eq. (7.50).

The weak-disorder kinetic equation for a particle in a random potential is of course

immediately obtained from classical mechanics, granted a stochastic treatment of the

impurity scattering, giving the collision integral

(1)

It [f ] = ’ {W (p , p)f (p, t) ’ W (p, p )f (p , t)} , (7.57)

p

where W (p , p) is the classical transition rate between momentum states, the classical

scattering cross section. In classical mechanics the distribution function concept is

unproblematic because we can simultaneously specify position and momentum, and

the terms on the left-hand side of Eq. (7.55) are simply the streaming terms in phase

space for the situation in question.

In the quantum case we have, in the Born approximation for the transition rate

between momentum states,

2πni

|Vimp (p ’ p )|2 δ( ’

W (p , p) = p)

p

V

2π

ni V | p|Vimp (ˆ )|p |2 δ( ’

= p) . (7.58)

x p

We note that in the Born approximation we always have W (p , p) = W (p, p ).12

We note that the expression W (p , p) in Eq. (7.58) is Fermi™s Golden Rule ex-

pression for the transition probability per unit time from momentum state p to

momentum state p (or vice versa) caused by the scattering o¬ an impurity, times

the number of impurities. The two terms in the collision integral thus have a sim-

ple interpretation because they describe the scattering in and out of a momentum

state. For example, the ¬rst term in the collision integral of the Boltzmann equation,

Eq. (7.56), is a loss term, and gives the rate of change of occupation of a phase space

volume due to the scattering of an electron from momentum p to momentum p by

the random potential. The probability per unit time of being scattered out of the

phase space volume around p, and into a volume around p , is the product of three

probabilities: (the probability that an electron is in that phase space volume to be

available for scattering) — (the transition probability per unit time for the transition

from state p to p ) — (the probability that there is an impurity in the space volume

to scatter). Similarly we have the interpretation of the other term as a scattering-in

term.

The obtained equation is a quasi-classical equation because, in between collisions

with impurities, the electrons move along straight lines just as in classical mechan-

12 In general, potential scattering is time-reversal invariant, and we always have W (p , p) =

W (’p, ’p ). If, in addition, the potential is invariant with respect to space inversion, we have

W (p , p) = W (’p , ’p), and thereby W (p , p) = W (p, p ).

190 7. Quantum kinetic equations

ics, but the scattering cross section is the quantum mechanical one.13 Besides the

inherent quantum statistics, this is the only quantum feature surviving in the weak

disorder limit, / F „ 1, where „ is the characteristic time scale for the dynamics,

the momentum relaxation time, soon to be discussed. The presented diagrammatic

method for deriving transport equations is capable of going beyond the Markov pro-

cess described by the classical kinetic equation, to include quantum e¬ects. One

can construct a kinetic equation determining the ¬rst quantum correction, the weak

localization e¬ect, but it is easier to employ linear response theory as described in

Chapter 11.

Let us study the simplest non-equilibrium situation where the distribution is out

of momentum equilibrium for only a single momentum state on the Fermi surface

fp (t) = f0 ( ) + δfp (t) δp,p (7.59)

p

and we assume no external ¬elds. The Boltzmann equation then reduces to

‚δfp (t) δfp

=’ (7.60)

‚t „p

whose solution describes the exponential relaxation to equilibrium

fp (t) = f0 ( p ) + δfp (t = 0) e’t/„p (7.61)

and the momentum relaxation time (which for the considered isotropic Fermi surface

does not depend on the direction of the momentum)

1

= Wp ,p (7.62)

„

p (=p)

is seen to be identical to the imaginary part of the retarded self-energy for E = F

1 1 dp

|Vimp (pF ’ p )|2 δ( ’

= = 2πni F) . (7.63)

p

3

„ „ ( F) (2π)

We noted above that the collision integral rendered the kinetic equation a stochas-

tic equation for the momentum, Pauli™s master equation. In the case where „ (p) can

be considered independent of the momentum p, „ is the phenomenological parameter

of the Drude theory of conduction, and ”t/„ (p) is, according to Eq. (7.61), the prob-

ability that an electron with momentum p in the time span ”t will su¬er a collision

with total loss of momentum direction memory. Such an assumption is not valid in

the quantum mechanical description as the scattering of a wave sets up correlations

that can not lead to a total memory loss in general, as we shall discuss in detail in

Chapter 11.

One might miss Pauli blocking factors in the expression for the collision integral,

Eq. (7.56), but they need not, as just shown, appear in the considered case of potential

13 If we go beyond the considered Born approximation and include multiple scattering, we en-

counter the exact cross section for scattering o¬ an impurity as expressed by the t-matrix. For a

discussion see chapter 3 in reference [1].

7.4. Impurity scattering 191

scattering. If one uses the Kadano¬“Baym form of the kinetic equation, Eq. (5.136),

Pauli blocking factors would then appear in intermediate results. Another lesson to

learn is that the form of the appearance of the quantum statistics, here the Fermi“

Dirac distribution function or other forms, depends on the type of Green™s functions

one employs; a case in question is our choice leading to the distribution function in

Eq. (7.49) and Eq. (7.50).

For the sole purpose of obtaining the weak-disorder kinetic equation, the use of

quantum ¬eld theoretic methods and Feynman diagrams is hardly necessary. How-

ever, it allows us in a simple way to assess the validity criterion for the classical kinetic

description, and to go beyond the classical limit and study quantum corrections. In

view of the neglected diagrams, the validity of the Boltzmann equation requires

/l, where l = vF „ is the mean free path.14 In

/ F„ 1, or equivalently pF

addition for the gradient approximation to be valid, the characteristic frequency and

wave vector of the perturbation must satisfy the weak restrictions ω < F , q < kF .

There can be some satisfaction in deriving the Boltzmann equation, in particular to

establish validity criteria, i.e. to establish the Landau criterion and not instead the

devastating for applications Peierls criterion, ω < kT , which an argument based

on a simple quasi-particle picture would suggest. But for the sake of deriving clas-

sical kinetic equations, the venture into quantum ¬eld theory is over-kill. The more

so, that in practice it is di¬cult to go beyond the linear regime systematically and

study nonlinear e¬ects. However, there exists a successful technique that leads to an

exception to this state of a¬airs, viz. the so-called quasi-classical Green™s function

technique. We consider this technique applied in the normal state in Section 7.5, and

its even more important application to superconductivity will be studied in Chapter

8.

Exercise 7.3. Show that the continuity equation is obtained by integrating the