and

|ˆ · gp | |gs | .

pK K

(7.180)

Inserting into the kinetic equation we get the relation

gp (E, R, T ) = ’l ∇R gs (E, R, T )

K K

(7.181)

and using the expressions for the current and density, Eq. (7.108) and Eq. (7.109),

we obtain their relationship

j(R, T ) = ’D0 ∇R ρ(R, T ) + σ0 E(R, T ) , (7.182)

where we have used the Einstein relation, σ0 = 2e2 N0 D0 , relating conductivity and

the di¬usion constant.

In the absence of the electric ¬eld, the kinetic equation becomes the di¬usion

equation for the s-wave component

(‚T ’ D0 ∇2 ) gs (E, R, T ) = 0 .

K

(7.183)

R

Exercise 7.9. Show that by introducing the distribution function

1K

ˆ

h(p, R, T ) = g (E = ξp + e•(R, T ), p, R, T ) (7.184)

2

the kinetic equation assumes the standard Boltzmann form, Eq. (7.55).

7.6 Beyond the quasi-classical approximation

The importance of the quasi-classical description is the very weak restrictions for its

applicability. However, it has two severe limitations. It relies on the assumption of

particle“hole symmetry and is thus unable to treat thermo-electric e¬ects, and since

212 7. Quantum kinetic equations

momenta are tied to the Fermi surface the e¬ect of the Lorentz force is lost and the

quasi-classical Green™s function technique is unable to describe magneto-transport.

In this section we shall show how these restrictions can be avoided following previous

works of Langreth [27] and Altshuler [28]. As an example, in Section 7.6.1 we consider

thermo-electric e¬ects in a magnetic ¬eld, the Nernst“Ettingshausen e¬ect.

A distribution function is introduced according to

GK = GR — h ’ h — GA , (7.185)

which upon insertion into the quantum kinetic equation, Eq. (7.3), and by use of

the equations of motion for the retarded and advanced Green™s functions, and the

property that the composition — is associative leads to the equation

GR — B ’ B — GA = 0 , (7.186)

where

1—

B[h] = [G’1 ’ eΣ — h]’ + [“ , h]+ ’ i ΣK .

, (7.187)

0

2

In the gradient approximation we then have

(GR ’ GA )B + [B, eG]p = 0 . (7.188)

Inserting the solution of the equation

(GR ’ GA )B = 0 (7.189)

into Eq. (7.188), we observe that the second term on the left in Eq. (7.187) has

the form of a double Poisson bracket and thus should be dropped in the gradient

approximation. The quantum kinetic equation therefore takes the form

B[h] = 0 (7.190)

and expressions in Eq. (7.187) should be evaluated in the gradient approximation.

Since the introduced distribution function is not gauge invariant, we shall not

succeed in obtaining an appropriate kinetic equation with the usual expression for

the Lorentz force unless the kinetic momentum is introduced instead of the canonical

one.27 Performing a gradient expansion of the term in Eq. (7.187) containing G’1 ,

0

we obtain in the mixed or Wigner coordinates

’i[G’1 — h]p = [E ’ e• ’ ξp’eA , h]p ,

, (7.191)

0

where (•, A) are the potentials describing the electromagnetic ¬eld.

˜

Within the gradient approximation a gauge-invariant distribution function h can

thus be introduced

˜

h(©, P, R, T ) = h(E, p, R, T ) (7.192)

de¬ned by the change of variables

P = p ’ eA(R, T ) , © = E ’ e•(R, T ) . (7.193)

27 Describingthe kinetics in the momentum representation assumes that we are not in the quantum

limit where Landau level quantization is of importance, ωc kT .

7.6. Beyond the quasi-classical approximation 213

We observe the identity (now indicating the variables involved in the Poisson brackets

by subscripts)

˜˜ ˜ ˜ ˜ ˜ ˜ ˜

[A, B]p,E = [A, B]P,© + eE · (‚© A∇P B ’ ‚© B∇P A) + eB · (∇P A — ∇P B) ,

(7.194)

where E = ’∇• ’ ‚T A and B = ∇ — A are the electric and magnetic ¬elds, respec-

˜ ˜

tively, and A and B are related to A and B by equations analogous to Eq. (7.192).

Using this identity, the following driving terms then appear in the gradient approxi-

mation

’i[G’1 ’ eΣ, h]p,E = [© ’ ξP ’ eΣ, h]P,©

˜

0

eE · ((1 ’ ‚© eΣ)∇P h) + v— ‚© h) + ev— — B · ∇P h ,

˜ ˜ ˜

˜

+ (7.195)

where we have introduced28

v— = ∇P (ξP + eΣ(©, P, R, T )) .

˜ (7.196)

As a result of the transformation Eq. (7.193), the kinematic and not the canonical

momentum enters the kinetic equation, and a gauge invariant kinetic equation is

obtained as desired.

We could equally well have obtained the kinetic equation on gauge invariant form

by choosing to introduce the mixed representation according to

G(X, p) ≡ dxe’ir·(p+eA(X))+it(E+e•(X)) G(X, x) (7.197)

whereupon, in accordance with Eq. (7.194), the Poisson bracket can be expressed as

= ‚E A {‚T + u · ∇R + (eE · u ’ (‚E A)’1 ‚T A)‚E

[A, B]p,E

+ (eE + ev — B + (‚E A)’1 ∇R A) · ∇p } B (7.198)

with

u = (‚E A)’1 ∇p A . (7.199)

The kinetic equation thus takes the form

+ ‚T eΣ ‚E + v— · (∇R + eE ‚E )

{(1 ’ ‚E eΣ)‚T

+ (eE + ev— — B) · ∇p } h = I[h] (7.200)

where the collision integral is given by

I[h] = iΣK ’ “ h . (7.201)

28 As long as inter-band transitions can be neglected, band structure e¬ects can be included as

shown in reference [3].

214 7. Quantum kinetic equations

Exercise 7.10. Consider the case of an instantaneous two-particle interaction be-

tween fermions, V (x), such as Coulomb interaction between electrons,

U R (x, t, x , t ) = V (x ’ x ) δ(t ’ t ) = U A (x, t, x , t ) (7.202)

and U K (x, t, x , t ) = 0. The Hartree“Fock self-energy skeleton diagrams, the dia-

grams in Figure 5.4, do not contribute to the collision integral owing to the instan-

taneous character of the interaction. The lowest-order self-energy skeleton diagrams

contributing to the collision integral are thus speci¬ed by the third and fourth dia-

grams in Figure 5.5.

Show that the corresponding electron“electron collision integral becomes

Ie’e [f ] = ’2π dp1 dp2 dp3 (U R (p ’ p))2 δ(p + p2 ’ p1 ’ p3 )

— δ(ξp + ξp2 ’ ξp1 ’ ξp3 )

— (fp fp2 (1 ’ fp1 )(1 ’ fp3 ) ’ (1 ’ fp ) (1 ’ fp2 )fp1 fp3 )) , (7.203)

where fp is the electron distribution function which in equilibrium reduces to the

Fermi function. If one uses the the real-time formulation in terms of the Green™s

functions GRAK , the canceling terms fp fp1 fp2 fp3 do not appear explicitly but have

to be added and subtracted.

Show that the decay of a momentum or energy state for the above collision integral

is given by the following energy relaxation rate

1

’2π dp1 dp2 dp3 (U R (p ’ p)2 δ(p + p2 ’ p1 ’ p3 )

=

„e’e (p)

— δ(ξp1 + ξp+p2 ’p1 ’ ξp ’ ξp2 )

— (fp2 (1 ’ fp1 )(1 ’ fp3 ) + fp3 (1 ’ fp2 ) fp ) , (7.204)

where the short notation has been introduced for the Fermi function, fp = f0 (ξp ).

Assume that the interaction is due to screened Coulomb interaction

2

e2

2

V (p)

(U R (p))2 = 0

= , (7.205)

’2 p2 + κ2

(p) s

where κ2 = 2N0 e2 / 0 is the screening wave vector.

s

Show that the electron“electron collision rate for an electron on the Fermi surface

has the temperature dependence

§

π 2 e2

⎪ 32 0 v2 κs 3 (kT )2 κs kF

⎨

1 F

= (7.206)

⎪

„e’e (T ) © 3 (kT )2

π

κs kF .

16 F

7.6. Beyond the quasi-classical approximation 215

The life time is seen to be determined by the phase-space restriction owing to Pauli™s

exclusion principle. The long lifetime of excitations near the Fermi surface due to

the exclusion principle is the basis of Landau™s phenomenological Fermi liquid theory

of strongly interacting degenerate fermions, and its microscopic Green™s function

foundation.

7.6.1 Thermo-electrics and magneto-transport

As an example of electron“phonon renormalization of a static transport coe¬cient, we

consider the Nernst“Ettingshausen e¬ect, viz. the high-¬eld Nernst“Ettingshausen

coe¬cient, which relates the current density to the vector product of the temperature

gradient and the magnetic ¬eld. For now, we shall neglect any momentum depen-

dence of the self-energy. The system is driven out of equilibrium by a temperature

gradient. The magnetic ¬eld is assumed to satisfy the condition

γ ωc , (7.207)

where ωc = |e|B/m is the Larmor or cyclotron frequency and γ is the collision rate.

The collision integral can then be neglected, and the kinetic equation reduces to

(v · ∇R + e(v — B) · ∇p ) h = 0 . (7.208)

In the gradient approximation, the electric current density is according to Eq. (7.16)

∞

dp

j(R, T ) = ’e dE v (Ah ’ [ eG, h]pE ) . (7.209)

(2π)3 ’∞

According to Eq. (7.207) and Eq. (7.208), the last term vanishes since

’∇p eG · ∇R h + eB · (∇p eG — ∇p h)

[ eG, h]pE =

‚ eG

’ (v · ∇R h + e(v — B) · ∇p h) = 0 .

= (7.210)

‚ξ

Inserting the solution of Eq. (7.208)

|∇T | ‚h0

h = h0 ’ py E (7.211)

eBT ‚E

into the current expression and performing a Sommerfeld expansion gives

(1 + »)S0 ‚ eΣ

∇T — B »=’

j= , (7.212)

B2 ‚E

E=0,p=pF

where S0 is the free electron entropy which in a degenerate electron gas is identical

to the speci¬c heat. In the jellium model one has » = g 2 N0 . Thus the enhancement

of the high-¬eld thermo-electric current is seen to be identical to the enhancement

of the equilibrium speci¬c heat.

216 7. Quantum kinetic equations

Taking into account a possible momentum dependence of the self-energy leads

to non-equilibrium contributions to the spectral weight function which, however, are

di¬cult to calculate. A calculation within the context of Landau“Boltzmann Fermi-

liquid theory leads to the appearance of two ∇p eΣ-dependent terms that exactly

cancel each other, thus suggesting the above result to be generally valid [9].

Thermopower measurements agree with the calculated mass enhancement accord-

ing to Eq. (7.212), see references [29, 30].

7.7 Summary

In this chapter the quantum kinetic equation approach to transport using the real-

time approach has been considered. The examples studied were condensed matter

systems, but the approach is useful in application to many physical systems, say in

nuclear physics in connection with nuclear reactions and heavy ion collisions, as dis-

cussed for example in reference [31]. We have also realized the di¬culties involved in

describing general non-equilibrium states. Since no universality of much help is avail-

able in guiding approximations, cases must be dealt with on an individual basis. Here

the use of the skeleton diagrammatic representation of the self-energy, just as for equi-

librium states, can be a powerful tool to assess controlled approximations in nontrivial

expansion parameters as we demonstrated for the case of electron“phonon interac-

tion. This allowed establishing, for example, that the classical Landau“Boltzmann

equation has a much wider range of applicability than to be expected a priori. The

general problem is the vast amount of information encoded in the one-particle Green™s

functions, truncated objects with boundless information of correlations expressed by

higher-order Green™s functions. It is therefore necessary to eliminate the informa-

tion in the equations of motion which do not in¬‚uence the studied properties, to

get rid of any excess information. The quasi-classical Green™s function technique

being such a successful scheme when it comes to understand the transport prop-

erties of metals, except for e¬ects depending on particle“hole asymmetry such as

thermo-electric e¬ects. The quasi-classical Green™s function technique allowed ana-

lytical calculation of mass renormalization e¬ects typical of interactions in quantum

systems, and are in general susceptible to numerical treatment.29 The quasi-classical

Green™s function technique is the basic tool for studying non-equilibrium properties

of the low-temperature superconducting state, a topic we turn to in the next chapter.

In fact, the quasi-classical Green™s function technique is a corner stone for describing

many quantum phenomena in condensed matter, being the systematic starting point

for treating quantum corrections to classical kinetics, and we shall exploit this to our

advantage when discussing the weak localization e¬ect in Chapter 11.

29 Despite brave e¬orts, little progress has, to my knowledge, been made using numerics to extend

solutions of the general quantum kinetic equation to include higher than second-order correlations.

This ¬eld will undoubtedly be studied in the future using numerics.

8

Non-equilibrium

superconductivity

Superconductivity was discovered in 1911 by H. Kamerlingh Onnes. Having suc-

ceeded in liquefying helium, transition temperature 4.2 K, this achievement in cryo-

genic technology was used to cool mercury to the man-made temperature that at

that time was closest to absolute zero. He reported the observation that mercury at

4.2 K abruptly entered a new state of matter where the electrical resistance becomes

vanishingly small. This extraordinary phenomenon, coined superconductivity, eluted

a microscopic understanding until the theory of Bardeen, Cooper and Schrie¬er in

1957 (BCS-theory).1 The mechanism responsible for the phase transition from the

normal state to the superconducting state at a certain critical temperature is that

an e¬ective attractive interaction between electrons makes the normal ground state

unstable. As far as conventional or low-temperature superconductors are concerned,

the attraction between electrons follows from the form of the phonon propagator,

Eq. (5.45), viz. that the electron“phonon interaction is attractive for frequencies less

than the Debye frequency, and in fact can overpower the screened Coulomb repulsion

between electrons, leading to an e¬ective attractive interaction between electrons.2

The original BCS-theory was based on a bold ingenious guess of an approximate

ground state wave function and its low-energy excitations describing the essentials

of the superconducting state. Later the diagrammatic Green™s function technique

was shown to be useful to describe more generally the properties of superconduc-

tors, such as under conditions of spatially varying magnetic ¬elds and especially for

general non-equilibrium conditions.

In terms of Green™s functions and the diagrammatic technique, the transition

from the normal state to the superconducting state shows up as a singularity in the

1 For an important review of the attempts to understand the phenomena of superconductivity

and its truly de¬ning state characteristic, the Meissner-e¬ect, i.e. the expulsion of a magnetic ¬eld

from a piece of material in the superconducting state, we refer the reader to the article by Bardeen

[32], written on the brink of the monumental discovery of the theoretical understanding of the new

state of matter discovered almost half a century earlier.

2 In high-temperature superconductors, the attractive interaction is not caused by the ionic back-

ground ¬‚uctuations but by spin ¬‚uctuations.

217

218 8. Non-equilibrium superconductivity

e¬ective interaction vertex. The e¬ect of a particular class of scattering processes

in the normal state drives the singularity. In diagrammatic terms certain vertex

corrections, capturing the e¬ect of the particular scattering process, corresponding

to re-summation of an in¬nite class of diagrams, become singular. In the case of

superconductivity, the particle“particle ladder self-energy vertex corrections, a typ-

ical member of which is depicted in Figure 8.1, where the wiggly line represents

the e¬ective attractive electron-electron interactions (in the simplest model simply

the electron“phonon interaction) becomes divergent in the normal state, signalling a

phase transition at a critical temperature Tc .

Figure 8.1 Cooper instability diagram.

Although the set of diagrams according to Migdal™s theorem by diagrammatic

estimation is formally of the order of ωD / F , where ωD is the Debye energy, which is

typically two orders of magnitude smaller than the Fermi energy, the particle“particle

ladder sums up a geometric series to produce a denominator which by vanishing

produces a singularity.3 In the simplest, longitudinal-only electron“phonon model,

the critical temperature is given by (see Exercise 8.3 on page 221)

ωD e’1/» ,

kTc (8.1)

where » = N0 g 2 is the dimensionless electron“phonon coupling constant in the jellium

model (recall Section 7.5.1). We note that the critical temperature is non-analytic

in the coupling constant, precisely such non-perturbative e¬ects are captured by

re-summation of an in¬nite class of diagrams. The singularity signals a transition

between two states, leading at zero temperature to a ground state that is very di¬erent

from the normal ground state, and in general at temperatures below the critical one

to properties astoundingly di¬erent from those of the normal state.

The signifying feature of the superconducting state is, as stressed by Yang [33],

that it possesses o¬-diagonal long-range order, i.e. for pair-wise far away separated

3 The story goes that Landau delayed the publication of Migdal™s result for several years, be-

cause it is in blatant contradiction to the existence of superconductivity (mediated by phonons).

Nowadays we are familiar with the status of diagrammatic estimates such as Migdal™s theorem (as

discussed in Section 7.5.1). They are not immune to the existence of singularities in certain in¬nite

re-summations of a particular set of diagrams. The situation is formally quite analogous to the