ńņš. 9 |

where N0 is the density of states at the Fermi energy, and the current density is given

by

1 dĖ

p

j(R, T ) = ā’ eN0 dE vF g K (E, p, R, T ) .

Ė (7.109)

2 4Ļ

18 Here and in the following, we assume for simplicity a spherical Fermi surface. For a general

Fermi surface one decomposes according to

dĪ¾p dsF

dp Ė

= ,

3 vF (2Ļ)3

(2Ļ)

where dĪ¾p /vF is the length of the momentum increment measured from the Fermi surface in the

Ė

directions Ā±Ė , and dsF is the corresponding Fermi surface area element.

p

200 7. Quantum kinetic equations

The Ī¾-integration does not respect the proper order of integrations, momentum in-

tegration being last because of the convergence property of the Greenā™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 Greenā™s function.

7.5.1 Electronā“phonon interaction

Here we apply the quasi-classical technique to the case of strong electronā“phonon

interaction, thereby obtaining the kinetic equation for the electrons that includes the

renormalization eļ¬ects.

Let us ļ¬rst make sure that the electronā“phonon self-energy is susceptible to Ī¾-

integration, i.e. it can be expressed solely in terms of the Ī¾-integrated Greenā™s func-

tion or quasi-classical Greenā™s function. Migdalā™s theorem states that the electronā“

phonon self-energy diagrams for the electron Greenā™s function where phonon lines

cross are small in the parameter ĻD / F , where ĻD 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 Electronā“phonon self-energy.

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

F

(eā’ph)

(1, 1 ) = ig 2 Ī³ii Gi j (1, 1 ) Dkk (1, 1 ) Ī³j j .

k

Ėk

Ī£ij (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 Migdalā™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 electronā“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 Greenā™s function technique 201

the arguments on the left (R, p, t1 , t1 ))21

dp

(eā’ph)

Gi j (R, p , t1 , t1 ) Dkk (R, p ā’ p , t1 , t1 ) Ī³j j . (7.111)

= ig 2 Ī³ii

k

Ėk

Ī£ij

(2Ļ)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 dĖ dĖ

p p

= dĪ¾ N (Ī¾) = N0 dĪ¾ (7.112)

(2Ļ)3 4Ļ 4Ļ

and the last equality is valid when particleā“hole symmetry applies.22 Using the fact

that the Debye energy is small compared with the Fermi energy,23 the various electron

Greenā™s function are tied to the Fermi surface, and we obtain the electronā“phonon

matrix self-energy expressed in terms of the quasi-classical matrix electron Greenā™s

function

Ī»k

(eā’ph)

(R, p, t1 , t1 ) = Ī³ii dĖ gi j (R, p , t1 , t1 ) Dkk (R, pF (Ė ā’ p ), t1 , t1 )Ėj j ,

Ī³k

Ė Ė pĖ

Ļij p

4

(7.113)

where Ī» = g 2 N0 is the dimensionless electronā“phonon coupling constant. The matrix

components of the matrix self-energy are therefore

Ī»

R(A)

g K (R, p , t1 , t1 ) DR(A) (R, pF (Ė ā’ p ), t1 , t1 )

Ė Ė pĖ

Ļeā’ph (R, p, t1 , t1 ) = dĖ

p

8

g R(A) (R, p , t1 , t1 ) DK (R, pF (Ė ā’ p ), t1 , t1 )

Ė pĖ

+ (7.114)

and

Ī»

dĖ (g R (R, p , t1 , t1 ) DR (R, pF (Ė ā’ p ), t1 , t1 )

K

Ė Ė pĖ

Ļeā’ph (R, p, t1 , t1 ) = p

8

+ g A (R, p , t1 , t1 ) DA (R, pF (Ė ā’ p ), t1 , t1 )

Ė pĖ

+ g K (R, p , t1 , t1 ) DK (R, pF (Ė ā’ p ), t1 , t1 ))

Ė pĖ (7.115)

or equivalently

Ī»

dĖ (g R ā’ g A )(DR ā’ DA ) + g K DK

K

Ļeā’ph = (7.116)

p

8

21 Inthe case of impurity scattering, the self-energy is expressed in terms of the quasi-classical

Greenā™s function according to

p

i dĖ

Ļimp (E, R, T ) = ā’ g(Ė , E, R, T ) ,

p

2Ļ„ 4Ļ

where the high-energy cut-oļ¬ 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 eļ¬ects

due to particleā“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 Ī¾-

variable, the momentum dependence of the self-energy can be neglected, and the

leftā“right subtracted Dyson equations, Eq. (7.1), can be integrated with respect to

Ī¾, giving the quantum kinetic equation

ā’1

g0 + i eĻ ā—¦ g K = 2iĻ K ā’ i (Ļ R ā’ Ļ A ) ā—¦ g K

, , , (7.118)

ā’ +

where

ā’1 ā’1

g0 (R, p, t1 , t1 ) = g0 (R, p, t1 ) Ī“(t1 ā’ t1 )

Ė Ė (7.119)

and

e2 2

ā’1

= ā‚t1 + vF Ā· (āx1 ā’ ieA(R, t1 )) + ieĻ(R, t1 ) ā’

Ė

g0 (R, p, t1 ) A (R, t1 ). (7.120)

2m

Here vF = pF /m, the Fermi velocity, speciļ¬es the Fermi surface direction, and ā—¦

implies matrix multiplication in the time variable. We have considered the case

where, say, the electrons in a metal are subject to electromagnetic ļ¬elds.

From the spectral representation

ā

dE A(E , p, R, T )

R(A)

G (E, p, R, T ) = (7.121)

ā’ā 2Ļ E ā’ E (ā’) i0

+

it follows that Ī¾-integrating eG gives a state independent constant and the last

term on the left-hand side in Eq. (7.3) vanishes upon Ī¾-integration.

ā’1

The form of g0 follows from the following observation where for deļ¬niteness we

focus on the scalar potential term. First transform to the mixed spatial coordinates

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

Since the Greenā™s function G(R, r) is a wildly oscillating function in the relative

ā’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

Ļ•(x1 , , t1 ) G(x1 , t1 , x1 , t1 ) Ļ•(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(RĀ·P+rĀ·p) Ļ•(k) G(P ā’ k, p ā’ k/2) , (7.124)

Ļ•(R + r/2) G(R, r) =

where the shifts of variables, P + k ā’ P and P + k/2 ā’ 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 Greenā™s function technique 203

Fermi wave vector, k kF , so that G(Pā’k, pā’k/2) G(Pā’k, p), again leading to

the stated result, Eq. (7.123). In the quasi-classical approximation the eļ¬ect of the

Lorentz force is lost, and for a perturbing electric ļ¬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 eļ¬ects of the Lorentz force in the normal state in Section 7.6. However, we

note that the inļ¬‚uence on the phase of the Greenā™s function is fully incorpotated in

the quasi-classical approximation, a fact we shall exploit when considering the weak

localization eļ¬ect in Chapter 11.

A simpliļ¬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 Ī¾-integrated retarded and advanced Greenā™s

functions

Ī“(t1 ā’ t1 ) ,

g R(A) (R, p, t1 , t1 ) = +

g R(A) (R, p, E, T ) = +

Ė Ė 1 (7.125)

(ā’) (ā’)

and they thus contain no information since particleā“hole asymmetry eļ¬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 , Ļ . (7.126)

F

These criteria are well satisļ¬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 diļ¬erent approximation schemes are

immediately available: one consists of linearization with respect to a perturbation

such as an electric ļ¬eld, allowing frequencies restricted only by the Fermi energy,

Ļ < F , to be considered, but of course restricted to weak ļ¬elds. The other assumes

perturbations to be suļ¬ciently slowly varying in time that a lowest-order expansion

in the time derivative is valid

[A ā—¦ B]ā’ ā‚E A ā‚T B ā’ ā‚T A ā‚E B .

A B A B

, (7.127)

For deļ¬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 ā’ ā‚e eĻ)ā‚T + ā‚T eĻ ā‚E + vF Ā· āR + eā‚T Ļ• ā‚E ) g K = Ieā’ph , (7.128)

where the collision integral is

Ieā’ph = 2iĻ K ā’ Ī³g K , (7.129)

204 7. Quantum kinetic equations

where

Ī³ = i(Ļ R ā’ Ļ 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 ā’E

dĖ

p

Ī³(E, p) = ā’Ļ dE Ī¼(pF (Ė ā’ p ), E ā’ E ) coth ā’ h(E , p )

Ė pĖ Ė

4Ļ 2T

(7.131)

and

E ā’E

dĖ

p

iĻeā’ph (E, p) = ā’Ļ dE Ī¼(pF (Ė ā’ p ), E ā’ E ) h(E , p ) coth ā’1 ,

K

Ė pĖ Ė

4Ļ 2T

(7.132)

where we have introduced the distribution function

1K

Ė Ė

h(E, p, R, T ) = g (E, p, R, T ) (7.133)

2

and

iN0 |gq |2

(DR (E, q) ā’ DA (E, q))

Ī¼(q, E) = (7.134)

2Ļ

is the Eliashberg spectral weight function. Here we have allowed for a more gen-

eral longitudinal electronā“phonon coupling than the jellium model. The coupling is

denoted gq , corresponding to momentum transfer q. The connection to the jellium

model is |gq | = g Ļq /2, where Ļq = c q is the energy of a phonon with momentum

q, c being the sound velocity.

We have further assumed that the phonons are in thermal equilibrium at temper-

ature T ,24 and have therefore used the ļ¬‚uctuationā“dissipation relation for bosons,

Eq. (5.103),

E

DR (E, p) ā’ DA (E, p)

DK (E, p) = coth . (7.135)

2kT

We note that the variables in the distribution function are quite diļ¬erent from

that of the classical Boltzmann equation for electronā“phonon interaction, which is

the Wigner coordinates (p, R, T ). Here an energy variable and a position on the

Fermi surface appear separately (besides space and time). This feature of the quasi-

classical equation reļ¬‚ects the fact that we do not rely on a deļ¬nite relation between

24 This is not necessary, but would otherwise lead to the requirement of considering the kinetic

equation for the phonons also. For typical transport situations in a metal, the approximation, viz.

considering the phonons a heat reservoir, is applicable.

7.5. Quasi-classical Greenā™s function technique 205

the energy and momentum variables as is the case in the quasi-particle approximation

of Section 7.3.2.

Introducing the Fermi and Bose type distribution functions

1

1 ā’ h(E, p, R, T )

Ė Ė

f (E, p, R, T ) = (7.136)

2

and

1 E

n(E) = ā’ 1 ā’ coth (7.137)

2 2kT

the collision integral takes the more familiar form

dĖ

p

dE Ī¼(pF (Ė ā’ p ), E ā’ E ) RE pĖ ,

EĖ

Ieā’ph = ā’2Ļ pĖ (7.138)

p

4Ļ

where

RE pĖ

EĖ

(1 + n(E ā’ E ))f (E, p)(1 ā’ f (E , p ))

Ė Ė

=

p

ā’ n(E ā’ E )(1 ā’ f (E, p))f (E , p ) .

Ė Ė (7.139)

Finally, introducing a gauge invariant distribution function by the substitution

f (E) ā’ f (E ā’ Ļ•(R, T )) (7.140)

we obtain the quantum kinetic equation

((1 ā’ ā‚E eĻ)ā‚T + ā‚T eĻ ā‚E + vF Ā· (āR + eE(R, T )) ā‚E ) f = Ieā’ph [f ] ,

(7.141)

where E(R, T ) = ā’āR Ļ•(R, T ) is the perturbing electric ļ¬eld. We note that the

self-energy terms on the right-hand side in the kinetic equation describe collision

processes, and we now turn to show that the self-energies on the left describe renor-

malization eļ¬ects, in particular mass renormalization due to the electronā“phonon

interaction.

From the kinetic equation, Eq. (7.141), Prange and Kadanoļ¬ [23] drew the con-

clusion that many-body eļ¬ects can be seen only in time-dependent transport prop-

erties and that static transport coeļ¬cients, such as d.c. conductivity and thermal

conductivity are correctly given by the usual Boltzmann results. However, there is a

restriction to the generality of this statement, viz. that in deriving the quasi-classical

equation of motion particleā“hole symmetry was assumed. Within the quasi-classical

scheme, all thermoelectric coeļ¬cients therefore vanish, and no conclusion can be

drawn about many-body eļ¬ects on the thermo-electric properties. In Section 7.6.1,

we shall by not employing the quasi-classical scheme consider how thermo-electric

properties do get renormalized by the electronā“phonon interaction.

206 7. Quantum kinetic equations

7.5.2 Renormalization of the a.c. conductivity

As an example of electronā“phonon renormalization of time-dependent transport co-

eļ¬cients we shall consider the a.c. conductivity in the frequency range ĻĻ„eā’ph 1,

where 1/Ļ„eā’ph is the clean-limit electronā“phonon scattering rate for an electron on

the Fermi surface

ā§ 7ĻĪ¶(3) (kT )3

āŖ Ī» (pF c)2 kT 2pF c

āŖ

āØ 2

1

= (7.142)

āŖ

Ļ„ ( F, T ) āŖ

ā©

2ĻĪ» kT kT 2pF c

Ī¶ being Riemannā™s zeta function.25 For deļ¬niteness we also consider the temperature

to be low compared to the Debye temperature ĪøD . In this high-frequency limit the

collision integral can be neglected and the linearized kinetic equation takes the simple

form

(1 ā’ ā‚E eĻ)ā‚T h + ā‚T eĻ ā‚E h0 + evF Ā· E(T ) ā‚E h0 = 0 (7.143)

for a spatially homogeneous electric ļ¬eld. Except for a real constant, just renormal-

izing the chemical potential, we have according to the Feynman rules

1 dĖ

p

dE |gpF ā’pF |2 h(E , p , R, T ) eD(pF ā’ pF , E ā’ E ) , (7.144)

Ė

eĻ = N0

2 4Ļ

where gpF ā’pF denotes the electronā“phonon coupling, and

1R

(D + DA ) .

eD = (7.145)

2

For an applied monochromatic ļ¬eld, E(t) = E0 exp{ā’iĻt}, the solution can be

sought in the form

h1 = aeE Ā· vF ā‚E h0 , (7.146)

where the constant a remains to be determined. Inserting Eq. (7.146) into the kinetic

equation we obtain

1

a= , (7.147)

ā’iĻ(1 + Ī»ā— )

where

|gpF ā’pF |2

dĖ

p

Ī»ā— = 2N0 (1 ā’ p Ā· p ) .

ĖĖ

dE (7.148)

4Ļ ĻpF ā’pF

The current can now be evaluated and we obtain for the frequency dependence

of the conductivity

ne2

Ļ(Ļ) = (7.149)

ā’iĻmopt

25 For

a calculation of the collision rate see Exercise 8.8 on page 237 and Section 11.3.1, and, for

example, chapter 10 of reference [1].

7.5. Quasi-classical Greenā™s function technique 207

where the optical mass is renormalized according to

mopt = m(1 + Ī»ā— ) (7.150)

a result originally obtained by Holstein using a diļ¬erent approach, viz. linear response

theory [26]. We note that it is the non-equilibrium electron contribution to the real

part of the self-energy that makes the optical mass renormalization diļ¬erent from

the speciļ¬c heat mass renormalization, m ā’ (1 + Ī»)m (see also the result of Exercise

8.8 on page 237).

As a consequence of electronā“phonon interaction, the physically observed mass of

the electron is not the mass or band structure eļ¬ective mass of the electron, but it has

been changed owing to the interaction.26 Furthermore, we note that the magnitude

of the mass renormalization depends on how the system is probed, the optical mass

being diļ¬erent from the speciļ¬c heat mass.

7.5.3 Excitation representation

The quasi-classical theory leads to equations which are more general than the Boltz-

mann equation, and the kinetic equation looks quite diļ¬erent. We have shown that

the basic variables, besides space and time, are the energy variable and the momen-

tum position on the Fermi surface. Although the electronā“phonon interaction does

not permit the quasi-particle approximation a priori, we recapitulate the deriva-

tion of reference [23] showing that it is still possible to cast the electronā“phonon

transport theory into the standard Landauā“Boltzmann form. We start by deļ¬ning a

quasi-particle energy Ep , which is deļ¬ned implicitly by (we suppress the space-time

variables and use the short notation Ep = E(p, R, T ))

Ė

Ep = Ī¾p + eĻ(Ep + eĻ•(R, T ), p, R, T ) (7.151)

thereby satisfying the equations

āp Ep = Zp āp Ī¾p (7.152)

and

āR Ep = Zp (eāR Ļ• ā‚E eĻ + āR eĻ) (7.153)

E=Ep +eĻ•(R,T )

and

ā‚T Ep = Zp (eā‚T Ļ•ā‚E eĻ + ā‚T eĻ) (7.154)

E=Ep +eĻ•(R,T )

where in Eq. (7.151), assuming for simplicity a spherical Fermi surface, any angular

dependence of the real part of the self-energy has been neglected, and the so-called

26 Thus interaction causes renormalization of observable quantities. This point of view is the

rationale for avoiding the ubiquitous inļ¬nities occurring in quantum ļ¬eld theories such as QED,

and being taken to an extreme since there the unobservable bare mass (and the bare coupling

constant, the bare electron charge) is taken, it turns out, to be inļ¬nite in order to provide the ļ¬nite

and accurate predictions of QED by phenomenologically introducing the observed mass and charge.

208 7. Quantum kinetic equations

wave-function renormalization constant

Zp = (1 ā’ ā‚E eĻ)ā’1 (7.155)

E=Ep +eĻ•(R,T )

has been introduced.

The energy variable E in the kinetic equation is now set equal to Ep + eĻ• and

we introduce the distribution function (again suppressing the space-time variables)

Ė

np = f (E, p, R, T ) . (7.156)

E=Ep +eĻ•(R,T )

Using the relations

āp n = āp Ep (ā‚E f ) (7.157)

E=Ep +eĻ•(R,T )

and

āR n = (āR f + āR (Ep + eĻ•(R, T ))ā‚E f ) (7.158)

E=Ep +eĻ•(R,T )

and

ā‚T n = (ā‚T f + ā‚T (Ep + eĻ•(R, T ))ā‚E f ) (7.159)

E=Ep +eĻ•(R,T )

and Eqs. (7.152ā“7.154), we obtain the kinetic equation of the form

ā’1 Ė

Zp (ā‚T + āp Ep Ā· āR ā’ āR (Ep + Ļ•(R, T )) Ā· āp ) n(p, R, T ) = Ieā’ph (7.160)

with the electronā“phonon collision integral

2Ļ dp p

Ė

Ė

Ieā’ph = ā’ Zp Ī¼(p ā’ p ) Rp ,

Ė (7.161)

Ė

3

N0 (2Ļ)

where

p

Ė

Rp = (1 + N (Ep ā’ Ep ))np (1 ā’ np ) ā’ N (Ep ā’ Ep )(1 ā’ np )np (7.162)

Ė

and

iN0 |gpā’p |2 R

Ī¼(p ā’ p ) = (D (p ā’ p , Ep ā’ Ep ) ā’ DA (p ā’ p , Ep ā’ Ep )) . (7.163)

Ė

2Ļ

In transforming the collision integral we have utilized the substitution

dĖ dĖ dEp dp

p p

dE ā’ N0 ā’

N0 dĪ¾p Zp . (7.164)

(2Ļ)3

4Ļ 4Ļ dĪ¾p

Since the sound velocity is much smaller than the Fermi velocity, the phonon

damping is negligible, and the phonon spectral weight function has delta function

character

Ī¼(p ā’ p ) = N0 |gpā’p |2 (Ī“(Ep ā’ Ep ā’ Ļpā’p ) ā’ Ī“(Ep ā’ Ep + Ļpā’p )) . (7.165)

Ė

7.5. Quasi-classical Greenā™s function technique 209

The kinetic equation can then be written in the ļ¬nal form

(ā‚T + āp Ep Ā· āR ā’ āR (Ep + Ļ•(R, T )) Ā· āp ) n(p, R, T ) = Ieā’ph , (7.166)

where the electronā“phonon collision integral is

dp p

Ė

Ieā’ph = ā’2Ļ Zp Zp |gpā’p |2 Rp (Ī“(Ep ā’ Ep ā’ Ļpā’p ).

Ė

3

(2Ļ)

ā’ Ī“(Ep ā’ Ep + Ļpā’p )) . (7.167)

This has the form of the familiar Landauā“Boltzmann equation, except for the fact

that the transition matrix elements are renormalized.

We stress that only the quasi-classical approximation was used to derive the above

kinetic equation. In particular, we have not assumed any relation between the life-

time of a electron in a momentum state at the Fermi surface and the temperature.

This would have been necessary for invoking a quasi-particle description in order

to justify the existence of long-lived electronic momentum states. It has thus been

established from microscopic principles that the validity of the Landauā“Boltzmann

description of the electronā“phonon system is determined not by the Peierls criterion

(stating the upper bound is not the Fermi energy but the temperature), but by the

Landau criterion

F. (7.168)

Ļ„ ( F, T )

This is of importance for the validity of the Boltzmann description of transport in

semiconductors, for which the Peierls criterion would be detrimental.

7.5.4 Particle conservation

That an approximation for the quasi-classical Greenā™s function respects conservation

laws, say particle number conservation, is not in general as easily stated as for the

microscopic Greenā™s function. We therefore establish it here explicitly. The collision

integral, Eq. (7.167), has the invariant

dp

Ieā’ph = 0 , (7.169)

(2Ļ)3

which we shall see expresses the conservation of the number of particles, here the

electrons in question. Integrating the kinetic equation, Eq. (7.166), with respect to

momentum we obtain the continuity equation

ā‚T n + āR Ā· j = 0 , (7.170)

where

dp

n(R, T ) = 2 n(p, R, T ) (7.171)

(2Ļ)3

210 7. Quantum kinetic equations

and

dp

āp Ep n(p, R, T )

j(R, T ) = 2 (7.172)

(2Ļ)3

are the Landauā“Boltzmann expressions for the density and current density and the

factor of two accounts for the spin of the electron.

In order to establish that these are indeed the correctly identiļ¬ed densities (in

the excitation representation), we should connect one of them with the microscopic

expression. Assuming that |eĻ•| F , the microscopic expression for the density,

Eq. (7.108), is (suppressing space-time variables in quantities, here in Ļ•)

ā

dĖ

p

n(R, T ) = ā’2N0 dE f (E + eĻ•, p) . (7.173)

4Ļ ā’ā

In order to compare the density expression in the particle representation with the

excitation representation we transform Eq. (7.171) to the particle representation

ā

dp dĖ

p

dE (1 ā’ ā‚E eĻ) f (E, p).

Ė

n(R, T ) = 2 n(p, R, T ) = 2N0 (7.174)

(2Ļ)3 4Ļ ā’ā

Since Eq. (7.173) and Eq. (7.174) appear to be diļ¬erent, Eq. (7.172) is also trans-

formed to the particle representation

ā

dp dĖ

p

āp Ep n(p, R, T ) = 2N0 Ė

2 dE vF f (E, p) . (7.175)

(2Ļ)3 4Ļ ā’ā

Comparing the expression in Eq. (7.175) to that of Eq. (7.109), we observe that it

is identical to the quasi-classical current-density expression. The only possibility for

the above-mentioned apparent discrepancy not to lead to a violation of the continuity

equation is the existence of the identity

ā

Ė

ā‚T dĖ dE f (E, p) ā‚E eĻ = 0 (7.176)

p

ā’ā

which we now prove. Inserting the expression from Eq. (7.144) into the left side of

Eq. (7.176) we are led to consider

dĖ dE dĖ dE |gpF ā’pF |2 ( eD(pF ā’ pF , E ā’ E )

p p

ā‚T f (E, p) ā‚E f (E , p ) ā’ ā‚E f (E, p) ā‚T f (E , p )) = 0

Ė Ė Ė Ė (7.177)

Ė Ė

which by interchanging the variables E, p and E , p is seen to vanish, and the identity

Eq. (7.176) is thus established. We have thus established that the approximations

made do not violate particle conservation.

7.6. Beyond the quasi-classical approximation 211

7.5.5 Impurity scattering

For electrons interacting with impurities in a conductor, the self-energy is given by

the diagram in Eq. (7.51), F Ļ„ , and we can immediately implement the quasi-

classical approximation. The equation for the kinetic component of the quasi-classical

Greenā™s function in the presence of an electric ļ¬eld becomes

1 dĖ K

p

(ā‚T + vF Ā· āR + eā‚T Ļ• ā‚E ) g K = ā’ g K (E, p, R, T ) +

Ė Ė

g (E, p, R, T ) ,

Ļ„ 4Ļ

(7.178)

where for simplicity we have assumed that the momentum dependence of the impurity

potential can be neglected.

In the diļ¬usive limit the quasi-classical kinetic Greenā™s function will be almost

isotropic, and an expansion in spherical harmonics needs to keep only the s- and

p-wave parts

g K (E, p, R, T ) = gs (E, R, T ) + p Ā· gp (E, R, T )

K

ĖK

Ė (7.179)

and

|Ė Ā· gp | |gs | .

pK K

(7.180)

Inserting into the kinetic equation we get the relation

gp (E, R, T ) = ā’l āR 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 ) = ā’D0 āR Ļ(R, T ) + Ļ0 E(R, T ) , (7.182)

where we have used the Einstein relation, Ļ0 = 2e2 N0 D0 , relating conductivity and

the diļ¬usion constant.

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

equation for the s-wave component

(ā‚T ā’ D0 ā2 ) gs (E, R, T ) = 0 .

K

(7.183)

R

Exercise 7.9. Show that by introducing the distribution function

1K

Ė

h(p, R, T ) = g (E = Ī¾p + eĻ•(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

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

212 7. Quantum kinetic equations

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

quasi-classical Greenā™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 eļ¬ects in a magnetic ļ¬eld, the Nernstā“Ettingshausen eļ¬ect.

A distribution function is introduced according to

GK = GR ā— h ā’ h ā— 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 Greenā™s functions, and the

property that the composition ā— is associative leads to the equation

GR ā— B ā’ B ā— GA = 0 , (7.186)

where

1ā—

B[h] = [Gā’1 ā’ eĪ£ ā— h]ā’ + [Ī“ , h]+ ā’ i Ī£K .

, (7.187)

0

2

In the gradient approximation we then have

(GR ā’ GA )B + [B, eG]p = 0 . (7.188)

Inserting the solution of the equation

(GR ā’ 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 Gā’1 ,

0

we obtain in the mixed or Wigner coordinates

ā’i[Gā’1 ā— h]p = [E ā’ eĻ• ā’ Ī¾pā’eA , h]p ,

, (7.191)

0

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

Ė

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

thus be introduced

Ė

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

deļ¬ned by the change of variables

P = p ā’ eA(R, T ) , Ī© = E ā’ eĻ•(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, Ļc kT .

7.6. Beyond the quasi-classical approximation 213

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

by subscripts)

ĖĖ Ė Ė Ė Ė Ė Ė

[A, B]p,E = [A, B]P,Ī© + eE Ā· (ā‚Ī© AāP B ā’ ā‚Ī© BāP A) + eB Ā· (āP A Ć— āP B) ,

(7.194)

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

Ė Ė

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

ā’i[Gā’1 ā’ eĪ£, h]p,E = [Ī© ā’ Ī¾P ā’ eĪ£, h]P,Ī©

Ė

0

eE Ā· ((1 ā’ ā‚Ī© eĪ£)āP h) + vā— ā‚Ī© h) + evā— Ć— B Ā· āP h ,

Ė Ė Ė

Ė

+ (7.195)

where we have introduced28

vā— = āP (Ī¾P + eĪ£(Ī©, P, R, T )) .

Ė (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) ā” dxeā’irĀ·(p+eA(X))+it(E+eĻ•(X)) G(X, x) (7.197)

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

= ā‚E A {ā‚T + u Ā· āR + (eE Ā· u ā’ (ā‚E A)ā’1 ā‚T A)ā‚E

[A, B]p,E

+ (eE + ev Ć— B + (ā‚E A)ā’1 āR A) Ā· āp } B (7.198)

with

u = (ā‚E A)ā’1 āp A . (7.199)

The kinetic equation thus takes the form

+ ā‚T eĪ£ ā‚E + vā— Ā· (āR + eE ā‚E )

{(1 ā’ ā‚E eĪ£)ā‚T

+ (eE + evā— Ć— B) Ā· āp } h = I[h] (7.200)

where the collision integral is given by

I[h] = iĪ£K ā’ Ī“ h . (7.201)

28 As long as inter-band transitions can be neglected, band structure eļ¬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 ā’ x ) Ī“(t ā’ t ) = U A (x, t, x , t ) (7.202)

and U K (x, t, x , t ) = 0. The Hartreeā“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 speciļ¬ed by the third and fourth dia-

grams in Figure 5.5.

Show that the corresponding electronā“electron collision integral becomes

Ieā’e [f ] = ā’2Ļ dp1 dp2 dp3 (U R (p ā’ p))2 Ī“(p + p2 ā’ p1 ā’ p3 )

Ć— Ī“(Ī¾p + Ī¾p2 ā’ Ī¾p1 ā’ Ī¾p3 )

Ć— (fp fp2 (1 ā’ fp1 )(1 ā’ fp3 ) ā’ (1 ā’ fp ) (1 ā’ 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 Greenā™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

ā’2Ļ dp1 dp2 dp3 (U R (p ā’ p)2 Ī“(p + p2 ā’ p1 ā’ p3 )

=

Ļ„eā’e (p)

Ć— Ī“(Ī¾p1 + Ī¾p+p2 ā’p1 ā’ Ī¾p ā’ Ī¾p2 )

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

where the short notation has been introduced for the Fermi function, fp = f0 (Ī¾p ).

Assume that the interaction is due to screened Coulomb interaction

2

e2

2

V (p)

(U R (p))2 = 0

= , (7.205)

ā’2 p2 + Īŗ2

(p) s

where Īŗ2 = 2N0 e2 / 0 is the screening wave vector.

s

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

has the temperature dependence

ā§

Ļ 2 e2

āŖ 32 0 v2 Īŗs 3 (kT )2 Īŗs kF

āØ

1 F

= (7.206)

āŖ

Ļ„eā’e (T ) ā© 3 (kT )2

Ļ

Īŗs 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 Pauliā™s

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

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

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

foundation.

7.6.1 Thermo-electrics and magneto-transport

As an example of electronā“phonon renormalization of a static transport coeļ¬cient, we

consider the Nernstā“Ettingshausen eļ¬ect, viz. the high-ļ¬eld Nernstā“Ettingshausen

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

gradient and the magnetic ļ¬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 ļ¬eld is assumed to satisfy the condition

Ī³ Ļc , (7.207)

where Ļc = |e|B/m is the Larmor or cyclotron frequency and Ī³ is the collision rate.

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

(v Ā· āR + e(v Ć— B) Ā· āp ) h = 0 . (7.208)

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

ā

dp

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

(2Ļ)3 ā’ā

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

ā’āp eG Ā· āR h + eB Ā· (āp eG Ć— āp h)

[ eG, h]pE =

ā‚ eG

ā’ (v Ā· āR h + e(v Ć— B) Ā· āp h) = 0 .

= (7.210)

ā‚Ī¾

Inserting the solution of Eq. (7.208)

|āT | ā‚h0

h = h0 ā’ py E (7.211)

eBT ā‚E

into the current expression and performing a Sommerfeld expansion gives

(1 + Ī»)S0 ā‚ eĪ£

āT Ć— B Ī»=ā’

j= , (7.212)

B2 ā‚E

E=0,p=pF

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

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

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

of the equilibrium speciļ¬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

diļ¬cult to calculate. A calculation within the context of Landauā“Boltzmann Fermi-

liquid theory leads to the appearance of two āp eĪ£-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 diļ¬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 electronā“phonon interac-

tion. This allowed establishing, for example, that the classical Landauā“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 Greenā™s

functions, truncated objects with boundless information of correlations expressed by

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

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

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

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

erties of metals, except for eļ¬ects depending on particleā“hole asymmetry such as

thermo-electric eļ¬ects. The quasi-classical Greenā™s function technique allowed ana-

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

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

Greenā™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 Greenā™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 eļ¬ect in Chapter 11.

29 Despite brave eļ¬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 ļ¬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 Schrieļ¬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 eļ¬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 electronā“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 eļ¬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 Greenā™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 ļ¬elds and especially for

general non-equilibrium conditions.

In terms of Greenā™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 deļ¬ning state characteristic, the Meissner-eļ¬ect, i.e. the expulsion of a magnetic ļ¬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 ļ¬‚uctuations but by spin ļ¬‚uctuations.

217

218 8. Non-equilibrium superconductivity

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

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

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

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

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

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

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

the electronā“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 Migdalā™s theorem by diagrammatic

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

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

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

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

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

ĻD eā’1/Ī» ,

kTc (8.1)

where Ī» = N0 g 2 is the dimensionless electronā“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 eļ¬ects are captured by

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

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

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

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

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

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

3 The story goes that Landau delayed the publication of Migdalā™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 Migdalā™s theorem (as

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

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

singularity involved in Andersonā™s metalā“insulator transition. In revealing the physics in this case,

diagrammatic techniques are also useful, as we shall discuss in Chapter 11.

8.1. BCS-theory 219

spatial arguments, the two-particle correlation function is non-vanishing

ā

ā

lim ĻĪ± (x4 ) ĻĪ² (x3 ) ĻĪ³ (x2 ) ĻĪ“ (x1 ) = 0 , (8.2)

|x1 ,x2 ā’x3 ,x4 |ā’ā

i.e. when the spatial arguments x1 and x2 are chosen arbitrarily far away from the

spatial arguments x3 and x4 , the two-particle correlation function nevertheless stays

non-vanishing, contrary to the case of the normal state. An order parameter function,

Ī”Ī³Ī“ (x, x ), expressing this property, can therefore be introduced according to

ā

ĻĪ± (x4 ) ĻĪ² (x3 ) ĻĪ³ (x2 ) ĻĪ“ (x1 ) = Ī”ā— (x4 , x3 ) Ī”Ī³Ī“ (x1 , x2 )

ā

lim (8.3)

Ī±Ī²

|x1 ,x2 ā’x3 ,x4 |ā’ā

and we speak of BCS-pairing.

8.1 BCS-theory

In this section we consider the BCS-theory, but shall not go into any details of

BCS-ology since instead we shall use the Greenā™s function technique to describe

and calculate properties of the superconducting state.4 The part of the interaction

responsible for the instability is captured by keeping in the Hamiltonian only the

so-called pairing interaction. In a conventional and clean superconductor, pairing

takes place between momentum and spin states (p, ā‘) and (ā’p, ā“), each others time-

reversed states,5 and we encounter orbital s-wave and spin-singlet pairing and the

BCS-Hamiltonian becomes6

Vpp cā cā cā’p ā“ cp ā‘ ,

ā

Hpairing = p cpĻ cpĻ + (8.4)

pā‘ ā’pā“

p,Ļ pp

where the eļ¬ective attractive interaction Vpp is only non-vanishing for momentum

states in the tiny region around the Fermi surface set by the Debye energy, ĻD , for the

case where the attraction is caused by electronā“phonon interaction. The parameters

specifying the boldly guessed BCS-ground state7

(up + vp cā cā ) |0

|BCS = (8.5)

pā‘ ā’pā“

p

4 The properties of the BCS-state are described in numerous textbooks, e.g. reference [34].

5 In a disordered superconductor, pairing takes place between an exact impurity eigenstate and

its time reversed eigenstate.

6 Other types of pairing occur in Nature. In 3 He p-wave pairing occurs, and high-temperature

superconductors have d-wave pairing.

7 The BCS-ground state is seen to be a state that is not an eigenstate of the total number

operator, i.e. it does not describe a state with a deļ¬nite number of electrons (recall Exercise 1.7

on page 20). For massless bosons, such as photons, a number-violating state is not an unphysical

state, but for an assembly of fermions having a ļ¬nite chemical potential and interactions obeying

particle conservation it certainly is, and only the enormous explanatory power of the BCS-theory

makes it decent to use a formulation that violates the most sacred of conservation laws. In other

words, the superconducting state can also be described in terms that do not violate gauge invariance

such as when staying fully in the electronā“phonon model, but the BCS-theory correctly describes

the oļ¬-diagonal long-range order, and is a very eļ¬cient way for incorporating and calculating the

order parameter, characterizing the superconducting state, and its consequences. Quantum ļ¬eld

theory is therefore also convenient, but the superconducting state can be described without its use

and instead formulated in terms of the one- and two-particle density matrices.

220 8. Non-equilibrium superconductivity

are then obtained by the criterion of minimizing the average energy in the grand

canonical ensemble, i.e. the average value of BCS|Hpairing ā’ Ī¼N|BCS , the pairing

Hamiltonian with energies measured from the chemical potential, which at zero tem-

perature is the Fermi energy, Ī¾p = p ā’ F . This leads to a gap in the single-particle

spectrum close to the Fermi surface. We shall not dwell on BCS-ology as we soon

introduce the mean-ļ¬eld approximation at the level of Greenā™s functions, and instead

oļ¬er it as exercises.

Exercise 8.1. Assume up and vp real so that (recall Exercise 1.7 on page 20) the

angle Ļp parameterizes the amplitudes, up = sin Ļp and vp = cos Ļp . Show that

1

BCS|Hpairing ā’ F N|BCS = Ī¾p (1 + cos 2Ļp ) + Vpp sin 2Ļp sin 2Ļp

4

p,Ļ pp

(8.6)

resulting in the minimum condition of the average grand canonical energy to be

2Ī¾p tan 2Ļp = Vpp sin 2Ļp . (8.7)

p

Using simple geometric relations, 2up vp = sin 2Ļp and vp ā’ u2 = cos 2Ļp , and

2

p

introducing the quantities Ī”p = ā’ p Vpp up vp and Ep = Ī¾p + Ī”2 , show that

2

p

the minimum condition becomes the self-consistency condition

1 Ī”p

Ī”p = ā’ Vpp (8.8)

2 Ep

p

for the BCS-energy gap in the excitation spectrum.

Exercise 8.2. Besides the normal state solution, Ī”p = 0, for an attractive inter-

action the self-consistency condition, Eq. (8.8) has a nontrivial solution, Ī”p = 0.

Assuming, as dictated by electronā“phonon interaction, that the interaction is at-

tractive only in a tiny region around the Fermi energy set by the Debye energy,

ĻD , the interaction is modeled by a constant attraction in this region, Vpp =

ā’V Īø(ĻD ā’ |Ī¾p |) Īø(ĻD ā’ |Ī¾p |). Show that in this model the self-consistency equa-

tion has the solution Ī”p = ā’Ī” Īø(ĻD ā’ |Ī¾p |), where the constant Ī”, the energy gap,

is determined by (the prime indicates that the summation is restricted)

V 1

1= , (8.9)

2 Ī¾p + Ī”2

2

p

which for weak coupling, N0 V 1 (N0 being the density of momentum states of the

electron gas at the Fermi energy), gives

2 ĻD eā’1/N0 V .

Ī” (8.10)

8.1. BCS-theory 221

Show that in this model

2

Ī”2 Ī¾p

BCS| Hpairing ā’ |BCS =ā’ Ī¾p ā’

FN + (8.11)

V Ep

p

and thereby that the energy diļ¬erence per unit volume between the state with Ī” = 0

and the normal state, where states up to the Fermi surface are ļ¬lled according to

Eq. (1.105), is given by ā’N0 Ī”2 /2.8 The state with Ī” = 0 is thus favored as the

ground state by the pairing interaction.

Exercise 8.3. Introduce new operators by the Bogoliubovā“Valatin transformation9

Ī³pā‘ = up cā ā’ vp cā’pā“

ā

Ī³ā’pā“ = up cā + vp cpā‘

ā

ā— ā—

, (8.12)

pā‘ ā’pā“

and their adjoints, leaving them canonical as the normalization condition, |up |2 +

|vp |2 = 1, is insisted, assuring the anti-commutation relations

ā ā

{Ī³pā‘ , Ī³p ā‘ } = Ī“pp = {Ī³pā“ , Ī³p ā“ } (8.13)

as well as

ā

{Ī³pā‘ , Ī³p ā“ } = 0 = {Ī³pā“ , Ī³p ā“ } {Ī³pā‘ , Ī³p ā‘ } = 0 = {Ī³pā‘ , Ī³p ā“ } .

, (8.14)

Show that Hpairing ā’ F N is diagonalized by the transformation to the Hamilto-

nian, up to an irrelevant constant term,

ā

Ėā Ė

Hpairing ā’ FN = Ep (Ī³pā‘ Ī³pā‘ + Ī³pā“ Ī³pā“ ) , (8.15)

p

provided 2Ī¾p up vp + (vp ā’ u2 )Ī”p = 0, where Ī”p satisļ¬es the self-consistency equa-

2

p

tion Eq. (8.8) (assuming for simplicity real amplitudes). Equivalently, noting the

coeļ¬cients can be chosen real,

1 Ī¾p 1 Ī¾p Ī”p

1ā’

u2 = 2

1+ , vp = , up vp = . (8.16)

p

2 Ep 2 Ep 2Ep

This provides a general description of the BCS-Hamiltonian in terms of free fermionic

quasi-particles with energy dispersion Ep = Ī¾p + Ī”2 , and an energy gap in the

2

p

spectrum has appeared.

Show the |BCS -state is the vacuum state for the Ī³-operators, Ī³p |BCS = 0.

At ļ¬nite temperatures Pauliā™s exclusion principle for the BCS-quasi-particles,

which is equivalent to the anti-commutation properties of the Ī³-operators, gives that

8 This so-called condensation energy is typically seven orders of magnitude smaller than the

average Coulomb energy, and for the pairing Hamiltonian to make sense it is implicitly assumed

that the Coulomb energy for an electron is the same in the two states, which the success of the

BCS-theory then indicates.

9 Recall the particleā“hole symmetry of the BCS-state discussed in Exercise 2.8. on page 39.

222 8. Non-equilibrium superconductivity

at temperature T the probability of occupation of energy state Ep is given by the

Fermi function

1

ā ā

Ī³pā‘ Ī³pā‘ = E /kT = Ī³ā’pā“ Ī³ā’pā“ . (8.17)

e +1

p

Show consequently that the energy gap is temperature dependent as determined

self-consistently by the gap equation

1 Ī”p Ep

Ī”p = ā’ Vpp tanh . (8.18)

2 Ep 2kT

p

Show in the simple model considered in the previous exercise, that the energy

gap vanishes at the critical temperature, Tc , given by

ĻD eā’1/N (0)V .

kTc (8.19)

The BCS-theory is a mean ļ¬eld self-consistent theory with anomalous terms as

speciļ¬ed by the oļ¬-diagonal long-range order. The eļ¬ective Hamiltonian of the su-

perconducting state can therefore also be arrived at by the following argument. The

eļ¬ective two-body interaction is short ranged, of the order of the Fermi wavelength,

the inter-atomic distance, and can be approximated by the eļ¬ective local two-body

interaction, a delta potential characterized by a coupling strength Ī³ (in the electronā“

phonon model Ī³ is the square of the electronā“phonon coupling constant, Ī³ = g 2 ).

The attractive two-body interaction term then becomes

1 ā

ā

V =ā’ Ī³ dx ĻĪ± (x) ĻĪ² (x) ĻĪ² (x) ĻĪ± (x) , (8.20)

2

Ī±,Ī²

assuming a spin-independent interaction. This is of course still a hopelessly com-

plicated many-body problem. The BCS-theory is a self-consistent theory where the

interaction term is substituted according to

1 ā

dx (Ī”ā— (x, x) ĻĪ² (x) ĻĪ± (x) + Ī”Ī²Ī± (x, x)ĻĪ± (x) ĻĪ² (x))

ā

V ā’ā’ Ī³ (8.21)

Ī±Ī²

2

Ī±,Ī²

a manageable quadratic form, however with anomalous terms. The implicit assump-

tion for a self-consistent theory is thus that the ļ¬‚uctuations in the states of interest

of the diļ¬erence between the two operators in Eq. (8.20) and Eq. (8.21) are small.

This is analogous to the Hartreeā“Fock treatment of the electronā“electron interaction

in the normal state. These normal terms should also be considered, but in a con-

ventional superconductor such as a metal like tin, these eļ¬ects lead to only a tiny

renormalization of the electron mass, and we can think of them as included through

the dispersion relation. In a strongly interacting degenerate Fermi system such as

3

He, these interactions need to be taken into account and must be dealt with in terms

of Landauā™s Fermi liquid theory, a quasi-particle description (for details see reference

8.1. BCS-theory 223

[35] and for the application of the quasi-classical Greenā™s function technique see ref-

erence [36]). One should be aware that the BCS-approximation is quite a bold move

since the BCS-Hamiltonian breaks a sacred conservation law, viz. particle number

conservation, or equivalently, gauge invariance is spontaneously broken.10

For conventional superconductors we encounter orbital s-wave and spin-singlet

pairing where the interaction part of the Hamiltonian is

ā ā

VBCS = ā’Ī³ dx (Ī”ā— (x) Ļā‘ (x) Ļā“ (x) + Ī”(x) Ļā“ (x) Ļā‘ (x)) (8.22)

as the superconducting order parameter is11

Ī”(x) = Ļā‘ (x) Ļā“ (x) . (8.23)

Of importance is the feature of self-consistency, i.e. the bracket means average with

respect to the order-parameter dependent BCS-Hamiltonian

2

1 ā‚

ā

ā’ eA(x, t) ā’ Ī¼ ĻĪ± (x)

HBCS = dx ĻĪ± (x)

2m i ā‚x

Ī±=ā“,ā‘

ā ā

Ī³ dx (Ī”ā— (x) Ļā‘ (x) Ļā“ (x) + Ī”(x) Ļā“ (x) Ļā‘ (x))

ā’ (8.24)

and Eq. (8.24) and Eq. (8.23) thus represent a complicated set of coupled equations.

We have placed the superconductor in an electromagnetic ļ¬eld represented by a

vector potential which, except for weak ļ¬elds or for temperatures near the critical

temperature, through self-consistency leads to unquenchable analytic intractabilities.

Only for simple and highly symmetric situations can the order parameter be speciļ¬ed

a priori, thereby opening up for analytical tractability.

In the Heisenberg picture, the equation of motion governed by the BCS Hamilto-

nian is for the spin-up electron ļ¬eld component

ā‚Ļā‘ (x, t) 1 ā

(ā’iāx ā’ eA(x, t))2 ā’ Ī¼ Ļā‘ (x, t) + Ī³Ī”(x, t) Ļā“ (x, t) (8.25)

i =

ā‚t 2m

and for the spin-down adjoint component

ā

ā‚Ļā“ (x, t) 1 ā

(iāx ā’ eA(x, t)) ā’ Ī¼ Ļā“ (x, t) ā’ Ī³Ī”ā— (x, t) Ļā‘ (x, t). (8.26)

2

ā’i =

ā‚t 2m

The BCS-Hamiltonian therefore leads to a set of coupled equations of motion for the

single-particle time-ordered Greenā™s function

ā

G(x, t; x , t ) = ā’i T (Ļā‘ (x, t) Ļā‘ (x , t )) (8.27)

10 In the electronā“phonon model, the Hamiltonian is gauge invariant.

11 In the case of p-wave or d-wave pairing, the order parameter has additional spin dependence.

224 8. Non-equilibrium superconductivity

and the anomalous or Gorkov Greenā™s function

ā ā

F (x, t; x , t ) = ā’i T (Ļā“ (x, t) Ļā‘ (x , t )) , (8.28)

viz. the Gorkov equations12

ā‚ 1 2

ā’ (ā’iāx ā’ eA(x, t)) + Ī¼ G(x, t, x , t ) + Ī³Ī”(x, t) F (x, t, x , t )

i

ā‚t 2m

= Ī“(x ā’ x )Ī“(t ā’ t ) (8.29)

and

ā‚ 1

(iāx1 ā’ eA(x1 , t1 ))2 + Ī¼ F (1, 1 ) + Ī³Ī”ā— (1) G(1, 1 ) = 0 , (8.30)

ā’i ā’

ā‚t1 2m

where in the latter equation we have introduced the usual condensed notation. The

spin labeling of the functions is irrelevant since no spin-dependent interactions, such

as spin ļ¬‚ip interactions due to magnetic impurities, are presently included and spin

up and down are therefore equivalent, except for the singlet feature of the anomalous

Greenā™s function as we consider s-wave pairing. The order parameter is speciļ¬ed by

the equal space and time anomalous Greenā™s function

ā ā

Ī”ā— (x, t) = i F (x, t+ ; x, t) = Ļā“ (x, t)Ļā‘ (x, t) . (8.31)

When the eļ¬ect of pairing is taken into account, the Feynman diagrammatics in

the electronā“phonon or BCS-model is modiļ¬ed by the presence of lines describing

the additional channel due to the non-vanishing of the anomalous Greenā™s function.

However, as the order parameter is small compared with the Fermi energy in a con-

ventional superconductor (as well as in superļ¬‚uid He-3), this new scale is irrelevant

for diagram estimation, and Migdalā™s theorem is then again valid (as ļ¬rst noted by

Eliashberg [37]). The peaked structure at the Fermi momentum of the Greenā™s func-

tions thus remains as in the normal state, and the argument for the validity of the

Migdal approximation now becomes identical for the superļ¬‚uid case once it is based

on the correct ground state, i.e. the anomalous self-energy terms are included. A

theory of strong coupling superconductivity, Eliashbergā™s theory, is thus available of

which the BCS-theory is the weak coupling limit, kTc ĻD , in accordance with

Eq. (8.1). It is convenient to collect the equations of motion for the normal and

anomalous Greenā™s functions into a single matrix equation of motion, and this is

done by introducing the Nambu ļ¬eld, by which the BCS-Hamiltonian is turned into

a quadratic form. Furthermore, we shall introduce the contour ordered and not just

the time ordered Greenā™s functions in order to describe the non-equilibrium states of

a superconductor.

12 Had we used the canonical ensemble, the chemical potential would be absent in Eq. (8.29), and

since

ńņš. 9 |