<<

. 9
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>>

2 4π
where N0 is the density of states at the Fermi energy, and the current density is given
by
1 dˆ
p
j(R, T ) = ’ eN0 dE vF g K (E, p, R, T ) .
ˆ (7.109)
2 4π
18 Here and in the following, we assume for simplicity a spherical Fermi surface. For a general
Fermi surface one decomposes according to
dξp dsF
dp ˆ
= ,
3 vF (2π)3
(2π)
where dξp /vF is the length of the momentum increment measured from the Fermi surface in the
ˆ
directions ±ˆ , and dsF is the corresponding Fermi surface area element.
p
200 7. Quantum kinetic equations


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

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




1 1
Figure 7.2 Electron“phonon self-energy.



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

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

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


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

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

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

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

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

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


since

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

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

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

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


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

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

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

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

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

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

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


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

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

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

q kF , ω . (7.126)
F

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

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

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

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

where the collision integral is

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


where

γ = i(σ R ’ σ A ) . (7.130)

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

E ’E

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

(7.131)

and
E ’E

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

where we have introduced the distribution function
1K
ˆ ˆ
h(E, p, R, T ) = g (E, p, R, T ) (7.133)
2
and
iN0 |gq |2
(DR (E, q) ’ DA (E, q))
μ(q, E) = (7.134)

is the Eliashberg spectral weight function. Here we have allowed for a more gen-
eral longitudinal electron“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 ωq /2, where ωq = 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 ¬‚uctuation“dissipation relation for bosons,
Eq. (5.103),

E
DR (E, p) ’ DA (E, p)
DK (E, p) = coth . (7.135)
2kT
We note that the variables in the distribution function are quite di¬erent from
that of the classical Boltzmann equation for electron“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 re¬‚ects the fact that we do not rely on a de¬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 Green™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 ’ h(E, p, R, T )
ˆ ˆ
f (E, p, R, T ) = (7.136)
2
and
1 E
n(E) = ’ 1 ’ coth (7.137)
2 2kT
the collision integral takes the more familiar form


p
dE μ(pF (ˆ ’ p ), E ’ E ) RE pˆ ,

Ie’ph = ’2π pˆ (7.138)
p


where

RE pˆ

(1 + n(E ’ E ))f (E, p)(1 ’ f (E , p ))
ˆ ˆ
=
p


’ n(E ’ E )(1 ’ f (E, p))f (E , p ) .
ˆ ˆ (7.139)

Finally, introducing a gauge invariant distribution function by the substitution

f (E) ’ f (E ’ •(R, T )) (7.140)

we obtain the quantum kinetic equation

((1 ’ ‚E eσ)‚T + ‚T eσ ‚E + vF · (∇R + eE(R, T )) ‚E ) f = Ie’ph [f ] ,
(7.141)

where E(R, T ) = ’∇R •(R, T ) is the perturbing electric ¬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 e¬ects, in particular mass renormalization due to the electron“phonon
interaction.
From the kinetic equation, Eq. (7.141), Prange and Kadano¬ [23] drew the con-
clusion that many-body e¬ects can be seen only in time-dependent transport prop-
erties and that static transport coe¬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 particle“hole symmetry was assumed. Within the quasi-classical
scheme, all thermoelectric coe¬cients therefore vanish, and no conclusion can be
drawn about many-body e¬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 electron“phonon interaction.
206 7. Quantum kinetic equations


7.5.2 Renormalization of the a.c. conductivity
As an example of electron“phonon renormalization of time-dependent transport co-
e¬cients we shall consider the a.c. conductivity in the frequency range ω„e’ph 1,
where 1/„e’ph is the clean-limit electron“phonon scattering rate for an electron on
the Fermi surface
§ 7πζ(3) (kT )3
⎪ » (pF c)2 kT 2pF c

⎨ 2
1
= (7.142)

„ ( F, T ) ⎪
©
2π» kT kT 2pF c
ζ being Riemann™s zeta function.25 For de¬niteness we also consider the temperature
to be low compared to the Debye temperature θD . In this high-frequency limit the
collision integral can be neglected and the linearized kinetic equation takes the simple
form
(1 ’ ‚E eσ)‚T h + ‚T eσ ‚E h0 + evF · E(T ) ‚E h0 = 0 (7.143)
for a spatially homogeneous electric ¬eld. Except for a real constant, just renormal-
izing the chemical potential, we have according to the Feynman rules
1 dˆ
p
dE |gpF ’pF |2 h(E , p , R, T ) eD(pF ’ pF , E ’ E ) , (7.144)
ˆ
eσ = N0
2 4π
where gpF ’pF denotes the electron“phonon coupling, and
1R
(D + DA ) .
eD = (7.145)
2
For an applied monochromatic ¬eld, E(t) = E0 exp{’iωt}, the solution can be
sought in the form
h1 = aeE · vF ‚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)
’iω(1 + »— )
where
|gpF ’pF |2

p
»— = 2N0 (1 ’ p · p ) .
ˆˆ
dE (7.148)
4π ωpF ’pF
The current can now be evaluated and we obtain for the frequency dependence
of the conductivity
ne2
σ(ω) = (7.149)
’iωmopt
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 Green™s function technique 207


where the optical mass is renormalized according to

mopt = m(1 + »— ) (7.150)

a result originally obtained by Holstein using a di¬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 di¬erent from
the speci¬c heat mass renormalization, m ’ (1 + »)m (see also the result of Exercise
8.8 on page 237).
As a consequence of electron“phonon interaction, the physically observed mass of
the electron is not the mass or band structure e¬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 di¬erent from the speci¬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 di¬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 electron“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 electron“phonon
transport theory into the standard Landau“Boltzmann form. We start by de¬ning a
quasi-particle energy Ep , which is de¬ned implicitly by (we suppress the space-time
variables and use the short notation Ep = E(p, R, T ))

ˆ
Ep = ξp + eσ(Ep + e•(R, T ), p, R, T ) (7.151)

thereby satisfying the equations

∇p Ep = Zp ∇p ξp (7.152)

and
∇R Ep = Zp (e∇R • ‚E eσ + ∇R eσ) (7.153)
E=Ep +e•(R,T )

and
‚T Ep = Zp (e‚T •‚E eσ + ‚T eσ) (7.154)
E=Ep +e•(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 in¬nities occurring in quantum ¬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 in¬nite in order to provide the ¬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 ’ ‚E eσ)’1 (7.155)
E=Ep +e•(R,T )

has been introduced.
The energy variable E in the kinetic equation is now set equal to Ep + e• and
we introduce the distribution function (again suppressing the space-time variables)
ˆ
np = f (E, p, R, T ) . (7.156)
E=Ep +e•(R,T )

Using the relations
∇p n = ∇p Ep (‚E f ) (7.157)
E=Ep +e•(R,T )

and
∇R n = (∇R f + ∇R (Ep + e•(R, T ))‚E f ) (7.158)
E=Ep +e•(R,T )

and
‚T n = (‚T f + ‚T (Ep + e•(R, T ))‚E f ) (7.159)
E=Ep +e•(R,T )

and Eqs. (7.152“7.154), we obtain the kinetic equation of the form
’1 ˜
Zp (‚T + ∇p Ep · ∇R ’ ∇R (Ep + •(R, T )) · ∇p ) n(p, R, T ) = Ie’ph (7.160)
with the electron“phonon collision integral
2π dp p
ˆ
˜
Ie’ph = ’ Zp μ(p ’ p ) Rp ,
˜ (7.161)
ˆ
3
N0 (2π)
where
p
ˆ
Rp = (1 + N (Ep ’ Ep ))np (1 ’ np ) ’ N (Ep ’ Ep )(1 ’ np )np (7.162)
ˆ

and
iN0 |gp’p |2 R
μ(p ’ p ) = (D (p ’ p , Ep ’ Ep ) ’ DA (p ’ p , Ep ’ Ep )) . (7.163)
˜

In transforming the collision integral we have utilized the substitution
dˆ dˆ dEp dp
p p
dE ’ N0 ’
N0 dξp Zp . (7.164)
(2π)3
4π 4π dξp
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

μ(p ’ p ) = N0 |gp’p |2 (δ(Ep ’ Ep ’ ωp’p ) ’ δ(Ep ’ Ep + ωp’p )) . (7.165)
˜
7.5. Quasi-classical Green™s function technique 209


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

(‚T + ∇p Ep · ∇R ’ ∇R (Ep + •(R, T )) · ∇p ) n(p, R, T ) = Ie’ph , (7.166)

where the electron“phonon collision integral is

dp p
ˆ
Ie’ph = ’2π Zp Zp |gp’p |2 Rp (δ(Ep ’ Ep ’ ωp’p ).
ˆ
3
(2π)


’ δ(Ep ’ Ep + ωp’p )) . (7.167)

This has the form of the familiar Landau“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 Landau“Boltzmann
description of the electron“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)
„ ( 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 Green™s function respects conservation
laws, say particle number conservation, is not in general as easily stated as for the
microscopic Green™s function. We therefore establish it here explicitly. The collision
integral, Eq. (7.167), has the invariant

dp
Ie’ph = 0 , (7.169)
(2π)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

‚T n + ∇R · j = 0 , (7.170)

where
dp
n(R, T ) = 2 n(p, R, T ) (7.171)
(2π)3
210 7. Quantum kinetic equations


and
dp
∇p Ep n(p, R, T )
j(R, T ) = 2 (7.172)
(2π)3

are the Landau“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 identi¬ed densities (in
the excitation representation), we should connect one of them with the microscopic
expression. Assuming that |e•| F , the microscopic expression for the density,
Eq. (7.108), is (suppressing space-time variables in quantities, here in •)


p
n(R, T ) = ’2N0 dE f (E + e•, p) . (7.173)
4π ’∞

In order to compare the density expression in the particle representation with the
excitation representation we transform Eq. (7.171) to the particle representation

dp dˆ
p
dE (1 ’ ‚E eσ) f (E, p).
ˆ
n(R, T ) = 2 n(p, R, T ) = 2N0 (7.174)
(2π)3 4π ’∞

Since Eq. (7.173) and Eq. (7.174) appear to be di¬erent, Eq. (7.172) is also trans-
formed to the particle representation

dp dˆ
p
∇p Ep n(p, R, T ) = 2N0 ˆ
2 dE vF f (E, p) . (7.175)
(2π)3 4π ’∞

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

ˆ
‚T dˆ dE f (E, p) ‚E eσ = 0 (7.176)
p
’∞

which we now prove. Inserting the expression from Eq. (7.144) into the left side of
Eq. (7.176) we are led to consider

dˆ dE dˆ dE |gpF ’pF |2 ( eD(pF ’ pF , E ’ E )
p p


‚T f (E, p) ‚E f (E , p ) ’ ‚E f (E, p) ‚T f (E , p )) = 0
ˆ ˆ ˆ ˆ (7.177)

ˆ ˆ
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 „ , and we can immediately implement the quasi-
classical approximation. The equation for the kinetic component of the quasi-classical
Green™s function in the presence of an electric ¬eld becomes
1 dˆ K
p
(‚T + vF · ∇R + e‚T • ‚E ) g K = ’ g K (E, p, R, T ) +
ˆ ˆ
g (E, p, R, T ) ,
„ 4π
(7.178)

where for simplicity we have assumed that the momentum dependence of the impurity
potential can be neglected.
In the di¬usive limit the quasi-classical kinetic Green™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 · gp (E, R, T )
K
ˆK
ˆ (7.179)

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
singularity involved in Anderson™s metal“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


lim ψ± (x4 ) ψβ (x3 ) ψγ (x2 ) ψδ (x1 ) = 0 , (8.2)
|x1 ,x2 ’x3 ,x4 |’∞

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,
”γδ (x, x ), expressing this property, can therefore be introduced according to

ψ± (x4 ) ψβ (x3 ) ψγ (x2 ) ψδ (x1 ) = ”— (x4 , x3 ) ”γδ (x1 , x2 )

lim (8.3)
±β
|x1 ,x2 ’x3 ,x4 |’∞

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 Green™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, ‘) and (’p, “), each others time-
reversed states,5 and we encounter orbital s-wave and spin-singlet pairing and the
BCS-Hamiltonian becomes6

Vpp c† c† c’p “ cp ‘ ,

Hpairing = p cpσ cpσ + (8.4)
p‘ ’p“
p,σ pp

where the e¬ective attractive interaction Vpp is only non-vanishing for momentum
states in the tiny region around the Fermi surface set by the Debye energy, ωD , for the
case where the attraction is caused by electron“phonon interaction. The parameters
specifying the boldly guessed BCS-ground state7

(up + vp c† c† ) |0
|BCS = (8.5)
p‘ ’p“
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 de¬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 ¬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 electron“phonon model, but the BCS-theory correctly describes
the o¬-diagonal long-range order, and is a very e¬cient way for incorporating and calculating the
order parameter, characterizing the superconducting state, and its consequences. Quantum ¬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 ’ μN|BCS , the pairing
Hamiltonian with energies measured from the chemical potential, which at zero tem-
perature is the Fermi energy, ξp = p ’ 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-¬eld approximation at the level of Green™s functions, and instead
o¬er it as exercises.

Exercise 8.1. Assume up and vp real so that (recall Exercise 1.7 on page 20) the
angle φp parameterizes the amplitudes, up = sin φp and vp = cos φp . Show that

1
BCS|Hpairing ’ F N|BCS = ξp (1 + cos 2φp ) + Vpp sin 2φp sin 2φp
4
p,σ pp
(8.6)
resulting in the minimum condition of the average grand canonical energy to be

2ξp tan 2φp = Vpp sin 2φp . (8.7)
p


Using simple geometric relations, 2up vp = sin 2φp and vp ’ u2 = cos 2φp , and
2
p

introducing the quantities ”p = ’ p Vpp up vp and Ep = ξp + ”2 , show that
2
p
the minimum condition becomes the self-consistency condition
1 ”p
”p = ’ Vpp (8.8)
2 Ep
p

for the BCS-energy gap in the excitation spectrum.


Exercise 8.2. Besides the normal state solution, ”p = 0, for an attractive inter-
action the self-consistency condition, Eq. (8.8) has a nontrivial solution, ”p = 0.
Assuming, as dictated by electron“phonon interaction, that the interaction is at-
tractive only in a tiny region around the Fermi energy set by the Debye energy,
ωD , the interaction is modeled by a constant attraction in this region, Vpp =
’V θ(ωD ’ |ξp |) θ(ωD ’ |ξp |). Show that in this model the self-consistency equa-
tion has the solution ”p = ’” θ(ωD ’ |ξp |), where the constant ”, the energy gap,
is determined by (the prime indicates that the summation is restricted)

V 1
1= , (8.9)
2 ξp + ”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 ωD e’1/N0 V .
” (8.10)
8.1. BCS-theory 221


Show that in this model
2
”2 ξp
BCS| Hpairing ’ |BCS =’ ξp ’
FN + (8.11)
V Ep
p

and thereby that the energy di¬erence per unit volume between the state with ” = 0
and the normal state, where states up to the Fermi surface are ¬lled according to
Eq. (1.105), is given by ’N0 ”2 /2.8 The state with ” = 0 is thus favored as the
ground state by the pairing interaction.


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

γp‘ = up c† ’ vp c’p“

γ’p“ = up c† + vp cp‘

— —
, (8.12)
p‘ ’p“

and their adjoints, leaving them canonical as the normalization condition, |up |2 +
|vp |2 = 1, is insisted, assuring the anti-commutation relations
† †
{γp‘ , γp ‘ } = δpp = {γp“ , γp “ } (8.13)

as well as

{γp‘ , γp “ } = 0 = {γp“ , γp “ } {γp‘ , γp ‘ } = 0 = {γp‘ , γp “ } .
, (8.14)

Show that Hpairing ’ F N is diagonalized by the transformation to the Hamilto-
nian, up to an irrelevant constant term,


˜† ˜
Hpairing ’ FN = Ep (γp‘ γp‘ + γp“ γp“ ) , (8.15)
p


provided 2ξp up vp + (vp ’ u2 )”p = 0, where ”p satis¬es the self-consistency equa-
2
p
tion Eq. (8.8) (assuming for simplicity real amplitudes). Equivalently, noting the
coe¬cients can be chosen real,

1 ξp 1 ξp ”p
1’
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 = ξp + ”2 , and an energy gap in the
2
p
spectrum has appeared.
Show the |BCS -state is the vacuum state for the γ-operators, γp |BCS = 0.
At ¬nite temperatures Pauli™s exclusion principle for the BCS-quasi-particles,
which is equivalent to the anti-commutation properties of the γ-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 particle“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
† †
γp‘ γp‘ = E /kT = γ’p“ γ’p“ . (8.17)
e +1
p


Show consequently that the energy gap is temperature dependent as determined
self-consistently by the gap equation

1 ”p Ep
”p = ’ 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

ωD e’1/N (0)V .
kTc (8.19)




The BCS-theory is a mean ¬eld self-consistent theory with anomalous terms as
speci¬ed by the o¬-diagonal long-range order. The e¬ective Hamiltonian of the su-
perconducting state can therefore also be arrived at by the following argument. The
e¬ective two-body interaction is short ranged, of the order of the Fermi wavelength,
the inter-atomic distance, and can be approximated by the e¬ective local two-body
interaction, a delta potential characterized by a coupling strength γ (in the electron“
phonon model γ is the square of the electron“phonon coupling constant, γ = g 2 ).
The attractive two-body interaction term then becomes
1 †

V =’ γ dx ψ± (x) ψβ (x) ψβ (x) ψ± (x) , (8.20)
2
±,β

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 †
dx (”— (x, x) ψβ (x) ψ± (x) + ”β± (x, x)ψ± (x) ψβ (x))

V ’’ γ (8.21)
±β
2
±,β

a manageable quadratic form, however with anomalous terms. The implicit assump-
tion for a self-consistent theory is thus that the ¬‚uctuations in the states of interest
of the di¬erence between the two operators in Eq. (8.20) and Eq. (8.21) are small.
This is analogous to the Hartree“Fock treatment of the electron“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 e¬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 Landau™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 Green™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

† †
VBCS = ’γ dx (”— (x) ψ‘ (x) ψ“ (x) + ”(x) ψ“ (x) ψ‘ (x)) (8.22)

as the superconducting order parameter is11

”(x) = ψ‘ (x) ψ“ (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 ‚

’ eA(x, t) ’ μ ψ± (x)
HBCS = dx ψ± (x)
2m i ‚x
±=“,‘



† †
γ dx (”— (x) ψ‘ (x) ψ“ (x) + ”(x) ψ“ (x) ψ‘ (x))
’ (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 ¬eld represented by a
vector potential which, except for weak ¬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 speci¬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 ¬eld component

‚ψ‘ (x, t) 1 †
(’i∇x ’ eA(x, t))2 ’ μ ψ‘ (x, t) + γ”(x, t) ψ“ (x, t) (8.25)
i =
‚t 2m

and for the spin-down adjoint component

‚ψ“ (x, t) 1 †
(i∇x ’ eA(x, t)) ’ μ ψ“ (x, t) ’ γ”— (x, t) ψ‘ (x, t). (8.26)
2
’i =
‚t 2m

The BCS-Hamiltonian therefore leads to a set of coupled equations of motion for the
single-particle time-ordered Green™s function

G(x, t; x , t ) = ’i T (ψ‘ (x, t) ψ‘ (x , t )) (8.27)
10 In the electron“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 Green™s function
† †
F (x, t; x , t ) = ’i T (ψ“ (x, t) ψ‘ (x , t )) , (8.28)

viz. the Gorkov equations12

‚ 1 2
’ (’i∇x ’ eA(x, t)) + μ G(x, t, x , t ) + γ”(x, t) F (x, t, x , t )
i
‚t 2m

= δ(x ’ x )δ(t ’ t ) (8.29)

and
‚ 1
(i∇x1 ’ eA(x1 , t1 ))2 + μ F (1, 1 ) + γ”— (1) G(1, 1 ) = 0 , (8.30)
’i ’
‚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 ¬‚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
Green™s function as we consider s-wave pairing. The order parameter is speci¬ed by
the equal space and time anomalous Green™s function
† †
”— (x, t) = i F (x, t+ ; x, t) = ψ“ (x, t)ψ‘ (x, t) . (8.31)

When the e¬ect of pairing is taken into account, the Feynman diagrammatics in
the electron“phonon or BCS-model is modi¬ed by the presence of lines describing
the additional channel due to the non-vanishing of the anomalous Green™s function.
However, as the order parameter is small compared with the Fermi energy in a con-
ventional superconductor (as well as in super¬‚uid He-3), this new scale is irrelevant
for diagram estimation, and Migdal™s theorem is then again valid (as ¬rst noted by
Eliashberg [37]). The peaked structure at the Fermi momentum of the Green™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 super¬‚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, Eliashberg™s theory, is thus available of
which the BCS-theory is the weak coupling limit, kTc ωD , in accordance with
Eq. (8.1). It is convenient to collect the equations of motion for the normal and
anomalous Green™s functions into a single matrix equation of motion, and this is
done by introducing the Nambu ¬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 Green™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

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