Kopnin N.B. Theory of Nonequilibrium Superconductivity

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where

(8.9)

and

(8.10)

Collecting now all orders of the perturbation expansion, we find that the

analytical continuation gives

(8.11)

where we define the total time-dependent Green function

(8.12)

It is this function which determines all the physical properties of a nonstationary

system in a real time. We assign the superscript K to the total function to

indicate that it coincides with the so-called Keldysh Green function introduced by

Keldysh (1964). We shall prove this later.

The retarded and advanced functions are defined in the usual way as

We assume integration over all

and k

. These functions only contain all

retarded or all advanced zero-order functions G

(0)

, respectively, and are obtained

end p.148

by summation of all the diagrams exactly as for the Matsubara functions. They

satisfy the Gor’kov equations with real frequencies

The anomalous function is

where

(8.13)

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It necessarily contains at least one incoming retarded together with one outgoing

advanced function G

(0)

The regular functions enter eqn (8.12) in combinations with the equilibrium

distribution, eqn (8.9), which is

(8.14)

is the Fermi function. It is the anomalous function which describes the deviation

of the system from equilibrium. The summation over all N gives

(8.15)

This simple expression for is only valid if there are no interactions with

impurities, phonons, and particle–particle interactions. We now consider these

interactions in turn.

8.1.2 Impurities

Consider first impurity scattering. We shall not write down all possible diagrams,

instead, we concentrate on one typical example which demonstrates the general

structure of the result of the analytical continuation. Consider the term which is

of the third order in the field and of the second order in the impurity

potential u. We have, similarly to the previous case,

where, for example,

Note that the Green functions on both sides of the impurity potential U have the

same frequencies because the Impurity scattering is elastic and does not change

end p.149

the particle energy. Proceeding in the same way as before, we obtain for this

particular term of the third order in

and of the second order in U, after

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summation over all cuts:

After averaging over positions of impurity atoms, we can present this term as

It is clear that, collecting all orders in and in the impurity potential, we again

obtain eqn (8.12) where the anomalous function now is

(8.16)

Sometimes, we shall use a symbolic form

(8.17)

where the frequencies and momenta are not shown, and the integration over all

internal variables is assumed. Equation (8.16) differs from eqn (8.15) by the

presence of the anomalous impurity self-energy

(a)

which necessarily contains

both retarder and advanced functions

(8.18)

Here is the angle between P and P . The self-energy is obtained from

in the same way as the self-energy is obtained from the regular Green

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functions :

end p.150

Top

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The regular functions are determined by the sum of diagrams of all orders in

and , therefore, they satisfy the usual equations

(8.19)

where the inverse operator is determined by eqn (8.3). Equation

(8.19) in the symbolic form is

(8.20)

One can obtain an equation, for the anomalous function as well. Let us

apply the operator in eqn (8.3) to eqn (8.16) from the left. We obtain, in a

symbolic form,

(8.21)

Using the definition of the total time-dependent function

and equations for

the retarded and advanced functions one can prove that the total function

satisfies a homogeneous equation of the type of eqn (8.21). Indeed, let us take

the combination

(8.22)

and apply to it the operator from the left. We get

(8.23)

Adding eqns (8.21) and (8.23) together, we find

(8.24)

Kopnin, Nikolai, Senior Scientist, Low Temperature Laboratory, Helsinki University of

Technology, and L.D. Landau Institute for Theoretical Physics, Moscow

Theory of Nonequilibrium Superconductivity

Print ISBN 9780198507888, 2001

pp. [151]-[155]

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Here

(8.25)

is the total self-energy. It is clear that

Similarly, applying the operator from the right, we find

(8.26)

and

(8.27)

end p.151

8.1.3 Order parameter, current, and particle density

The self-consistency order parameter equation can easily be obtained using the

general equation for the order parameter in terms of the Matsubara Green

function and the rules we have derived here for calculating the sums over the

Matsubara frequencies in terms of real-frequency integrations, eqn (8.11). We

find for an s-wave pairing

(8.28)

In the same way, the current takes the form

(8.29)

The electron density becomes

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(8.30)

8.2 The phonon model

8.2.1 Self-energy

Consider how one can generalize this approach for the phonon model where the

pairing between electrons occurs due to their interaction with phonons. In the

phonon model by Eliashberg (1960), the order parameter itself is regarded not as

a “field” but rather as a self-energy in the Gor’kov Green function F. The starting

point is the equation for the Matsubara Green functions eqn (8.2) where

iT (n + 1/2). The inverse operator is defined as in eqn (8.3), the

difference being that the “field”

contains only the electromagnetic potentials.

The phonon self-energy has the same diagram structure as the impurity

self-energy shown in Fig. 4.2(b) where the dashed line stands for the phonon

Green function D while the crosses denote the matrix element g

of the

electron–phonon interaction:

(8.31)

Here – = 2 iTm. For simplicity, we do not include impurities here. We

consider phonons as “free” particles assuming that their interaction with

electrons

end p.152

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and of Dz– ." title="Fig. 8.3. The complex plane of frequencies with cuts

along the singular lines of

and of Dz– ." class="figure">

IG. 8.3. The complex plane of frequencies with cuts along the

singular lines of

and of D

z–

does not change their properties considerably (Migdal 1958). According to eqn

(2.57), the phonon Green function is

(8.32)

Consider a contribution of the N-th order in field to the self-energy in eqn (8.31)

and transform the sum over the Matsubara frequency

into the contour

integral. We have

We do not show integration over d

p , for brevity. Singularities of the function in

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the square brackets are at the lines Im (z –

) =0 where has its poles,

and at the line Im (z –

) = 0 where the poles of D are located (see Fig. 8.3). We

make cuts along these singular lines, and then unroll the contours along the

cuts. Doing this, we put z –

+ iIm

at those cuts which correspond to the

singular lines of

and z = + iIm

at the cut which corresponds to the

singular line of the function D. Note that both

= 2 iT (n + 1/2) and

iTk

are still imaginary, but is already a real variable. We obtain

end p.153

(8.33)

The notation shows the variables of the function ; In

particular, one can see that It does not depend on

Until now we did not include the interaction with phonons into the Green function

under the integral in eqn (8.33). However, we can make an important

observation. Since

is real, the singular lines of as a function, of

coincide with those of . This means that we can do the same analytical

continuation if we include the self-energies into the internal function

under the integral in eqn (8.33) and thus consider it as a total Green function

which contains all orders of the interaction with phonons.

Consider first analytical continuation over

which gives the regular

self-energies. Performing the continuation from the region Im

> Im ( – ) > 0

we get the retarded self-energy. We now shift the integration variable

at each cut belonging to

(N)

and also the shift

at the cut for D. We have

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where is defined by eqn (8.8). Collecting all orders in N, we obtain

(8.34)

Similarly, continuing eqn (8.33) over from the region Im ( – ) < Im < 0

and collecting all orders in N, we obtain the advanced self-energy

(8.35)

The phonon functions depend on the momentum transfer P –

P , while the electron functions have the variables .

The total

end p.154

function is determined by the same equations (8.12) and (8.16) as for

the case with impurities

The retarded and advanced functions are defined as sums of all the contributions

which contain only the retarded G

and or only the advanced ones, G

and

, respectively. The equation for retarded and advanced functions can thus be

derived by the analytical continuation of the corresponding equation for the

Matsubara functions from the region Im

> Im( – ) > 0 and Im( – ) < Im

< 0, respectively:

(8.36)

The total self-energy is defined exactly as in eqn (8.25):

(8.37)

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