†

σ=‘,“

(8.34)

where

1

(’i∇x1 ’ eA(x1 , t1 ))2 + eφ(x1 , t1 ) ’ μ

h(1) = (8.35)

2m

and the presence of coupling of the electrons to a classical electromagnetic ¬eld has

been included.

Consider the quadratic form in terms of the Nambu ¬eld

h(1) ”(1)

H = dx1 Ψ† (1) Ψ(1) , (8.36)

” (1) ’h— (1)

—

where h— (1) denotes complex conjugate of the single-particle Hamiltonian

1

h— (1) =

2

(i∇x1 ’ eA(x1 , t1 )) + eφ(x1 , t1 ) ’ μ . (8.37)

2m

The o¬-diagonal terms are identical to the ones in the BCS-Hamiltonian, but only

the ¬rst of the diagonal terms

† †

dx1 (ψ‘ (1)h(1)ψ‘ (1) ’ ψ“ (1)h— (1)ψ“ (1)) (8.38)

gives the corresponding kinetic energy term. In the second term partial integrations

are performed, giving

† †

dx1 ψ“ (x1 , t1 )h— (1)ψ“ (1) = dx1 h(x1 , t1 )ψ“ (1 )ψ“ (1) . (8.39)

x1 =x1

Using the equal-time anti-commutation relations for the electron ¬elds produces the

wanted order of the operators but an additional delta function

†

dx1 h(x1 , t1 ) δ(x1 ’ x1 ) ’ ψ“ (1)ψ“ (1 ) , (8.40)

x1 =x1

226 8. Non-equilibrium superconductivity

which, however, is just a state independent (in¬nite) constant that has no in¬‚uence

on the dynamics and can be dropped. We have thus shown that the BCS-Hamiltonian

can equivalently be written in terms of the Nambu ¬eld as

h(1) ”(1)

HBCS = dx1 Ψ† (1) Ψ(1). (8.41)

”— (1) ’h— (1)

Two by two (2 — 2) matrices are introduced in Nambu space according to

ψ‘ (1) †

Ψ(1) Ψ† (1 ) ≡ (ψ‘ (1 ), ψ“ (1 ))

†

ψ“ (1)

†

ψ‘ (1)ψ‘ (1 ) ψ‘ (1)ψ“ (1 )

= (8.42)

† † †

ψ“ (1)ψ‘ (1 ) ψ“ (1)ψ“ (1 )

or, in Nambu index notation,

(Ψ(1) Ψ† (1 ))ij ≡ Ψi (1) Ψ† (1 ) . (8.43)

j

For the opposite sequence we de¬ne the 2—2-matrix

Ψ† (1) Ψi (1 )

(Ψ† (1) Ψ(1 ))ij ≡ j

†

ψ‘ (1)ψ‘ (1 ) ψ“ (1)ψ‘ (1 )

= . (8.44)

† † †

ψ‘ (1)ψ“ (1 ) ψ“ (1)ψ“ (1 )

The Nambu ¬eld is seen to satisfy the canonical anti-commutation rules

[Ψ(x) , Ψ† (x ))]+ = δ(x ’ x ) 1 (8.45)

and

[Ψ† (x) , Ψ† (x ))]+ = 0 = [Ψ(x) , Ψ(x ))]+ (8.46)

where 1 and 0 are the unit and zero matrices in Nambu space, respectively.

The contour-ordered Green™s function is de¬ned in particle“hole or Nambu space

according to

G(1, 1 ) = ’i Tct (ΨH (1) Ψ† (1 )) , (8.47)

H

>

where the subscript indicates the ¬eld is in the Heisenberg picture. For t1 c t1 the

contour-ordered Green™s function becomes

’i (ΨH (1) Ψ† (1 ))

G> (1, 1 ) = H

†

ψ‘ (1)ψ‘ (1 ) ψ‘ (1)ψ“ (1 )

’i

= . (8.48)

† † †

ψ“ (1)ψ‘ (1 ) ψ“ (1)ψ“ (1 )

In Nambu index notation the greater Green™s function simply becomes

G> (1, 1 ) = ’i Ψi (1) Ψ† (1 ) . (8.49)

j

ij

8.1. BCS-theory 227

The lesser Green™s function is then

i (Ψ† (1 ) ΨH (1))

G< (1, 1 ) = H

†

ψ‘ (1 )ψ‘ (1) ψ“ (1 )ψ‘ (1)

= i (8.50)

† † †

ψ‘ (1 )ψ“ (1) ψ“ (1 )ψ“ (1)

and in matrix notation

G< (1, 1 ) = ’i Ψ† (1) Ψj (1 ) . (8.51)

i

ij

To acquaint ourselves with Nambu space we consider the dynamics of the Nambu

¬eld governed by the BCS-Hamiltonian. In the presence of classical electromagnetic

¬elds, the free one-particle Hamiltonian is

H0 (t) = dx Ψ† 0 (x, t) h(x, t) ΨH0 (x, t) , (8.52)

H

where

h(1) 0

≡ hij (1) .

h(1) = (8.53)

’h— (1)

0

From the equations of motion for the free Nambu ¬eld

i‚t1 Ψi (1) = hij (1)Ψj (1) (8.54)

and

i‚t1 Ψ† (1) = ’h— (1)Ψ† (1) , (8.55)

ij

i j

where the ¬elds are in the Heisenberg picture with respect to H0 (t), the equations of

motion for the free Nambu Green™s functions become

(i‚t1 ’ h(1))G> (1, 1 ) = 0 (8.56)

0

and

←—

i‚t1 G> (1, 1 ) = ’G> (1, 1 ) h (1 ) , (8.57)

0 0

where the arrow indicates that the spatial di¬erential operator operates to the left.

<

Identical equations of motion are obtained for G0 (1, 1 ).

The presence of the pairing interaction then leads to the appearance of a self-

energy which is purely o¬-diagonal in Nambu space

0 ”(1)

ΣBCS = . (8.58)

—

” (1) 0

In order to get more symmetric equations we perform the transformation

G(1, 1 ) ’ „3 G(1, 1 ) ≡ G , (8.59)

228 8. Non-equilibrium superconductivity

where „3 denotes the third Pauli matrix in Nambu space. The equations of motion

for the free Nambu Green™s functions then become

> >

i„3 ‚t1 G< (1, 1 hN (1)G< (1, 1

)= ) (8.60)

0 0

and

←—

> >

’G< (1, 1

G< (1, 1

i„3 ‚t1 )= ) (1 ) , (8.61)

hN

0 0

where

1 ’2

hN (1) = h(1)„3 = ’ ‚ (1) + eφ(1) ’ μ (8.62)

2m

and ’

‚ (1) = ∇x1 ’ ie„3 A(x1 , t1 ) . (8.63)

The BCS-self-energy, describing the pairing interaction, then becomes

’”(1)

0

ΣBCS = . (8.64)

—

” (1) 0

Introducing the Nambu ¬eld facilitates the description of the particle“hole coher-

ence in a superconductor. Next we introduce the real-time formalism for describing

non-equilibrium states as discussed in Chapter 5, here for the purpose of describing

non-equilibrium superconductivity. For a superconductor this means adding to the

Nambu-indices of Green™s functions the additional Schwinger“Keldysh or dynamical

indices.

Exercise 8.4. Show that, in equilibrium, the retarded Nambu Green™s function has

the form (unit matrices in Nambu space are suppressed)

GR (E, p) = (E „3 ’ ξ(p) ’ ΣR (E, p))’1 . (8.65)

In a strong coupling superconductor the self-energy has, according to the electron“

phonon model, the form

ΣR (E) = (1 ’ Z R (E))E „3 ’ i¦R (E) „1 , (8.66)

and show as a consequence that the retarded Nambu Green™s function becomes

’E Z R (E) „3 ’ ξ(p) ’ i¦R (E) „1

R

G (E, p) = . (8.67)

(ξ(p))2 ’ E 2 (Z R (E))2 + (¦R (E))2

8.1.2 Equations of motion in Nambu“Keldysh space

The contour-ordered Green™s function in Nambu space is de¬ned according to

GC (1, 1 ) = ’i„3 Tct (ΨH (1) Ψ† (1 )) (8.68)

H

and is mapped into real-time dynamical or Schwinger“Keldysh space according to

the usual rule, Eq. (5.1),

G11 (1, 1 ) G12 (1, 1 )

GC (1, 1 ) ’ G(1, 1 ) ≡ , (8.69)

’G21 (1, 1 ) ’G22 (1, 1 )

8.1. BCS-theory 229

where the Schwinger“Keldysh components now are Nambu matrices

Ψ† (1 ))

G11 (1, 1 ) = ’i „3 T (ΨH (1) (8.70)

H

and G-lesser

†

G12 (1, 1 ) = G< (x1 , t1 , x1 , t1 ) = i „3 ψH (x1 , t1 ) ψH (x1 , t1 ) (8.71)

and G-greater

†

G21 (1, 1 ) = G> (x1 , t1 , x1 , t1 ) = ’i „3 ψH (x1 , t1 ) ψH (x1 , t1 ) (8.72)

and

Ψ† (1 ))

˜

G22 (1, 1 ) = ’i „3 T (ΨH (1) (8.73)

H

and the Pauli matrix appears because of the convention, Eq. (8.59).

The information contained in the various Schwinger“Keldysh components of the

matrix Green™s function is rather condensed and it can be useful to have explicit

expressions for two independent components, say G-lesser and G-greater, from which

all other relevant Green™s functions can be constructed.

Exercise 8.5. Show that, in terms of the electron ¬eld, we have

†

ψ‘ (1) ψ‘ (1 ) ψ‘ (1) ψ“ (1 )

G (1, 1 ) = ’i„3

>

(8.74)

† † †

ψ“ (1) ψ‘ (1 ) ψ“ (1) ψ“ (1 )

and

†

ψ‘ (1 ) ψ‘ (1) ψ“ (1 ) ψ‘ (1)

<

G (1, 1 ) = i„3 . (8.75)

† † †

ψ‘ (1 ) ψ“ (1) ψ“ (1 ) ψ“ (1)

The matrix Green™s function on triagonal form

GR GK

G= (8.76)

GA

0

has, according to the construction in Section 5.3, the retarded and advanced compo-

nents

†

’iθ(t ’ t ) „3 [ψH (x, t) , ψH (x , t )]+

GR (x, t, x , t ) =

θ(t ’ t ) G> (x, t, x , t ) ’ G< (x, t, x , t )

= (8.77)

and

†

iθ(t ’ t) „3 [ψH (x, t) , ψH (x , t )]+

GA (x, t, x , t ) =

’θ(t ’ t) G> (x, t, x , t ) ’ G< (x, t, x , t )

= (8.78)

230 8. Non-equilibrium superconductivity

and the Keldysh or kinetic Green™s function

†

’i„3 [ψH (x, t) , ψH (x , t )]’

GK (x, t, x , t ) =

G> (x, t, x , t ) + G< (x, t, x , t ) .

= (8.79)

Exercise 8.6. Write down the retarded, advanced and kinetic components of the

Nambu Green™s function in terms of the electron ¬eld.

The equation of motion, the non-equilibrium Dyson equation, for the matrix

Green™s function becomes

(G’1 — G)(1, 1 ) = δ(1 ’ 1 ) + (Σ — G)(1, 1 ) (8.80)

0

and for the conjugate equation

(G — G’1 )(1, 1 ) = δ(1 ’ 1 ) + (G — Σ)(1, 1 ) , (8.81)

0

where the inverse free matrix Green™s function in Nambu“Keldysh space

G’1 (1, 1 ) = G’1 (1) δ(1 ’ 1 ) (8.82)

0 0

is speci¬ed in the triagonal representation by

G’1 (1) = (i„ (3) ‚t1 ’ h(1)) , (8.83)

0

and

„3 0

„ (3) = (8.84)

0 „3

is the 4—4-matrix, diagonal in Keldysh indices and „3 the third Pauli matrix in

Nambu space. Here

12

h(1) = ’ ‚ (1) + eφ(1) ’ μ , (8.85)

2m

where

‚(1) = ∇x1 ’ ie„ (3) A(x1 , t1 ) . (8.86)

Written out in components the matrix equation, Eq. (8.80), this gives

G’1 (1) GR(A) (1, 1 ) = δ(1 ’ 1 ) + (ΣR(A) — GR(A) )(1, 1 ) (8.87)

0

and

G’1 (1) GK (1, 1 ) = (ΣR — GK )(1, 1 ) + (ΣK — GA )(1, 1 ) . (8.88)

0

Subtracting the left and right Dyson equations, Eq. (8.80) and Eq. (8.81), we

obtain an equation identical in form to that of the normal state, Eq. (7.1), an equation

for the spectral weight function and a quantum kinetic equation of the form Eq. (7.3).

However, they are additionally matrix equations in Nambu space. Generally they are

8.1. BCS-theory 231

too complicated to be analytically tractable. It is therefore of importance that the

quasi-classical approximation works for the superconducting state, at least excellently

in low temperature superconductors where the superconducting coherence length,

ξ0 = vF /π”, is much longer than the Fermi wavelength. In other words, the small

distance information in the above equations is irrelevant and should be removed.

After discussing the gauge transformation properties of the Nambu Green™s functions,

we turn to describe the quasi-classical theory of non-equilibrium superconductors

which precisely does that.13

8.1.3 Green™s functions and gauge transformations

The ¬eld operator representing a charged particle transforms according to

˜

ψ(x, t) ’ ψ(x, t) eieΛ(x,t) ≡ ψ(x, t) (8.89)

under the gauge transformation

‚Λ(x, t)

•(x, t) ’ •(x, t) + A(x, t) ’ A(x, t) ’ ∇x Λ(x, t) .

, (8.90)

‚t

The probability and current density of the particles will be invariant to this shift;

quantum mechanics is gauge invariant.

The matrix Green™s function therefore transforms according to

†

eie(Λ(1)’Λ(1 )) ψ‘ (1 )ψ‘ (1) eie(Λ(1)+Λ(1 )) ψ“ (1 )ψ‘ (1)

˜

G< (1, 1 ) =i † † †

’e’ie(Λ(1)+Λ(1 )) ψ‘ (1 )ψ“ (1) ’e’ie(Λ(1)’Λ(1 )) ψ“ (1 )ψ“ (1)

= eieΛ(1)„3 G< (1, 1 ) e’ieΛ(1 )„3 (8.91)

and similarly for

†