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

† † i

F N (x, t+ ; x , t) = ’i N + 2|ψ“ (x, t) ψ‘ (x , t)|N = e2 μt

F μ (x, t; x , t)

the term ’2μ F N (x, t; x , t ) would appear on the left in Eq. (8.30).

8.1. BCS-theory 225

8.1.1 Nambu or particle“hole space

In order to write the BCS-Hamiltonian, Eq. (8.24), in standard quadratic form of a

¬eld, we introduce with Nambu the pseudo-spinor ¬eld

ψ‘ (1) Ψ1 (1)

Ψ(1) ≡ ≡ , (8.32)

†

Ψ2 (1)

ψ“ (1)

where, introducing condensed notation, 1 ≡ (t1 , x1 ) comprises the spatial variable

and in the Heisenberg picture also the time variable. The adjoint Nambu ¬eld is

Ψ† (1) ≡ (ψ‘ (1), ψ“ (1)) ≡ (Ψ† (1), Ψ† (1)) ,

†

(8.33)

1 2

where in the last de¬nition we have introduced the matrix notation for the Nambu

or particle“hole space.

The BCS-Hamiltonian is

⎛ ⎞

HBCS = dx1 ⎝ ψσ (1)h(1)ψσ (1) + ”(1)ψ‘ (1)ψ“ (1) + ”— (1)ψ“ (1)ψ‘ (1))⎠ ,