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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)
|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“
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

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

2ξp tan 2φp = Vpp sin 2φp . (8.7)

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

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

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

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 ξp
BCS| Hpairing ’ |BCS =’ ξp ’
FN + (8.11)
V Ep

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)

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

1 ξp 1 ξp ”p
u2 = 2
1+ , vp = , up vp = . (8.16)
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
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
† †
γp‘ γp‘ = E /kT = γ’p“ γ’p“ . (8.17)
e +1

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

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)

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)

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
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
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)
’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 )
‚t 2m

= δ(x ’ x )δ(t ’ t ) (8.29)

‚ 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
† † 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)) ,

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))⎠ ,

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