Carsten Timm: Theory of superconductivity

It is instructive to first consider the normal state, for which ∆ k → 0. Then
{
0 for ξ k < 0,
|u k | 2 = 1 2
|v k | 2 = 1 2
(
1 + ξ )
k
|ξ k |
(
1 − ξ )
k
|ξ k |
=
1 for ξ k > 0,
{
1 for ξ k < 0,
=
0 for ξ k > 0.
(10.30)
(10.31)
We see that the Bogoliubov quasiparticles described by γ, γ † , are holes for energies below the Fermi energy
(ξ k < 0) and electrons for energies above (ξ k > 0). Their dispersion is E k = |ξ k |. For a parabolic normal
dispersion ξ k :
E k
hole
excitations
electron
excitations
0 k F
k
The excitation energies E k are always positive except at the Fermi surface—it costs energy to create a hole in
the Fermi sea and also to insert an electron into an empty √ state outside of the Fermi sea.
Superconductivity changes the dispersion to E k = ξk 2 + |∆ k| 2 :
E k
∆ kF
0 k F
k
Superconductivity evidently opens an energy gap of magnitude |∆ kF | in the excitation spectrum.
We should recall that in deriving H BCS we have ignored a constant, which we now reinsert,
H BCS = E BCS + ∑ kσ
E k γ † kσ γ kσ. (10.32)
The energy of the system is E BCS if no quasiparticles are present and is increased (by at least |∆ kF |) if quasiparticles
are excited. The state without any quasiparticles is the pure condensate. Since E BCS depends on temperature
through ⟨c −k,↓ c k↑ ⟩, the condensate is not generally the BCS ground state discussed previously. However, one can
show that it agrees with the ground state in the limit T → 0.
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