Bardeen-Cooper-Schrieffer (BCS) Theory
In a Nobel prize-winning paper Bardeen, Cooper and Schrieffer used a variational wavefunction to explain the superconducting state of metals at zero temperature. Consider the
following second-quantized Hamiltonian for a two-component Fermi gas (the two components are labelled by a spin index α =↑, ↓) with an attractive interaction V0 < 0 between
electrons with opposite spins, i.e.
Ĥ =
X
†
ψ̂k,α +
k ψ̂k,α
k,α
V0 X †
†
ψ̂K/2+k0 ,↑ ψ̂K/2−k
0 ,↓ ψ̂K/2−k,↓ ψ̂K/2+k,↑ .
V K,k,k0
(1)
The vector K denotes the momentum of the center of mass. We determine the groundstate energy using the wavefunction
|ΨBCS i =
Y
uk +
†
†
vk ψ̂k,↑
ψ̂−k,↓
|0i ,
(2)
k
where uk , vk are variational parameters that we for simplicity take to be real. This wavefunction describes physically a Bose-Einstein condensate of fermion pairs with opposite
spin and momentum, called Cooper pairs.
(a) Show that |ΨBCS i is normalized if u2k + vk2 = 1.
(b) Show that the normalized BCS wavefunction leads to the following expectation
values
†
hΨBCS |ψ̂k,↑
ψ̂k,↑ |ΨBCS i = vk2 ,
(3)
†
2
ψ̂k,↓ |ΨBCS i = v−k
,
hΨBCS |ψ̂k,↓
(4)
hΨBCS |ψ̂k,↓ ψ̂−k,↑ |ΨBCS i = u−k v−k .
(5)
The expectation value hΨBCS |Ĥ − µN̂ |ΨBCS i gives rise to the following terms (DO NOT
PROVE IT!), namely
hΨBCS |Ĥ − µN̂ |ΨBCS i = 2
X
(k − µ) vk2 +
k
V0 X
uk vk uk0 vk0 + . . . ,
V k,k0
(6)
where we do not consider any other possible terms, N̂ is the operator for the total number
of atoms and µ is the chemical potential.
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(c) Motivated by (a), we may write uk = sin θk and vk = cos θk . Express hΨBCS |Ĥ −
µN̂ |ΨBCS i as obtained from (6) in terms of θk , and minimize this result to find
0 = (k − µ) sin 2θk + cos 2θk ∆ ,
(7)
where ∆, which is also called the gap parameter, obeys the gap equation
∆=−
V0 X
sin 2θk0 .
2V k0
(8)
(d) Solve for θk and show that the gap equation can be rewritten as
1
1
1 X
p
.
=−
V0
V k 2 (k − µ)2 + ∆2
(9)
Later you will learn that the gap ∆ is interpreted as the energy needed to break up
a Cooper pair, which means that it costs a certain amount of energy to excite the BCS
ground state. As a result, an object moving through a condensate of Cooper pairs needs
to have a minimum kinetic energy to transfer momentum, i.e. to experience friction.
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