Carsten Timm: Theory of superconductivity

a restriction of our variational ansatz. Then we obtain
⟨ψ BCS |H| ψ BCS ⟩ = ∑ kσ
+ 1 N
ξ k ⟨0| ∏ q
∑
kk ′ V kk ′ ⟨0| ∏ q
(
u
∗
q + v ∗ q c −q,↓ c q↑
)
c
†
kσ c kσ
∏ (
)
u q ′ + v q ′ c † q ′ ↑ c† −q ′ ,↓
|0⟩
q ′
(
u
∗
q + v ∗ q c −q,↓ c q↑
)
c
†
k↑ c† −k,↓ c −k ′ ↓c k′ ↑
∏ (
)
u q ′ + v q ′ c † q ′ ↑ c† −q ′ ,↓
|0⟩
q ′
= ∑ ξ k ⟨0| |v k | 2 c −k,↓ c k↑ c † k↑ c k↑c † k↑ c† −k,↓ |0⟩ + ∑ ξ k ⟨0| |v −k | 2 c k↓ c −k,↑ c † k↓ c k↓c † −k,↑ c† k↓ |0⟩
k
k
+ 1 ∑
V kk ′ ⟨0| v
N
ku ∗ ∗ k ′u kv k ′ c −k,↓ c k↑ c † k↑ c† −k,↓ c −k ′ ,↓c k′ ↑c † k ′ ↑ c† −k ′ ,↓ |0⟩
kk ′
= ∑ 2ξ k |v k | 2 + 1 ∑
V kk ′ v
N
ku ∗ k u ∗ k ′v k ′ =: E BCS. (9.20)
k
kk ′
This energy should be minimized with respect to the u k , v k . For E BCS to be real, the phases of u k and v k must
be the same. But since E BCS is invariant under
u k → u k e iφ k
, v k → v k e iφ k
, (9.21)
we can choose all u k , v k real. The constraint from normalization then reads u 2 k + v2 k
= 1 and we can parametrize
the coefficients by
u k = cos θ k , v k = sin θ k . (9.22)
Then
We obtain the minimum from
∂E BCS
∂θ q
We replace q by k and parametrize θ k by
E BCS = ∑ 2ξ k sin 2 θ k + 1 ∑
V kk ′ sin θ k cos θ k sin θ k ′ cos θ k ′
N
k
kk ′
= ∑ ξ k (1 − cos 2θ k ) + 1 ∑ V kk ′
sin 2θ k sin 2θ k ′. (9.23)
N 4
k
kk ′
= 2ξ q sin 2θ q + 1 ∑ V qk ′
cos 2θ q sin 2θ k ′ + 1 ∑
N 2
N
k ′ k
= 2ξ q sin 2θ q + 1 ∑
V qk ′ cos 2θ q sin 2θ k ′
N
k ′
V kq
2 sin 2θ k cos 2θ q
!
= 0. (9.24)
sin 2θ k =:
∆ k
√
ξ
2
k
+ ∆ 2 k
(9.25)
and write
cos 2θ k =
ξ
√ k
. (9.26)
ξ
2
k
+ ∆ 2 k
The last equality is only determined by the previous one up to the sign. We could convince ourselves that the
other possible choice does not lead to a lower E BCS . Equation (9.24) now becomes
⇒
2
ξ k ∆
√ k
+ 1 ∑
ξ
2
k
+ ∆ 2 N
k
∑
∆ k = − 1 N
k ′ V kk ′
k ′ V kk ′
ξ k ∆ k ′
√
ξ
2
k
+ ∆ 2 k√
ξ
2
k ′ + ∆ 2 k ′ = 0 (9.27)
∆ k ′
2 √ ξ 2 k ′ + ∆ 2 k ′ . (9.28)
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