Carsten Timm: Theory of superconductivity

10
BCS theory
The variational ansatz of Sec. 9.2 has given us an approximation for the many-particle ground state |ψ BCS ⟩. While
this is interesting, it does not yet allow predictions of thermodynamic properties, such as the critical temperature.
We will now consider superconductors at non-zero temperatures within mean-field theory, which will also provide
a new perspective on the BCS gap equation and on the meaning of ∆ k .
10.1 BCS mean-field theory
We start again from the Hamiltonian
H = ∑ kσ
ξ k c † kσ c kσ + 1 N
∑
kk ′ V kk ′ c † k↑ c† −k,↓ c −k ′ ,↓c k′ ↑. (10.1)
A mean-field approximation consists of replacing products of operators A, B according to
Note that the error introduced by this replacement is
AB ∼ = ⟨A⟩ B + A ⟨B⟩ − ⟨A⟩ ⟨B⟩ . (10.2)
AB − ⟨A⟩ B − A ⟨B⟩ + ⟨A⟩ ⟨B⟩ = (A − ⟨A⟩)(B − ⟨B⟩), (10.3)
i.e., it is of second order in the deviations of A and B from their averages. A well-known mean-field approximation
is the Hartree or Stoner approximation, which for our Hamiltonian amounts to the choice A = c † k↑ c k ′ ↑, B =
c † −k,↓ c −k ′ ,↓. However, Bardeen, Cooper, and Schrieffer realized that superconductivity can be understood with
the help of a different choice, namely A = c † k↑ c† −k,↓ , B = c −k ′ ,↓c k′ ↑. This leads to the mean-field BCS Hamiltonian
H BCS = ∑ ξ k c † kσ c kσ + 1 ∑ (
)
V kk ′ ⟨c † k↑
N
c† −k,↓ ⟩ c −k ′ ,↓c k′ ↑ + c † k↑ c† −k,↓ ⟨c −k ′ ,↓c k′ ↑⟩ − ⟨c † k↑ c† −k,↓ ⟩⟨c −k ′ ,↓c k′ ↑⟩ .
kσ
kk ′ (10.4)
We define
∆ k := − 1 ∑
V kk ′ ⟨c −k
N
′ ,↓c k ′ ↑⟩ (10.5)
k ′
so that
∆ ∗ k = − 1 ∑
V kk ′ ⟨c † k
N
′ ↑ c† −k ′ ,↓⟩. (10.6)
k ′
At this point it is not obvious that the quantity ∆ k is the same as the one introduced in Sec. 9.2 for the special
case of the ground state. Since this will turn out to be the case, we nevertheless use the same symbol from the
start. We can now write
H BCS = ∑ kσ
ξ k c † kσ c kσ − ∑ k
∆ ∗ k c −k,↓ c k↑ − ∑ k
∆ k c † k↑ c† −k,↓
+ const. (10.7)
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