Carsten Timm: Theory of superconductivity

This gap is said to have s-wave symmetry in that it does not have lower symmetry than the lattice, unlike d-wave.
To emphasize the sign change, it is often called an s ± -wave gap. The simplest realization would be
12.4 Triplet superconductors and He-3
∆ k = ∆ 0 cos k x a cos k y a. (12.38)
So far, we have assumed that Cooper pairs are formed by two electrons with opposite spin so that the total
spin of pair vanishes (spin-singlet pairing). This assumption becomes questionable in the presence of strong
ferromagnetic interactions, which favor parallel spin alignment. If superconductivity is possible at all in such a
situation, we could expect to find spin-1 Cooper pairs. Since they would be spin triplets, one is talking of triplet
superconductors. This scenario is very likely realized in Sr 2 RuO 4 (which is, interestingly, isostructural to the
prototypical cuprate La 2 CuO 4 ), a few organic salts, and some heavy-fermion compounds. It is even more certain
to be responsible for the superfluidity of He-3, where neutral He-3 atoms instead of charged electrons form Cooper
pairs, see Sec. 2.2.
Formally, we restrict ourselves to a BCS-type mean-field theory. We generalize the effective interaction to
allow for an arbitrary spin dependence,
H = ∑ ξ k c † kσ c kσ + 1 ∑ ∑
V στσ′ τ
N
′(k, k′ ) c † kσ c† −k,τ c −k ′ ,τ ′c k ′ σ ′, (12.39)
kσ
kk ′ στσ ′ τ ′
where σ, τ, σ ′ , τ ′ =↑, ↓ are spin indices. In decomposing the interaction, we now allow the averages ⟨c −k,τ c kσ ⟩ to
be non-zero for all τ, σ. Thus the mean-field Hamiltonian reads
H MF = ∑ kσ
We define
so that
ξ k c † kσ c kσ + 1 N
∑
kk ′
∑
στσ ′ τ ′ V στσ′ τ ′(k, k′ )
k ′
(⟨ ⟩ )
c † kσ c† −k,τ
c −k′ ,τ ′c k ′ σ ′ + c† kσ c† −k,τ ⟨c −k ′ ,τ ′c k ′ σ ′⟩ + const.
(12.40)
∆ στ (k) := − 1 ∑ ∑
V στσ′ τ
N
′(k, k′ ) ⟨c −k′ ,τ ′c k ′ σ ′⟩ (12.41)
σ ′ τ ′
∆ ∗ στ (k) = − 1 N
∑ ∑
k ′
σ ′ τ ′ V σ′ τ ′ στ (k ′ , k)
⟨c † k ′ σ ′ c † −k ′ ,τ ′ ⟩
. (12.42)
Here, we have used that [
Vστσ ′ τ ′(k, k′ ) ] ∗
= Vσ ′ τ ′ στ (k ′ , k), (12.43)
which follows from hermiticity of the Hamiltonian H. Then
H MF = ∑ kσ
ξ k c † kσ c kσ − ∑ kστ
∆ ∗ στ (k) c −k,τ c kσ − ∑ ∆ στ (k) c † kσ c† −k,τ
+ const. (12.44)
kστ
The gap function ∆ στ (k) is now a matrix in spin space,
(
∆↑↑ (k) ∆ ˆ∆(k) = ↑↓ (k)
∆ ↓↑ (k) ∆ ↓↓ (k)
)
. (12.45)
The function ˆ∆(k) has an important symmetry property that follows from the symmetry of averages ⟨c −k,τ c kσ ⟩:
It is clear that
⟨c kσ c −k,τ ⟩ = −⟨c −k,τ c kσ ⟩. (12.46)
Furthermore, by relabeling σ ↔ τ, σ ′ ↔ τ ′ , k → −k, k ′ → −k ′ in the interaction term of H, we see that the
interaction strength must satisfy the relation
V στσ ′ τ ′(k, k′ ) = V τστ ′ σ ′(−k, −k′ ). (12.47)
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