Carsten Timm: Theory of superconductivity

we can write the gap matrix in a compact form as
ˆ∆(k) = (∆ k 1 + d(k) · σ) iσ y , (12.63)
where σ is the vector of Pauli matrices. The mean-field Hamiltonian H MF is diagonalized by a Bogoliubov
transformation and the gap function ˆ∆(k) is obtained selfconsistently from a gap equation in complete analogy
to the singlet case discussed in Sec. 10.1, except that ˆ∆(k) is now a matrix and that we require four coefficients
u k↑ , u k↓ , v k↑ , v k↓ . We do not show this here explicitly.
A few remarks on the physics are in order, though. The symmetry ˆ∆(−k) = − ˆ∆ T (k) implies
⇒
(∆ −k 1 + d(−k) · σ) iσ y = −i (σ y ) T
} {{ }
(
∆k 1 + d(k) · σ T ) (12.64)
iσ y
∆ −k 1 + d(−k) · σ = σ y ( ∆ k 1 + d(k) · σ T ) σ y
⎛
⎞
= ∆ k 1 + d(k) · ⎝ σy (σ x ) T σ y
σ y (σ y ) T σ y ⎠
⎛
σ y (σ z ) T σ y
⎞
σy σ x σ y
= ∆ k 1 + d(k) · ⎝ −σ y σ y σ y
σ y σ z σ y
⎠
= ∆ k 1 − d(k) · σ. (12.65)
Thus we conclude that ∆ k is even,
whereas d(k) is odd,
∆ −k = ∆ k , (12.66)
d(−k) = −d(k). (12.67)
Hence, the d-vector can never be constant, unlike the single gap in Sec. 10.1. Furthermore, we can expand ∆ k
into even basis functions of rotations in real (and k) space,
∆ k = ∆ s ψ s (k) + ∆ dx 2 −y 2
ψ dx 2 −y 2
(k) + ∆ d3z 2 −r 2
ψ d3z 2 −r 2
(k) + ∆ dxy ψ dxy (k) + ∆ dyz ψ dyz (k) + ∆ dzx ψ dzx (k) + . . . ,
(12.68)
and expand (the components of) d(k) into odd basis functions,
d(k) = d px ψ px (k) + d py ψ py (k) + d pz ψ pz (k) + . . . (12.69)
For the singlet case we have already considered the basis functions ψ s (k) = 1 for conventional superconductors,
ψ s (k) = cos k x a cos k y a for the pnictides, and ψ dx 2 −y 2
(k) = cos k x a − cos k y a for the cuprates. A typical basis
function for a triplet superconductor would be ψ px = sin k x a. He-3 in the so-called B phase realized at now too
high pressures (see Sec. 2.2) has the d-vector
d(k) = ˆx ψ px (k) + ŷ ψ py (k) + ẑ ψ pz (k), (12.70)
where the basis functions are that standard expressions for p x , p y , and p z orbitals (note that there is no Brillouin
zone since He-3 is a liquid),
√
3
ψ px (k) =
4π sin θ k cos ϕ k , (12.71)
√
3
ψ py (k) =
4π sin θ k sin ϕ k , (12.72)
√
3
ψ pz (k) =
4π cos θ k. (12.73)
This is the so-called Balian-Werthamer state.
134