6 p-Wave Superconductivity and d-Vector Representation
193
6.9.2 Symmetries
Only two one-dimensional representations are left, called A and B, due to the low
orthorhombic symmetry. In the paramagnetic state, the first one (A) looks very much
like the B-phase of superfluid
3 He
d A (k) ∝ u x ˆ
k x e x + u y ˆ
k y e y + u z ˆ
k z e z ,
(6.46)
where u x , u y , u z are real functions of k with full orthorhombic symmetry. So in the
paramagnetic state, we start from
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
↑
A = −u x ˆ
k x + iu y ˆ
k y
↓
A = +u x ˆ
k x + iu y ˆ
k y
0
A = u z ˆ
k z
.
(6.47)
In the ferromagnetic state, the amplitude (and possibly the phase) of the
↑
A and
↓
A components can differ. So the order parameter should read
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
↑
A = η
↑
(−u x ˆ
k x + iu y ˆ
k y ) = −η
↑
x
ˆ
k x + iη
↑
y
ˆ
k y
↓
A = η
↓
(+u x ˆ
k x + iu y ˆ
k y ) = η
↓
x
ˆ
k x + iη
↓
y
ˆ
k y
0
A = η
0 u z ˆ
k z = η
0
z
ˆ
k z
(6.48)
with no phase difference between the complex amplitudes η
↑ , η
↓ , η
0 . In terms of
d-vectors
d =
1
2ψ
[−
↑
( ˆ
n)(e x + ie y ) +
↓
( ˆ
n)(e x − ie y )] +
0 e z
=
1
2ψ
[(η
↑
x
ˆ
k x − iη
↑
y
ˆ
k y )(e x + ie y ) + (η
↓
x
ˆ
k x + iη
↓
y
ˆ
k y )(e x − ie y )] + η
0
z
ˆ
k z e z
=
1
2ψ
η
↑
x + η
↓
x
ˆ
k x − i
η
↑
y − η
↓
y
ˆ
k y
e x
+
η
↑
y + η
↓
y
ˆ
k y + i
η
↑
x − η
↓
x
ˆ
k x
e y + 2η
0
z
ˆ
k z e z
.
(6.49)
Similarly for the B-phase, the order parameter reads
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
↑
B = ζ
↑
z
ˆ
k z
↓
B = ζ
↓
z
ˆ
k z
0
B = ζ
0
x
ˆ
k x + iζ
0
y
ˆ
k y
(6.50)