6 p-Wave Superconductivity and d-Vector Representation
175
ˆ
=
⎛
⎝
cos ϕ
sin ϕ
0
⎞
⎠ ,
R (d) = sin(π/2) ˆ
∧ d =
1
ψ
⎛
⎝
sin ϕϕ 0
− cos ϕϕ 0
0
⎞
⎠ ,
↑ = ψ(−d x + id y ) = −ie
iϕ
0 ; ↓ = ψ(d x + id y ) = −ie
−iϕ
0 ,
which is indeed an ESP state with only ↑↑ and ↓↓ spin components. It is a good
exercise to check that, reciprocally, any ESP state with equal weight for the up- and
down-spin components can also be written as a pure S z = 0 state for some choice of
the quantization axis.
Exercise 6.2 Show that any ESP state with equal weight for the up- and down-spin
component can also be written as a pure |S z = 0 state. Solution in Sect. 6.11.
6.6 Some Uses of the d-Vector Representation
6.6.1 Amplitude of the d-Vector
As promised, let us say a few words on the question of normalization of the d-vector.
For s-wave superconductors, in the simplest cases, we know that the order parameter
can be taken as proportional to the gap. Of course, this is wrong in the general
case, e.g. gapless superconductivity exists (induced by a critical amount of magnetic
impurities for example). But the idea is that |ψ|
2 somehow represents the superfluid
density. For a spin-triplet superconductor, we can define this quantity as
| =
d
4π
α,β
ϕ
∗
βα ϕ αβ =
d
4π
tr(ϕ
∗
( ˆ
n)ϕ( ˆ
n))
= |ψ|
2
d
4π
tr(σ 2 (d
∗
· σ )(d · σ )σ 2 ) = 2|ψ|
2
d
4π
|d( ˆ
n)|
2
.
Note that the definition above is coherent with the fact that from the very beginning,
we did not normalize (by
1
√
2
) the | ↑↑ + ↓↓↓ component of |ψ in (6.3). |d( ˆ
n)|
2 can
be interpreted as the angular-dependent superconducting (or superfluid) density, and
by convention, one takes
d
4π
|d( ˆ
n)|
2
= 1 .
(6.23)
So on calculating averaged quantities |O|/|, one should remember that
| = 2|ψ|
2 .
175
ˆ
=
⎛
⎝
cos ϕ
sin ϕ
0
⎞
⎠ ,
R (d) = sin(π/2) ˆ
∧ d =
1
ψ
⎛
⎝
sin ϕϕ 0
− cos ϕϕ 0
0
⎞
⎠ ,
↑ = ψ(−d x + id y ) = −ie
iϕ
0 ; ↓ = ψ(d x + id y ) = −ie
−iϕ
0 ,
which is indeed an ESP state with only ↑↑ and ↓↓ spin components. It is a good
exercise to check that, reciprocally, any ESP state with equal weight for the up- and
down-spin components can also be written as a pure S z = 0 state for some choice of
the quantization axis.
Exercise 6.2 Show that any ESP state with equal weight for the up- and down-spin
component can also be written as a pure |S z = 0 state. Solution in Sect. 6.11.
6.6 Some Uses of the d-Vector Representation
6.6.1 Amplitude of the d-Vector
As promised, let us say a few words on the question of normalization of the d-vector.
For s-wave superconductors, in the simplest cases, we know that the order parameter
can be taken as proportional to the gap. Of course, this is wrong in the general
case, e.g. gapless superconductivity exists (induced by a critical amount of magnetic
impurities for example). But the idea is that |ψ|
2 somehow represents the superfluid
density. For a spin-triplet superconductor, we can define this quantity as
| =
d
4π
α,β
ϕ
∗
βα ϕ αβ =
d
4π
tr(ϕ
∗
( ˆ
n)ϕ( ˆ
n))
= |ψ|
2
d
4π
tr(σ 2 (d
∗
· σ )(d · σ )σ 2 ) = 2|ψ|
2
d
4π
|d( ˆ
n)|
2
.
Note that the definition above is coherent with the fact that from the very beginning,
we did not normalize (by
1
√
2
) the | ↑↑ + ↓↓↓ component of |ψ in (6.3). |d( ˆ
n)|
2 can
be interpreted as the angular-dependent superconducting (or superfluid) density, and
by convention, one takes
d
4π
|d( ˆ
n)|
2
= 1 .
(6.23)
So on calculating averaged quantities |O|/|, one should remember that
| = 2|ψ|
2 .