182
J.-P. Brison
6.8 d-vector Representation of Some Known (or Suspected)
p-Wave Superconductors
6.8.1 Phases of Superfluid 3 He
3 He has been the very first case for p-wave superconductivity (superfluidity, to be
precise!), and it is, beyond contest, a true paradigm for this state of matter. The reasons
are that, despite its very low superfluid transition temperature (≈ 1 mK on the melting
curve), the spin state and most of the superfluid properties could be identified and
studied with tremendous precision thanks to nuclear magnetic resonance (NMR): in
superfluid
3 He, the Cooper pairs spin is the nuclear spin of the
3 He atoms, which are
directly probed by NMR. Moreover, the system is rotationally invariant, so with the
simplest (spherical) possible Fermi surface, and with spherical harmonics as basis for
the irreducible representations of the superconducting order parameter. In k-space,
for a p-wave state, they read
Y 11 ( ˆ
k) = −
3
2
( ˆ
k x + i ˆ
k y ) ; Y 1−1 ( ˆ
k) =
3
2
( ˆ
k x − i ˆ
k y ) ; Y 10 ( ˆ
k) = −
√
3 ˆ
k z .
(6.32)
It can be explicitly checked that [see (6.22)]
ˆ
k z · L k Y 1m = ˆ
k z · (k ∧
1
i
∇ k Y 1m ) = m Y 1m .
(6.33)
Another reason for which superfluid
3 He is a paradigm of p-wave superconductors
is that it presents a rich phase diagram (shown in Fig. 6.3
1 ), with three well-identified
phases: two (called A and B) in the temperature–pressure plane, and an additional
(A 1 ) phase under magnetic fields, which are key references. A main topic these days is
that of the topological properties of some superconductors, which has been addressed
in great detail some tens of years ago for superfluid
3 He [5]. Let us examine them
quickly.
6.8.1.1 B-Phase
The B-phase of superfluid
3 He is simply characterized by
d(k) = ˆ
k ; or explicitely
|( ˆ
k) = ψ
(− ˆ
k x + i ˆ
k y )| ↑↑↑ + ( ˆ
k x + i ˆ
k y )| ↓↓↓ + ˆ
k z (| ↑↓↓ + | ↓↑↑)
.
(6.34)
1 Adapted from ‘Heliums egenskaper vid låga temperaturer’, P. Berglund, Kosmos 1988, s. 63
(Courtesy of the Swedish Physical Society).
J.-P. Brison
6.8 d-vector Representation of Some Known (or Suspected)
p-Wave Superconductors
6.8.1 Phases of Superfluid 3 He
3 He has been the very first case for p-wave superconductivity (superfluidity, to be
precise!), and it is, beyond contest, a true paradigm for this state of matter. The reasons
are that, despite its very low superfluid transition temperature (≈ 1 mK on the melting
curve), the spin state and most of the superfluid properties could be identified and
studied with tremendous precision thanks to nuclear magnetic resonance (NMR): in
superfluid
3 He, the Cooper pairs spin is the nuclear spin of the
3 He atoms, which are
directly probed by NMR. Moreover, the system is rotationally invariant, so with the
simplest (spherical) possible Fermi surface, and with spherical harmonics as basis for
the irreducible representations of the superconducting order parameter. In k-space,
for a p-wave state, they read
Y 11 ( ˆ
k) = −
3
2
( ˆ
k x + i ˆ
k y ) ; Y 1−1 ( ˆ
k) =
3
2
( ˆ
k x − i ˆ
k y ) ; Y 10 ( ˆ
k) = −
√
3 ˆ
k z .
(6.32)
It can be explicitly checked that [see (6.22)]
ˆ
k z · L k Y 1m = ˆ
k z · (k ∧
1
i
∇ k Y 1m ) = m Y 1m .
(6.33)
Another reason for which superfluid
3 He is a paradigm of p-wave superconductors
is that it presents a rich phase diagram (shown in Fig. 6.3
1 ), with three well-identified
phases: two (called A and B) in the temperature–pressure plane, and an additional
(A 1 ) phase under magnetic fields, which are key references. A main topic these days is
that of the topological properties of some superconductors, which has been addressed
in great detail some tens of years ago for superfluid
3 He [5]. Let us examine them
quickly.
6.8.1.1 B-Phase
The B-phase of superfluid
3 He is simply characterized by
d(k) = ˆ
k ; or explicitely
|( ˆ
k) = ψ
(− ˆ
k x + i ˆ
k y )| ↑↑↑ + ( ˆ
k x + i ˆ
k y )| ↓↓↓ + ˆ
k z (| ↑↓↓ + | ↓↑↑)
.
(6.34)
1 Adapted from ‘Heliums egenskaper vid låga temperaturer’, P. Berglund, Kosmos 1988, s. 63
(Courtesy of the Swedish Physical Society).