180
J.-P. Brison
of the Cooper pair (an object of a coherence length scale) and the total spin of the
Cooper pairs can be much weaker. And in real solid, it is very difficult to either
calculate (predict) or to measure this spin–orbit interaction. This question is very
important because it determines the symmetry group which has to be considered for
the classification of the different superconducting states. If spin–orbit is weak, the
relative orientation of spin and orbit should be decoupled: so, for example, one can
imagine that the spin could reorient ‘freely’ under the action of an external field.
If spin–orbit is strong, the orbital state (the gap nodes for example) expected to
be pinned on the crystal lattice will prevent such a reorientation of the spin state.
Therefore
• If the spin–orbit interaction is weak, the symmetry group considered for the classification of the possible superconducting states will be G ⊗ U (1) ⊗ T ⊗ R S ,
where R S are the (3D) rotations in spin space, G is the crystal point group, U (1)
the gauge symmetry (always broken in the superconducting state) and T the timereversal symmetry. Due to R S , d should reorient under field to minimize the
Zeeman energy.
• If the spin–orbit interaction is strong, the symmetry group considered for the classification of the possible superconducting states will be G ⊗ U (1) ⊗ T , meaning
that the d-vector is expected to be ‘pinned’ on the lattice. In such a case, additional spin anisotropy may appear in the superconducting state, possibly detected,
for example, by an anisotropy of the Knight-shift reduction below T SC , or by an
anisotropic paramagnetic limitation.
For most of the candidate p-wave superconductors, determining what is the best
of the two limits for the description of the system remains an open issue (see, for
example, the discussion on UPt 3 in Sects. 6.8.2.2 and 6.8.2.3).
6.7.2 Anisotropy of the Susceptibility for the Strong
Spin–Orbit Case
Experimentally, an important question when analysing the behaviour of a potential
triplet superconductor is the Pauli depairing effect and its anisotropy on the upper critical field, or equivalently, the anisotropy of the change of the Knight shift below T SC ,
both of which depending on the Cooper pairs spin susceptibility. Supposing that we
are in the strong spin–orbit limit, the question is to derive from the possible order
parameters, in which directions there will be no change of the electronic spin susceptibility between normal and superconducting phases, and in which directions, if
any, there will be at least a partial suppression of this spin susceptibility.
As a matter of fact, it is important to realize that even spin-triplet superconductors,
whatever the spin–orbit regime, can present a reduction of the susceptibility for all
orientations of the magnetic field. We will see below (Sect. 6.8.1) that superfluid
3 He
realize, in its B-phase, an A 1u state for which d ∝ k. This means that, on each point
of the Fermi surface, the order parameter is described by a pure |S z = 0 state if the
J.-P. Brison
of the Cooper pair (an object of a coherence length scale) and the total spin of the
Cooper pairs can be much weaker. And in real solid, it is very difficult to either
calculate (predict) or to measure this spin–orbit interaction. This question is very
important because it determines the symmetry group which has to be considered for
the classification of the different superconducting states. If spin–orbit is weak, the
relative orientation of spin and orbit should be decoupled: so, for example, one can
imagine that the spin could reorient ‘freely’ under the action of an external field.
If spin–orbit is strong, the orbital state (the gap nodes for example) expected to
be pinned on the crystal lattice will prevent such a reorientation of the spin state.
Therefore
• If the spin–orbit interaction is weak, the symmetry group considered for the classification of the possible superconducting states will be G ⊗ U (1) ⊗ T ⊗ R S ,
where R S are the (3D) rotations in spin space, G is the crystal point group, U (1)
the gauge symmetry (always broken in the superconducting state) and T the timereversal symmetry. Due to R S , d should reorient under field to minimize the
Zeeman energy.
• If the spin–orbit interaction is strong, the symmetry group considered for the classification of the possible superconducting states will be G ⊗ U (1) ⊗ T , meaning
that the d-vector is expected to be ‘pinned’ on the lattice. In such a case, additional spin anisotropy may appear in the superconducting state, possibly detected,
for example, by an anisotropy of the Knight-shift reduction below T SC , or by an
anisotropic paramagnetic limitation.
For most of the candidate p-wave superconductors, determining what is the best
of the two limits for the description of the system remains an open issue (see, for
example, the discussion on UPt 3 in Sects. 6.8.2.2 and 6.8.2.3).
6.7.2 Anisotropy of the Susceptibility for the Strong
Spin–Orbit Case
Experimentally, an important question when analysing the behaviour of a potential
triplet superconductor is the Pauli depairing effect and its anisotropy on the upper critical field, or equivalently, the anisotropy of the change of the Knight shift below T SC ,
both of which depending on the Cooper pairs spin susceptibility. Supposing that we
are in the strong spin–orbit limit, the question is to derive from the possible order
parameters, in which directions there will be no change of the electronic spin susceptibility between normal and superconducting phases, and in which directions, if
any, there will be at least a partial suppression of this spin susceptibility.
As a matter of fact, it is important to realize that even spin-triplet superconductors,
whatever the spin–orbit regime, can present a reduction of the susceptibility for all
orientations of the magnetic field. We will see below (Sect. 6.8.1) that superfluid
3 He
realize, in its B-phase, an A 1u state for which d ∝ k. This means that, on each point
of the Fermi surface, the order parameter is described by a pure |S z = 0 state if the