He-3B revisited

What happens to an anisotropic superconductor (e.g. helium-3) when you shrink the Cooper pairs? In the case of a normal (BCS) superconductor, nothing terribly interesting takes place: the system goes smoothly from a BCS-like condensate, in which the distance between an electron and its partner is larger than the spacing between them, to a regular condensate of diatomic molecules for which the reverse is true.

This result generalizes, e.g., to the case of p-wave pairing in a completely spin-polarized Fermi gas. (See Leggett, J. Phys. Colloques 41 (1980) C7-19-C7-26.) The situation of s-wave pairing in a spin-imbalanced Fermi gas is somewhat more interesting: at the BEC end the molecules don't really care about the presence of additional fermions and one therefore has a BEC with free fermions floating on top of it; at the BCS end, however, the spin-imbalance frustrates Cooper pairing -- which requires the spin-up and -down Fermi seas to be degenerate. For finite spin-imbalance, one therefore has a quantum phase transition (I believe this is also a thermal phase transition) instead of a crossover.

I was interested in generalizing these ideas to the case of equal-spin p-wave superconductors (or Fermi superfluids) like He-3A and He-3B. The basic reason one might expect something interesting to happen is that the superfluid states are extremely fancy: much fancier, in particular, than anything one expects from tightly bound diatomic molecules. (This is particularly true for the A phase, which is stable only because of a subtle feedback mechanism.) The question is whether there's a quantum phase transition between the BEC and BCS ends; the answer -- which, rather disappointingly, Volovik got to -- is that there is. Volovik constructs a topological invariant that's zero in the BEC limit and nonzero in the BCS limit; it follows, of course, that these limits cannot be adiabatically connected.

Topological invariant for superfluid 3He-B and quantum phase transitions
Authors: G.E. Volovik
http://arxiv.org/abs/0909.3084

Abstract: We consider topological invariant describing the vacuum states of superfluid 3He-B, which belongs to the special class of time-reversal invariant topological insulators and superfluids. Discrete symmetries important for classification of the topologically distinct vacuum states are discussed. One of them leads to the additional subclasses of 3He-B states and is responsible for the finite density of states of Majorana fermions living on the diffusive wall. Integer valued topological invariant is expressed in terms of the Green's function, which allows us to consider systems with interaction.