Stability conditions and Fermi surface topologies in a superconductor

Candidate homogeneous, isotropic superfluid or superconducting states of paired fermion species with different chemical potentials, can lead to quasiparticle excitation energies that vanish at either 0, 1, or 2 spheres in momentum space. With no zeros, we have a conventional BCS superconductor. The other two cases, "gapless" superconductors, appear in mean- field theory for sufficiently large mismatches and/or sufficiently large coupling strengths. Here we examine several stability criteria for those candidate phases. Positivity of number susceptibility appears to provide the most powerful constraint, and renders all the two-zero states that we have examined mechanically unstable. Our results should apply directly to ultracold fermionic atom systems.