Generalized Bardeen-Cooper-Schrieffer States and the Proposed Low-Temperature Phase of Liquid ${\mathrm{He}}^3$

Particle interactions in a Fermi gas may be such as to attract pairs near the Fermi surface more strongly in $l=1, 2, 3$ or higher states than in the simple spherically symmetrical $s$ state. In that case the Bardeen-Cooper-Schrieffer condensed state must be generalized, and the resulting state is an anisotropic superfluid. We have studied the properties of this type of state in considerable detail, especially for $l=1\mathrm{} \mathrm{and} 2$. We have derived expressions for the energy, the moment of inertia, the magnetic susceptibility and the specific heat. We also derive the density correlation function and the density-current density correlation; in some cases the latter implies that the liquid has net surface currents and a net orbital angular momentum. The ground state for $l=2\mathrm{}$ is different from those previously considered, and has cubic symmetry and no net angular momentum. A general method for replacing the possibly rather complicated potential by a simple scattering matrix is given. A brief discussion of possible collective effects is included. We apply our results to liquid ${\mathrm{He}}^3$; after correction for scattering by a method due to Suhl, it is found that the predicted transition should take place below 0.02°K. Other possible applications are suggested.