Phase separation and an upper bound for a generalized superfluid gap for cold fermi fluids in the unitary regime

An upper bound is derived for Delta for a cold dilute fluid of equal amounts of two species of fermion in the unitary limit k(f)a -> infinity where k(f) is the Fermi momentum, a is the scattering length, and Delta is a pairing energy: the difference in energy per particle between adding to the system a macroscopic number (but infinitesimal fraction) of particles of one species compared to adding equal numbers of both. The bound is delta <= 5/3 [2(2 xi)(2/5) - (2 xi)] where xi = epsilon/epsilon(FG), delta = 2 Delta/epsilon(FG), epsilon is the energy per particle and epsilon(FG) is the energy per particle of a noninteracting Fermi gas. If the bound is saturated, then systems with unequal densities of the two species will separate spatially into a superfluid phase with equal numbers of the two species and a normal phase with the excess. If the bound is not saturated, then Delta is the usual superfluid gap. If the superfluid gap exceeds the maximum allowed by the inequality, phase separation occurs.