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 Δ for a cold dilute fluid of equal amounts of two species of fermion in the unitary limit k_fa→∞ (where k_f is the Fermi momentum, a is the scattering length, and Δ 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 δ≤5/3[2(2ξ)^2/5-(2ξ)] where ξ=ϵ/ϵ_FG, δ=2Δ/ϵ_FG; ϵ is the energy per particle and ϵ_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 Δ is the usual superfluid gap. If the superfluid gap exceeds the maximum allowed by the inequality, phase separation occurs.