Phase Separation and an upper bound for $\Delta$ for Fermi fluids in the Phase Separation and an upper bound for $\Delta$ for Fermi fluids in the unitary regime

Abstract

An upper bound is derived for $\Delta$ for a cold dilute fluid of equal amounts of two species of fermion in the unitary regime $k_f a \to \infty$ (where $k_f$ is the Fermi momentum and $a$ 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 $\delta \leq {5/3} (2 (2 \xi)^{2/5} - (2 \xi))$ where $\xi=\epsilon/\epsilon_{\rm FG}$, $\delta= 2 \Delta/\epsilon_{\rm FG}$; $\epsilon$ is the energy per particle and $\epsilon_{\rm FG}$ 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.