An upper bound is derived for $\Delta$ for a cold dilute fluid of equal amounts of two species of fermion in the unitary limit $k_{f}a\to \infty$ (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 $\delta \le \frac{5}{3}[2(2\xi {)}^{2/5}-(2\xi )]$ where $\xi =ϵ/{ϵ}_{\mathrm{FG}}$, $\delta =2\Delta /{ϵ}_{\mathrm{FG}}$; $ϵ$ is the energy per particle and ${ϵ}_{\mathrm{FG}}$ 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.

• Received 5 January 2005

DOI:https://doi.org/10.1103/PhysRevLett.95.120403