Fermionic density functional at a Feshbach resonance

Abstract

We consider a dilute gas of neutral unpolarized fermionic atoms at zero temperature. The atoms interact via a short-range (tunable) attractive interaction. We demonstrate analytically a curious property of the gas at unitarity. Namely, the correlation energy of the gas, evaluated by second-order perturbation theory, has the same density dependence as the first-order exchange energy, and the two almost exactly cancel each other at a Feshbach resonance irrespective of the shape of the potential, provided $(\ensuremath{\mu}{r}_{s})\ensuremath{\gg}1$. Here $(\ensuremath{\mu}{)}^{\ensuremath{-}1}$ is the range of the two-body potential, and ${r}_{s}$ is defined through the number density, $n=3∕(4\ensuremath{\pi}{r}_{s}^{3})$. The implications of this result for universality are discussed.