# The Fermionic Density-functional at Feshbach Resonance

## TL;DR

This paper analytically explores the properties of a dilute fermionic gas at unitarity, revealing a near cancellation of correlation and exchange energies at Feshbach resonance, which has implications for the universality of such systems.

## Contribution

It demonstrates that at unitarity, the correlation energy and exchange energy nearly cancel each other, independent of potential shape, highlighting a universal property of fermionic gases.

## Key findings

- Correlation and exchange energies cancel at unitarity.
- The cancellation is independent of potential shape.
- Implications for universality of fermionic gases at 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 Feshbach resonance irrespective of the shape of the potential, provided $(\mu r_s) >> 1$. Here $(\mu)^{-1}$ is the range of the two-body potential, and $r_s$ is defined through the number density $n=3/(4\pi r_s^3)$. The implications of this result for universality is discussed.