Energetics of a strongly correlated Fermi gas

TL;DR

This paper derives a universal energy identity for a strongly correlated two-component Fermi gas with contact interactions, linking its energy to momentum distribution and introducing a novel mathematical approach to handle divergences.

Contribution

It presents a new universal identity for the energy of a Fermi gas with contact interactions and introduces a novel mathematical method for dealing with ultraviolet divergences.

Findings

Derived a linear functional of momentum distribution for the Fermi gas energy.

Analyzed short-range structure and pair correlations in the system.

Explored the dimer-fermion scattering length and other properties.

Abstract

The energy of the two-component Fermi gas with the s-wave contact interaction is a simple linear functional of its momentum distribution: $$E_\text{internal}=\hbar^2\Omega C/4\pi am+\sum_{\vect k\sigma}(\hbar^2 k^2/2m)(n_{\vect k\sigma}-C/k^4)$$ where the external potential energy is not included, $a$ is the scattering length, $\Omega$ is the volume, $n_{\vect k\sigma}$ is the average number of fermions with wave vector $\vect k$ and spin $\sigma$, and $C\equiv\lim_{\vect k\to\infty} k^4 n_{\vect k\up} =\lim_{\vect k\to\infty} k^4 n_{\vect k\down}$. This result is a \textit{universal identity}. Its proof is facilitated by a novel mathematical idea, which might be of utility in dealing with ultraviolet divergences in quantum field theories. Other properties of this Fermi system, including the short-range structure of the one-body reduced density matrix and the pair correlation function, and the dimer-fermion scattering length, are also studied.