Energetics of a strongly correlated Fermi gas

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.Comment: 28 pages, 1 figur