Dilute Fermi gas: kinetic and interaction energies

TL;DR

This paper derives low-density expansions for kinetic and interaction energies of a dilute 3D Fermi gas with repulsive interactions, revealing the significant role of pairwise interactions and their dependence on two characteristic lengths.

Contribution

It provides the first detailed third-order low-density expansions for energies in a Fermi gas, using two independent methods for validation and highlighting the impact of pairwise interactions.

Findings

Kinetic energy depends significantly on pairwise interactions.

Both energies depend on the scattering length and a related length b.

Results are validated through two independent calculation methods.

Abstract

A dilute homogeneous 3D Fermi gas in the ground state is considered for the case of a repulsive pairwise interaction. The low-density (dilution) expansions for the kinetic and interaction energies of the system in question are calculated up to the third order in the dilution parameter. Similar to the recent results for a Bose gas, the calculated quantities turn out to depend on a pairwise interaction through the two characteristic lengths: the former, $a$, is the well-known s-wave scattering length, and the latter, $b$, is related to $a$ by $b=a-m (\partial a/\partial m)$, where $m$ stands for the fermion mass. To take control of the results, calculations are fulfilled in two independent ways. The first involves the Hellmann-Feynman theorem, taken in conjunction with a helpful variational theorem for the scattering length. This way is used to derive the kinetic and interaction energies from the familiar low-density expansion of the total system energy first found by Huang and Yang. The second way operates with the in-medium pair wave functions. It allows one to derive the quantities of interest``from the scratch'', with no use of the total energy. An important result of the present investigation is that the pairwise interaction of fermions makes an essential contribution to their kinetic energy. Moreover, there is a complicated and interesting interplay of these quantities.