Analytical treatment of interacting Fermi gas in arbitrary dimensional harmonic trap

TL;DR

This paper provides exact analytical expressions for the energy and chemical potential of an interacting Fermi gas in arbitrary dimensions within a harmonic trap, demonstrating the validity of the Thomas-Fermi approximation across various conditions.

Contribution

It offers the first exact perturbative calculations of ground state properties of interacting Fermi gases in arbitrary dimensions and compares them with classical approximations.

Findings

Thomas-Fermi approximation matches exact results across dimensions.

Interaction energy decreases as T^(-d/2) at high temperatures.

Ground state interactions are analyzed in 2D and 3D systems.

Abstract

We study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground state energy and chemical potential, and obtain simple analytic expressions of the total energy and chemical potential. At zero temperature, we find that Thomas-Fermi approximation agrees well with exact results for any dimension even though system is dilute and small, i.e. when the Thomas-Fermi approximation is generally expected to fail. In the high temperature (classical) region, we find interaction energy decreases in proportion to T^(-d/2), where T is temperature and d is dimension of the system. Effect of interaction in the ground state in two and three-dimensional systems is also discussed.