Density functional theory of the trapped Fermi gas in the unitary regime

TL;DR

This paper develops a density-functional theory approach for trapped Fermi gases at unitarity, revealing limitations in 3D for small particle numbers and providing an exact result in 2D due to unique energy density properties.

Contribution

It reformulates a recent DFT approach using fractional exclusion statistics and systematically analyzes finite-N effects in 3D and 2D systems at unitarity.

Findings

Finite-N corrections in 3D lead to unphysical universal factor values.

In 2D, the universal factor is exactly 1 due to the Thomas-Fermi energy density-functional.

The simple DFT approach may not be suitable for small particle numbers in 3D.

Abstract

We investigate a density-functional theory (DFT) approach for an unpolarized trapped dilute Fermi gas in the unitary limit . A reformulation of the recent work of T. Papenbrock [Phys. Rev. A, {\bf 72}, 041602(R) (2005)] in the language of fractional exclusion statistics allows us to obtain an estimate of the universal factor, $\xi_{3D}$, in three dimensions (3D), in addition to providing a systematic treatment of finite-$N$ corrections. We show that in 3D, finite-$N$ corrections lead to unphysical values for $\xi_{3D}$, thereby suggesting that a simple DFT applied to a small number of particles may not be suitable in 3D. We then perform an analogous calculation for the two-dimensional (2D) system in the infinite-scattering length regime, and obtain a value of $\xi_{2D}=1$. Owing to the unique properties of the Thomas-Fermi energy density-functional in 2D our result, in contrast to 3D, is {\em exact} and therefore requires no finite-$N$ corrections.