Airy-averaged gradient corrections for two-dimensional fermion gases

TL;DR

This paper develops Airy-averaged gradient corrections for two-dimensional fermion gases, improving the semiclassical kinetic-energy functional and particle density descriptions, especially near classically forbidden regions, with applications to low-temperature Fermi gases.

Contribution

It introduces an Airy-averaged approach to gradient corrections, enabling smooth density transitions and improved kinetic-energy functionals for 2D fermion systems.

Findings

Excellent agreement with exact densities at low temperatures

Provides criteria for semiclassical expansion applicability

Derives a well-behaved ground-state kinetic-energy functional

Abstract

Building on the discussion in PRA 93, 042510 (2016), we present a systematic derivation of gradient corrections to the kinetic-energy functional and the one-particle density, in particular for two-dimensional systems. We derive the leading gradient corrections from a semiclassical expansion based on Wigner's phase space formalism and demonstrate that the semiclassical kinetic-energy density functional at zero temperature cannot be evaluated unambiguously. In contrast, a density-potential functional description that effectively incorporates interactions provides unambiguous gradient corrections. Employing an averaging procedure that involves Airy functions, thereby partially resumming higher-order gradient corrections, we facilitate a smooth transition of the particle density into the classically forbidden region of arbitrary smooth potentials. We find excellent agreement of the semiclassical Airy-averaged particle densities with the exact densities for very low but finite temperatures, illustrated for a Fermi gas with harmonic potential energy. We furthermore provide criteria for the applicability of the semiclassical expansions at low temperatures. Finally, we derive a well-behaved ground-state kinetic-energy functional, which improves on the Thomas-Fermi approximation.