Airy gas model: From three to reduced dimensions

TL;DR

This paper extends the Airy gas model from three dimensions to one and two dimensions, deriving explicit densities and kinetic energy functionals, and analyzing their relation to local density approximations and the Thomas-Fermi model.

Contribution

It provides explicit expressions for edge densities and kinetic energy densities in reduced dimensions, expanding the applicability of the Airy gas model.

Findings

Densities obey the local virial theorem.

KED functional reduces to Thomas-Fermi in the bulk limit.

Derived explicit formulas for densities in 1D and 2D.

Abstract

By using the propagator of linear potential as a main tool, we extend the Airy gas model, originally developed for the three-dimensional ($d=3$) edge electron gas, to systems in reduced dimensions ($d=2,1$). First, we derive explicit expressions for the edge particle density and the corresponding kinetic energy density (KED) of the Airy gas model in all dimensions. The densities are shown to obey the local virial theorem. We obtain a functional relationship between the positive KED and the particle density and its gradients and analyze the results inside the bulk as a limit of the local-density approximation. We show that in this limit the KED functional reduces to that of the Thomas-Fermi model in $d$ dimensions.