A manifestly Hermitian semiclassical expansion for the one-particle density matrix of a two-dimensional Fermi gas

TL;DR

This paper develops a Hermitian semiclassical expansion for the one-particle density matrix of a 2D Fermi gas, enabling improved energy functional calculations beyond local-density approximation with small, negative gradient corrections.

Contribution

It introduces a Hermitian-preserving semiclassical expansion for the density matrix and applies it to refine the dipole-dipole interaction energy functional in 2D Fermi gases.

Findings

Finite second-order gradient correction to Hartree-Fock energy.

Improved accuracy over local-density approximation for small particle numbers.

Gradient correction is small, negative, and significantly enhances energy estimates.

Abstract

The semiclassical $\hbar$-expansion of the one-particle density matrix for a two-dimensional Fermi gas is calculated within the Wigner transform method of Grammaticos and Voros, originally developed in the context of nuclear physics. The method of Grammaticos and Voros has the virture of preserving both the Hermiticity and idempotency of the density matrix to all orders in the $\hbar$-expansion. As a topical application, we use our semiclassical expansion to go beyond the local-density approximation for the construction of the total dipole-dipole interaction energy functional of a two-dimensional, spin-polarized dipolar Fermi gas. We find a {\em finite}, second-order gradient correction to the Hartree-Fock energy, which takes the form $\varepsilon (\nabla \rho)^2/\sqrt{\rho}$, with $\varepsilon$ being small ($|\varepsilon| \ll1$) and negative. We test the quality of the corrected energy by comparing it with the exact results available for harmonic confinement. Even for small particle numbers, the gradient correction to the dipole-dipole energy provides a significant improvement over the local-density approximation.