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

Abstract

The semiclassical $\hslash$-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 $\hslash$-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 $\epsilon (\nabla \rho {)}^2/\sqrt{\rho }$, with $\epsilon$ being small ($|\epsilon |\ll 1$) 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.

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References in corpus (4)

• Kirzhnits gradient expansion for a D-dimensional Fermi gas
• On the Kirzhnits gradient expansion in two dimensions
• Thomas-Fermi von Weizsäcker theory for a harmonically trapped, two-dimensional, spin-polarized dipolar Fermi gas
• A nonlocal kinetic energy functional for an inhomogeneous two-dimensional Fermi gas

Cited by in corpus (1)

• Preserving the Hermiticity of the One-Body Density Matrix for a Non-Interacting Fermi Gas