Correlation Energy of a Weakly Interacting Fermi Gas

TL;DR

This paper rigorously derives the leading order correlation energy of a high-density, weakly interacting Fermi gas, confirming the random-phase approximation and refining collective bosonization methods in three dimensions.

Contribution

It provides a rigorous derivation of the correlation energy in a specific scaling regime, improving the mathematical understanding of collective excitations in Fermi gases.

Findings

Verification of the random-phase approximation prediction.

Refinement of collective bosonization methods.

Approximate diagonalization of the effective Hamiltonian.

Abstract

We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree-Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy.