Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

TL;DR

This paper rigorously justifies the random phase approximation for the correlation energy of a Fermi gas in the mean-field regime, using bosonic approximations of particle-hole pairs and Bogoliubov theory.

Contribution

It introduces a novel approach to bounding the correlation energy by approximating particle-hole pairs with a bosonic quadratic Hamiltonian and applying Bogoliubov theory.

Findings

Established a rigorous upper bound matching the RPA prediction.

Validated the use of bosonic approximations for particle-hole excitations.

Confirmed the applicability of the RPA in the mean-field regime for repulsive interactions.

Abstract

While Hartree-Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree-Fock state given by plane waves and introduce collective particle-hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann-Brueckner-type upper bound to the ground state energy. Our result justifies the random phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.