The Gell-Mann$-$Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime

TL;DR

This paper establishes a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime, including Coulomb interactions, refining bosonization techniques to handle singular potentials and exchange effects.

Contribution

It provides the first rigorous derivation of the Gell-Mann–Brueckner formula for the correlation energy in the high-density limit for a broad class of potentials.

Findings

Proves an upper bound matching the Gell-Mann–Brueckner formula.

Extends analysis to Coulomb potential.

Refines bosonization to include exchange contributions.

Abstract

We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann$-$Brueckner formula $c_{1}\rho\log\left(\rho\right)+c_{2}\rho$ in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.