The Correlation Energy of the Electron Gas in the Mean-Field Regime

TL;DR

This paper establishes rigorous bounds on the correlation energy of interacting fermions in the mean-field regime, including Coulomb interactions, confirming an analogue of the Gell-Mann--Brueckner formula in the high-density limit.

Contribution

It provides the first rigorous lower bound matching the known upper bound for the correlation energy of the electron gas, extending analysis beyond mean-field scaling.

Findings

Established a lower bound on correlation energy for Coulomb interactions.

Confirmed the Gell-Mann--Brueckner formula in the high-density limit.

Extended analysis beyond mean-field scaling.

Abstract

We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained in \cite{ChrHaiNam-23b}, our result establishes an analogue of the Gell-Mann--Brueckner formula $c_{1}\rho\log\left(\rho\right)+c_{2}\rho$ for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials.