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

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.