The Ground State Energy of a Mean-Field Fermi Gas in Two Dimensions

We rigorously establish a formula for the correlation energy of a two-dimensional Fermi gas in the mean-field regime for potentials whose Fourier transform $\widehat{V}$ satisfies $\widehat{V}(\cdot )\mathrm{\mid }\cdot \mathrm{\mid }\in {\mathrm{\ell }}^1$. Further, we establish the analogous upper bound for $\widehat{V}(\cdot {)}^2\mathrm{\mid }\cdot {\mathrm{\mid }}^{1+\epsilon }\in {\mathrm{\ell }}^1$, which includes the Coulomb potential $\widehat{V}(k)\sim \mathrm{\mid }k{\mathrm{\mid }}^{-2}$. The proof is based on an approximate bosonization using slowly growing patches around the Fermi surface. In contrast to recent proofs in the three-dimensional case, we need a refined analysis of low-energy excitations, as they are less numerous, but carry larger contributions.