On the correlation energy of interacting fermionic systems in the mean-field regime

TL;DR

This paper rigorously analyzes the correlation energy of large interacting fermionic systems in three dimensions, providing optimal bounds and confirming second-order perturbation theory predictions in the mean-field regime.

Contribution

It introduces a novel combination of methods to establish optimal bounds and compute the leading order correlation energy for fermions in the mean-field limit.

Findings

Established optimal bounds for correlation energy.

Computed leading order correlation energy matching perturbation theory.

Validated theoretical predictions with rigorous mathematical proofs.

Abstract

We consider a system of $N\gg 1$ interacting fermionic particles in three dimensions, confined in a periodic box of volume $1$, in the mean-field scaling. We assume that the interaction potential is bounded and small enough. We prove upper and lower bounds for the correlation energy, which are optimal in their $N$-dependence. Moreover, we compute the correlation energy at leading order in the interaction potential, recovering the prediction of second order perturbation theory. The proof is based on the combination of methods recently introduced for the study of fermionic many-body quantum dynamics together with a rigorous version of second-order perturbation theory, developed in the context of non-relativistic QED.