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.

The Hartree-Fock ground state of a homogeneous electron gas is never translation invariant, even at high densities. As proved by Overhauser, the (paramagnetic) free Fermi gas is always unstable under the formation of spin- or charge-density waves. We give here an explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies a posteriori the use of the noninteracting Fermi gas as a reference state in the large-density expansion of the correlation energy of the homogeneous electron gas. We are also able to discuss the positive temperature phase diagram and prove that the Overhauser instability only occurs at temperatures which are exponentially small at high density. Our work sheds a new light on the Hartree-Fock phase diagram of the homogeneous electron gas.

We show that, within a linear approximation of BCS theory, a weak homogeneous magnetic field lowers the critical temperature by an explicit constant times the field strength, up to higher order terms. This provides a rigorous derivation and generalization of results obtained in the physics literature fromWHH theory of the upper critical magnetic field. A new ingredient in our proof is a rigorous phase approximation to control the effects of the magnetic field.

We study the critical temperature of a superconductive material in a weak external electric potential via a linear approximation of the BCS functional. We reproduce a similar result as in Frank et al. (Commun Math Phys 342(1):189–216, 2016) using the strategy introduced in Frank et al. (The BCS critical temperature in a weak homogeneous magnetic field), where we considered the case of an external constant magnetic field.