Résumé en anglais

We study the Bogoliubov-Dirac-Fock model that allows to consider relativistic electrons interacting with the vacuum in the presence of an external electrostatic field. It can be seen as a Hartree-Fock approximation of QED, where photons are neglected. A state is described by its one-body density matrix: an infinite rank, self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator. We are interested in the properties of minimizers of the BDF-energy in the presence of an external field with charge density $\nu\ge 0$ in the regime $\alpha$, $\alpha \log(\Lambda)$ and $\alpha \nu$ (in some norms) small where $\alpha$ is the coupling constant and $\Lambda$ the ultraviolet cut-off. We prove that the density of such minimizer is integrable and compute the effective charge of the system. We also ensure the existence of minimizers under charge constraint $M\in\mathbf{N}^*$ provided that there holds $M-1< \int \nu$ close to the nonrelativistic limit $\alpha\to 0$ with $\alpha\log(\Lambda)$ fixed to a small value. This contrasts with the assumptions of [Arch. Ration. Mech. Anal, 192(3):453-499(2009)] where $\Lambda$ is fixed. As a consequence, the nonrelativistic model we obtain in the limit keeps track of the charge renormalisation: it is different from the Hartree-Fock model obtained.