Properties of periodic Dirac--Fock functional and minimizers

Existence of minimizers for the Dirac--Fock model in crystals was recently proved by Paturel and S\'er\'e and the authors \cite{crystals} by a retraction technique due to S\'er\'e \cite{Ser09}. In this paper, inspired by Ghimenti and Lewin's result \cite{ghimenti2009properties} for the periodic Hartree--Fock model, we prove that the Fermi level of any periodic Dirac--Fock minimizer is either empty or totally filled when $\frac{\alpha}{c}\leq C_{\rm cri}$ and $\alpha>0$. Here $c$ is the speed of light, $\alpha$ is the fine structure constant, and $C_{\rm cri}$ is a constant only depending on the number of electrons and on the charge of nuclei per cell. More importantly, we provide an explicit upper bound for $C_{\rm cri}$. Our result implies that any minimizer of the periodic Dirac--Fock model is a projector when $\frac{\alpha}{c}\leq C_{\rm cri}$ and $\alpha>0$. In particular, the non-relativistic regime (i.e., $c\gg1$) and the weak coupling regime (i.e., $0<\alpha\ll1$) are covered. The proof is based on a delicate study of a second-order expansion of the periodic Dirac--Fock functional composed with the retraction used in \cite{crystals}.