Properties of periodic Dirac--Fock functional and minimizers

Existence of minimizers for the Dirac–Fock model in crystals was recently proved by Paturel and Séré and the authors \cite{crystals} by a retraction technique due to Séré \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 $\frac{\alpha }{c}\le C_{\mathrm{c}\mathrm{r}\mathrm{i}}$ and $\alpha >0$. Here $c$ is the speed of light, $\alpha$ is the fine structure constant, and $C_{\mathrm{c}\mathrm{r}\mathrm{i}}$ 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 $C_{\mathrm{c}\mathrm{r}\mathrm{i}}$. Our result implies that any minimizer of the periodic Dirac–Fock model is a projector when $\frac{\alpha }{c}\le C_{\mathrm{c}\mathrm{r}\mathrm{i}}$ and $\alpha >0$. In particular, the non-relativistic regime (i.e., $c\gg 1$) and the weak coupling regime (i.e., $0<\alpha \ll 1$) 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}.