Properties of periodic Dirac–Fock functional and minimizers

Existence of minimizers for the Dirac–Fock model for crystals was recently proved by Paturel and Séré and the authors [9]. In this paper, inspired by Ghimenti and Lewin's result [13] 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 $\frac{\alpha }{c}\le C_{\mathrm{cri}}$ and $\alpha >0$. Here c is the speed of light, α is the fine structure constant, and $C_{\mathrm{cri}}$ 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 $C_{\mathrm{cri}}$.

Our result implies that any minimizer of the periodic Dirac–Fock model is a projector when $\frac{\alpha }{c}\le C_{\mathrm{cri}}$ 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 a retraction that was introduced by Séré in [24] for atoms and molecules and later extended to the case of crystals in [9].