Properties of periodic Dirac--Fock functional and minimizers

TL;DR

This paper analyzes the properties of minimizers in the periodic Dirac--Fock model, establishing conditions under which the Fermi level is either empty or fully occupied, and showing minimizers are projectors in certain regimes.

Contribution

It proves that the Fermi level of periodic Dirac--Fock minimizers is either empty or totally filled under specific conditions and provides an explicit upper bound for the critical constant.

Findings

Fermi level is either empty or totally filled when /c _{cri} and lpha > 0

Minimizers are projectors in the non-relativistic and weak coupling regimes

Provides explicit bounds for the critical constant C_{cri}

Abstract

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}.