Existence of minimizers for the Dirac-Fock model of crystals

TL;DR

This paper introduces a relativistic mean-field model for crystals using periodic density matrices and proves the existence of ground states under certain conditions, extending non-relativistic models to the relativistic setting.

Contribution

It is the first to establish the existence of minimizers for a fully relativistic Dirac-Fock model of crystals.

Findings

Existence of a ground state for the relativistic crystal model.

Extension of non-relativistic Hartree-Fock to relativistic case.

Conditions under which the ground state exists.

Abstract

Whereas many different models exist in the mathematical and physics literature for ground states of non-relativistic crystals, the relativistic case has been much less studied and we are not aware of any mathematical result on a fully relativistic treatment of crystals. In this paper, we introduce a mean-field relativistic energy for crystals in terms of periodic density matrices. This model is inspired both from a recent definition of the Dirac-Fock ground state for atoms and molecules, due to one of us, and from the non-relativistic Hartree-Fock model for crystals. We prove the existence of a ground state when the number of electrons per cell is not too large.