Existence of minimizers for the Dirac-Fock Model of Crystals

Isabelle Catto (1) , Long Meng (2) , Eric Paturel (3) , Eric Séré (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 LMJL - Laboratoire de Mathématiques Jean Leray

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 existence of a ground-state when the number of electrons per cell is not too large.

Mots clés: Dirac-Fock; crystals; ground state; variational method

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique Quantique [quant-ph]

Isabelle Catto, Long Meng, Eric Paturel, Eric Séré. Existence of minimizers for the Dirac-Fock Model of Crystals. 2022. ⟨hal-03871041v1⟩