A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case

摘要

This article is concerned with the derivation and the mathematical study of anew mean-field model for the description of interacting electrons in crystalswith local defects. We work with a reduced Hartree-Fock model, obtained fromthe usual Hartree-Fock model by neglecting the exchange term. First, we recallthe definition of the self-consistent Fermi sea of the perfect crystal, whichis obtained as a minimizer of some periodic problem, as was shown by Catto, LeBris and Lions. We also prove some of its properties which were not mentionedbefore. Then, we define and study in details a nonlinear model for theelectrons of the crystal in the presence of a defect. We use formal analogiesbetween the Fermi sea of a perturbed crystal and the Dirac sea in QuantumElectrodynamics in the presence of an external electrostatic field. The latterwas recently studied by Hainzl, Lewin, S\'er\'e and Solovej, based on ideasfrom Chaix and Iracane. This enables us to define the ground state of theself-consistent Fermi sea in the presence of a defect. We end the paper byproving that our model is in fact the thermodynamic limit of the so-calledsupercell model, widely used in numerical simulations.