[工程科技]A new approach to the modelling of local defects in crystals the reduced Hartree-Fock case.pdf

A NEW APPROACH TO THE MODELING OF LOCAL DEFECTS IN CRYSTALS: THE REDUCED HARTREE-FOCK CASE ´ ` ´ ERIC CANCES, AMELIE DELEURENCE, AND MATHIEU LEWIN 8 0 0 Abstract. This article is concerned with the derivation and the mathematical 2 study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, r obtained from the usual Hartree-Fock model by neglecting the exchange term. p First, we recall the definition of the self-consistent Fermi sea of the perfect A crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties 7 which were not mentioned before. 1 Then, we define and study in detail a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the 3 Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynam- v ics in the presence of an external electrostatic field. The latter was recently 1 studied by Hainzl, Lewin, S´er´e and Solovej, based on ideas from Chaix and 7