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

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

2007-02-21

arxiv.org/abs/math-ph/0702071v3

Communications in Mathematical Physics 281, 1 (2008) 129-177

Physics

Mathematical Physics

Final version, to appear in Comm. Math. Phys

Scientific paper

10.1007/s00220-008-0481-x

This article is concerned with the derivation and the mathematical 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, obtained from the usual Hartree-Fock model by neglecting the exchange term. First, we recall the definition of the self-consistent Fermi sea of the perfect 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 which were not mentioned before. Then, we define and study in details a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, S\'er\'e and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the self-consistent Fermi sea in the presence of a defect. We end the paper by proving that our model is in fact the thermodynamic limit of the so-called supercell model, widely used in numerical simulations.

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