Communications in Mathematical Physics

, Volume 281, Issue 1, pp 129–177

A New Approach to the Modeling of Local Defects in Crystals: The Reduced Hartree-Fock Case


DOI: 10.1007/s00220-008-0481-x

Cite this article as:
Cancès, É., Deleurence, A. & Lewin, M. Commun. Math. Phys. (2008) 281: 129. doi: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 detail 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, Séré 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.

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Éric Cancès
    • 1
  • Amélie Deleurence
    • 1
  • Mathieu Lewin
    • 2
  1. 1.CERMICSEcole Nationale des Ponts et Chaussées (Paris Tech) & INRIA (Micmac Project)Champs-sur-MarneFrance
  2. 2.CNRS & Laboratoire de Mathématiques UMR 8088Université de Cergy-PontoiseCergy-Pontoise CedexFrance