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.
Chaix, P., Iracane, D.: From quantum electrodynamics to mean field theory: I. The Bogoliubov-Dirac-Fock formalism. J. Phys. B. 22, 3791–3814 (1989)Google Scholar
Chaix, P., Iracane, D., Lions, P.L.: From quantum electrodynamics to mean field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation. J. Phys. B. 22, 3815–3828 (1989)CrossRefADSGoogle Scholar
Dovesi, R., Orlando, R., Roetti, C., Pisani, C., Saunders, V.R.: periodic Hartree-Fock method and its implementation in the Crystal code. Phys. Stat. Sol. (b) 217, 63–88 (2000)CrossRefGoogle Scholar
Hainzl, Ch., Lewin, M., Séré, E.: Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation. Commun. Math. Phys. 257, 515–562 (2005)MATHCrossRefADSGoogle Scholar
Hainzl, Ch., Lewin, M., Séré, E.: Self-consistent solution for the polarized vacuum in a no-photon QED model. J. Phys. A: Math & Gen. 38(20), 4483–4499 (2005)MATHCrossRefADSGoogle Scholar
Hainzl, Ch., Lewin, M., Séré, E.: Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics. Arch. Rat. Mech. Anal. (to appear)Google Scholar
Hainzl, Ch., Lewin, M., Solovej, J.P.: The mean-field approximation in quantum electrodynamics. The no-photon case. Comm. Pure Applied Math. 60(4), 546–596 (2007)MATHCrossRefMathSciNetGoogle Scholar
Hainzl, C., Lewin, M., Séré, É., Solovej, J.P.: A Minimization Method for Relativistic Electrons in a Mean-Field Approximation of Quantum Electrodynamics. Phys. Rev. A 76, 052104 (2007)CrossRefADSGoogle Scholar
Hasler, D., Solovej, J.P.: The Independence on Boundary Conditions for the Thermodynamic Limit of Charged Systems. Commun. Math. Phys. 261(3), 549–568 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
Hunziker, W.: On the Spectra of Schrödinger Multiparticle Hamiltonians. Helv. Phys. Acta 39, 451–462 (1996)MathSciNetGoogle Scholar
Kittel, Ch.: Quantum Theory of Solids. Second Edition. Wiley, New York (1987)Google Scholar
Lieb, E.H., Solovej, J.P., Yngvason, J.: Asymptotics of heavy atoms in high magnetic fields. I. Lowest Landau band regions. Comm. Pure Appl. Math. 47(4), 513–591 (1994)MATHCrossRefMathSciNetGoogle Scholar
Lions, P.-L.: The concentration-compactness method in the Calculus of Variations. The locally compact case. Part. I: Anal. non linéaire, Ann. IHP 1 109–145, (1984); Part. II: Anal. non linéaire, Ann. IHP 1 223–283, (1984)Google Scholar
Pisani, C.: Quantum-mechanical treatment of the energetics of local defects in crystals: a few answers and many open questions. Phase Transitions 52, 123–136 (1994)CrossRefGoogle Scholar
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol I, Functional Analysis, Second Ed. New York: Academic Press, 1980Google Scholar
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol IV, Analysis of Operators. New York: Academic Press, 1978Google Scholar
Ruelle, D.: Statistical Mechanics. Rigorous results. Imperial College Press and World Scientific Publishing, Singapore (1999)Google Scholar
Seiler, E., Simon, B.: Bounds in the Yukawa2 Quantum Field Theory: Upper Bound on the Pressure, Hamiltonian Bound and Linear Lower Bound. Commun. Math. Phys. 45, 99–114 (1975)CrossRefADSMathSciNetGoogle Scholar
Simon, B.: Trace Ideals and their Applications. Vol 35 of London Mathematical Society Lecture Notes Series. Cambridge: Cambridge University Press, 1979Google Scholar