Derivation of Pekar's Polarons from a Microscopic Model of Quantum Crystals

Abstract

A polaron is an electron interacting with a polar crystal, which is able to form a bound state by using the distortions of the crystal induced by its own density of charge. In this paper we derive Pekar's famous continuous model for polarons (in which the crystal is replaced by a simple effective Coulomb self-attraction) by studying the macroscopic limit of the reduced Hartree-Fock theory of the crystal. The macroscopic density of the polaron converges to that of Pekar's nonlinear model, with a possibly anisotropic dielectric matrix. The polaron also exhibits fast microscopic oscillations which contribute to the energy at the same order, but whose characteristic length is small compared to the scale of the polaron. These oscillations are described by a simple periodic eigenvalue equation. Our approach also covers multi-polarons composed of several electrons, repelling each other by Coulomb forces.

References in corpus (9)

• Geometric methods for nonlinear many-body quantum systems
• A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case
• On Singularity formation for the L^2-critical Boson star equation
• Non-perturbative embedding of local defects in crystalline materials
• The Dielectric Permittivity of Crystals in the reduced Hartree-Fock approximation
• Bi-Polaron and N-Polaron Binding Energies
• The bipolaron in the strong coupling limit
• Bounds on the Minimal Energy of Translation Invariant $N$-Polaron Systems
• Binding of Polarons and Atoms at Threshold

Cited by in corpus (5)

• Supercell calculations in the reduced Hartree-Fock model for crystals with local defects
• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• On uniqueness and non-degeneracy of anisotropic polarons
• Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit
• Existence of Ground State of an Electron in the BDF Approximation