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

TL;DR

This paper derives Pekar's polaron model from a microscopic quantum crystal model, showing how macroscopic behavior emerges from microscopic interactions, including effects of anisotropy and multi-electron systems.

Contribution

It provides a rigorous derivation of Pekar's model from reduced Hartree-Fock theory, including microscopic oscillations and multi-polaron interactions.

Findings

Macroscopic density converges to Pekar's nonlinear model

Microscopic oscillations contribute to energy at the same order

Model covers multi-electron polarons with Coulomb repulsion

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.