The reduced Hartree-Fock model for short-range quantum crystals with defects

TL;DR

This paper develops a mathematical model for quantum crystals with defects using the reduced Hartree-Fock approach, proving existence of ground states and analyzing defect effects on electronic properties.

Contribution

It introduces a rigorous analysis of quantum crystals with defects, including existence proofs and decay properties of solutions under Yukawa interactions.

Findings

Existence of electronic ground state with defects

Decay properties of solutions for local defects

Expansion of density of states for low defect concentration

Abstract

In this article, we consider quantum crystals with defects in the reduced Hartree-Fock framework. The nuclei are supposed to be classical particles arranged around a reference periodic configuration. The perturbation is assumed to be small in amplitude, but need not be localized in a specific region of space or have any spatial invariance. Assuming Yukawa interactions, we prove the existence of an electronic ground state, solution of the self-consistent field equation. Next, by studying precisely the decay properties of this solution for local defects, we are able to expand the density of states of the nonlinear Hamiltonian of a system with a random perturbation of Anderson-Bernoulli type, in the limit of low concentration of defects. One important step in the proof of our results is the analysis of the dielectric response of the crystal to an effective charge perturbation.