A reduced Hartree-Fock model of slice-like defects in the Fermi sea

TL;DR

This paper introduces a mean-field reduced Hartree-Fock model for extended defects in metals, analyzing their electronic structure and demonstrating convergence of Yukawa to Coulomb interactions, with numerical evidence of Friedel oscillations.

Contribution

It develops a mathematical framework for extended defects in metals within the reduced Hartree-Fock model, including convergence results and numerical simulations.

Findings

Existence of minimizers for the model.

Convergence of Yukawa to Coulomb ground states.

Observation of Friedel oscillations in simulations.

Abstract

Studying the electronic structure of defects in materials is an important subject in condensed matter physics. From a mathematical point of view, nonlinear mean-field models of localized defects in insulators are well understood. We present here a mean-field model to study a particular instance of extended defects in metals. These extended defects typically correspond to taking out a slab of finite width in the three-dimensional homogeneous electron gas. We work in the framework of the reduced Hartree-Fock model with either Yukawa or Coulomb interactions. Using techniques developed in~[Frank2011, Frank2013] to study local perturbations of the free-electron gas, we show that our model admits minimizers, and that Yukawa ground state energies and density matrices converge to ground state Coulomb energies and density matrices as the Yukawa parameter tends to zero. We moreover present numerical simulations where we observe Friedel oscillations in the total electronic density.