# Localization for gapped Dirac Hamiltionians with random perturbations: Application to graphene antidot lattices

## TL;DR

This paper investigates how random impurities affect the electronic properties of graphene antidot lattices modeled by Dirac Hamiltonians, demonstrating localization phenomena at band edges due to disorder.

## Contribution

It establishes band edge localization for Dirac Hamiltonians with random perturbations in graphene-like systems, extending understanding of disorder effects in such materials.

## Key findings

- Proves existence of dense pure point spectrum near band edges
- Shows eigenfunctions decay exponentially, indicating localization
- Demonstrates dynamical localization in the perturbed system

## Abstract

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator $H_0 := D_S + V_0$ is the sum of a Dirac-like operator $D_S$ plus a periodic matrix valued potential $V_0$, and is assumed to have an open gap. The random potential $V_\omega$ is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family $H_\omega := H_0 + V_\omega$ is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.