Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

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.