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

In this paper we study random perturbations of firstorder elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator H0 := DS + V0 is the sum of a Dirac-like operator DS plus a periodic matrix-valued potential V0, and is assumed to have an open gap. The random potential Vω 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 Hω := H0 + Vω is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.