Honeycomb lattice potentials and Dirac points

Abstract

We prove that the two-dimensional Schrödinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities ( Dirac points ) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.