Honeycomb Lattice Potentials and Dirac Points

In this article we study the spectral properties of the Schrödinger operator HV “ ́Δ ` V pxq, x P R, where the potential, V , is periodic and has honeycomb structure symmetry. For general periodic potentials the spectrum of HV , considered as an operator on LpRq, is the union of closed intervals of continuous spectrum called the spectral bands. Associated with each spectral band are a band dispersion function, μpkq, and Floquet-Bloch states, upx;kq “ ppx;kqeik ̈x, where Hupx;kq “ μpkqupx;kq and ppx;kq is periodic with the periodicity of V pxq. The quasi-momentum, k, varies over B, the first Brillouin zone [10]. Therefore, the time-dependent Schrödinger equation has solutions of the form eipk ̈x ́μpkqtq ppx;kq. Furthermore, any finite energy solution of the initial value problem for the timedependent Schrödinger equation is a continuum weighted superposition, an integral dk, over such states. Thus, the time-dynamics are strongly influenced by the character of μpkq on the spectral support of the initial data. We investigate the properties of μpkq in the case where V “ Vh is a honeycomb lattice potential, i.e. Vh is periodic with respect to a particular lattice, Λh, and has honeycomb structure symmetry; see Definition 2.1. There has been intense interest within the fundamental and applied physics communities in such structures; see, for example, the survey articles [14, 16]. Graphene, a single atomic layer of carbon atoms, is a two-dimensional structure with carbon atoms located at the sites of a honeycomb structure. Most remarkable is that the associated dispersion surfaces are observed to have conical singularities at the vertices of Bh, which in this case is a regular hexagon. That is, locally about any such quasi-momentum vertex, k « K‹, one has