Wave Packets in Honeycomb Structures and Two-Dimensional Dirac Equations

Abstract

In a recent article (Fefferman and Weinstein, in J Am Math Soc 25:1169-1220, 2012), the authors proved that the non-relativistic Schrodinger operator with a generic honeycomb lattice potential has conical (Dirac) points in its dispersion surfaces. These conical points occur for quasi-momenta, which are located at the vertices of the Brillouin zone, a regular hexagon. In this paper, we study the time-evolution of wave-packets, which are spectrally concentrated near such conical points. We prove that the large, but finite, time dynamics is governed by the two-dimensional Dirac equations.