Abstract
In a recent article (Fefferman and Weinstein, in J Am Math Soc 25:1169–1220, 2012), the authors proved that the non-relativistic Schrödinger 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.
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Fefferman, C.L., Weinstein, M.I. Wave Packets in Honeycomb Structures and Two-Dimensional Dirac Equations. Commun. Math. Phys. 326, 251–286 (2014). https://doi.org/10.1007/s00220-013-1847-2
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DOI: https://doi.org/10.1007/s00220-013-1847-2