## Honeycomb lattice potentials and Dirac points

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- by Charles L. Fefferman and Michael I. Weinstein
- J. Amer. Math. Soc.
**25**(2012), 1169-1220 - DOI: https://doi.org/10.1090/S0894-0347-2012-00745-0
- Published electronically: June 25, 2012
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## 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.

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## Bibliographic Information

**Charles L. Fefferman**- Affiliation: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544
- MR Author ID: 65640
- Email: cf@math.princeton.edu
**Michael I. Weinstein**- Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027
- MR Author ID: 181490
- Email: miw2103@columbia.edu
- Received by editor(s): February 16, 2012
- Received by editor(s) in revised form: May 24, 2012
- Published electronically: June 25, 2012
- Additional Notes: The first author was supported in part by US-NSF Grant DMS-09-01040

The second author was supported in part by US-NSF Grant DMS-10-08855 - © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**25**(2012), 1169-1220 - MSC (2010): Primary 35Pxx
- DOI: https://doi.org/10.1090/S0894-0347-2012-00745-0
- MathSciNet review: 2947949