G. Berkolaiko, Texas A&M University

**Title:** Conical (Dirac) points in honeycomb structures

**Abstract:**

Many exciting physical properties of graphene can be traced to the

presence of conical singularities (“Dirac points”) in its dispersion

relation. Initial calculations (e.g. Wallace (1947)) were done within

the tight-binding model approximation (essentially, a discrete Laplace

operator on two vertices). More recently, Kuchment and Post (2007)

showed the presence of Dirac points in quantum graphs arranged to

resemble graphene (honeycomb) structure. Grushin (2009) considered

the Laplacian in R^2 with a weak potential having symmetries of the

honeycomb lattice; Fefferman and Weinstein (2012) proved the presence

of Dirac point for any potential with required symmetry. We will

present a very simple proof that works in all the above models: R^2

Laplacian, discrete Laplacian and quantum graph Laplacian.

In the talk we will discuss the statement of the problem, give a short

proof of the presence of the Dirac points using symmetries of the

operator and the Band-Parzanchevsky-BenShach quotient construction, and

a proof of stability of Dirac points which uses the Berry phase,

illustrated by animations.

The talk is based on joint work with A. Comech and discussions with

P. Kuchment and N. Do.