April 22, 2014 – Gregory Berkolaiko, Texas A&M University

G. Berkolaiko, Texas A&M University

Title: Conical (Dirac) points in honeycomb structures

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.