Quantum Systems: A Mathematical Journey from Few to Many Particles

Linear and nonlinear waves in honeycomb structures

Michael Weinstein

Columbia University

I will focus on the properties of waves in 2-dimensional honeycomb structures. Two areas where such structures have been explored extensively are (i) condensed matter physics (graphene) and (ii) photonics.

Many of the remarkable wave propagation properties of honeycomb structures are related to the presence of conical singularities (``Dirac points'') in the associated dispersion surfaces. Physical modeling (since Wallace, 1947) has centered on the tight-binding (discrete) approximation of the underlying partial differential equation (Schroedinger's eqn).

I'll present results (joint with C.L. Fefferman) on the existence of Dirac points for the non-relativistic Schroedinger equation with a generic honeycomb lattice potential. We also prove that the very long time dynamics of wave-packets is governed by an effective two-dimensional Dirac system. Finally, we discuss the propagation of nonlinear waves in honeycomb structures.