# Wave-packet dynamics in locally periodic media with a focus on the effects of Bloch band degeneracies

We study the dynamics of waves in media with a local periodic structure which varies adiabatically (over many periods of the periodic lattice) across the medium. We focus in particular on the case where symmetries of the periodic structure lead to degeneracies in the Bloch band dispersion surface. An example of such symmetry-induced degeneracies are the Dirac points of media with honeycomb lattice symmetry, such as graphene. Our results are as follows: (1) A systematic and rigorous derivation of the anomalous velocity of wave-packets due to the Bloch bands Berry curvature. The Berry curvature is large near to degeneracies, where it takes the form of a monopole. We also derive terms which do not appear in the works of Niu et al. which describe a field-particle coupling effect between the evolution of observables associated with the wave-packet and the evolution of the wave packet envelope. These terms are of the same order as the anomalous velocity. (2) Restricting to one spatial dimension, the derivation of the precise dynamics when a wave-packet is incident on a Bloch band degeneracy. In particular we derive the probability of an inter-band transition and show that our result is consistent with an appropriately interpreted Landau-Zener formula. I will present these results for solutions of a model Schr\{o}dinger equation; extending our results to systems described by Maxwell's equations is the subject of ongoing work. This is joint work with Michael Weinstein and Jianfeng Lu.