Mathematical and Physical Aspects of Topologically Protected States

Adiabatic cycles, topological transitions, and the Chern-Simons axion coupling

David Vanderbilt

Rutgers University


I will first discuss the behavior of the Chern-Simons axion coupling under a non-trivial adiabatic cycle and show how this corresponds to pumping of anomalous Hall conductivity across the sample. In particular, I will present a derivation of a "surface theorem" relating the the bulk theta to the excess Chern number of hybrid Wannier center sheets at the surface. I will end by discussing topological-to-normal transitions that are mediated by an intermediate Weyl semimetal phase, and discuss how this can be understood from the viewpoint of the Chern-Simons three-form and second Chern number.