Topology of electronic materials and their linear and nonlinear responses (Lecture 1)

Abstract

Certain natural geometric properties of electron wavefunctions in a crystal turn out to explain a vast range of experimentally relevant properties. The original example was the explanation of the integer quantum Hall effect by Thouless and co-workers in terms of the “Berry curvature” derived from Bloch states. We now understand that a kind of gauge field in the Brillouin zone is the key to many equilibrium and linear-response properties, and current work is seeking to generalize these results to nonlinear and non-equilibrium properties as well. This talk reviews the basic concepts of wavefunction geometry starting from basic notions of undergraduate quantum mechanics, then covers more recent applications to new topological states, with a particular focus on effects beyond the standard adiabatic limit.