Topology in Floquet-Bloch systems: physics beyond the topological insulators

Floquet theory provides a powerful framework for understanding the dynamics of periodically-driven quantum systems. When applied to systems with spatially periodic potentials, the emergence of Floquet-Bloch quasienergy bands suggests many exciting opportunities to explore topological phenomena through analogies to those known in equilibrium. Due to the periodicity of quasienergy, however, the topological classification of periodically-driven systems is richer than that of static systems. This opens the way for discovering truly new, non-equilibrium topological phenomena. For example, a one dimensional Floquet system may host chiral bands, while in two dimensions a system whose Floquet bands all have zero Chern numbers may support robust chiral edge states. In this talk I will address the question: when is a Floquet-Bloch system {\it not} like a static system? I will discuss both general considerations, based on the structure of the time-dependent Floquet-Bloch evolution operator, and specific examples which highlight the possibilities available in both non-interacting and interacting systems.