Non-adiabatic Berry phase description of Floquet topological states

We study Floquet topological states using non-adiabatic Berry phases. We show that these geometric quantities correspond to the centers of Wannier states in the frequency domain, in complete analogy with how conventional Berry phases in equilibrium systems correspond to Wannier centers in position space. This link reveals a natural way to understand the topology of periodically driven phases in terms of the behavior of Wannier centers in the frequency domain. We illustrate the versatility of this approach with a model of a driven Majorana wire. We show that the bulk topology is characterized by a Chern number computed using a non-adiabatic version of the Berry curvature. Furthermore, we find that the corresponding Majorana modes at the edge carry a quantized non-adiabatic polarization in the frequency domain. We discuss higher-dimensional systems, such as Floquet Chern insulators, as well as systems with interactions and disorder.