Unified theory to characterize Floquet topological phases by quench dynamics

The conventional characterization of periodically driven systems usually necessitates the time-domain information beyond Floquet bands, hence lacking universal and direct schemes of measuring Floquet topological invariants. Here we propose a unified theory based on quantum quenches to characterize generic $d$-dimensional ($d$D) Floquet topological phases, in which the topological invariants are constructed with only minimal information of the static Floquet bands. For a $d$D phase which is initially static and trivial, we introduce the quench dynamics by suddenly turning on the periodic driving, and show that the quench dynamics exhibits emergent topological patterns in ($d-1$)D momentum subspaces where Floquet bands cross, from which the Floquet topological invariants are directly obtained. This prediction provides a simple and unified characterization, in which one can not only extract the number of conventional and anomalous Floquet boundary modes, but also identify the topologically protected singularities in the phase bands. The applications are illustrated with 1D and 2D models which are readily accessible in cold atom experiments. Our study opens a new framework for the characterization of Floquet topological phases.