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 Floquet topological phases in which the topological invariants are constructed with only minimal information of the static Floquet bands. For a d-dimensional phase that is initially static and trivial, we introduce the quench dynamics by suddenly turning on the periodic driving. We show that the quench dynamics exhibits emergent topological patterns in (d - 1)-dimensional momentum subspaces where Floquet hands cross, from which the Floquet topological invariants are directly obtained. This result provides a simple and unified characterization in which one can extract the number of conventional and anomalous Floquet boundary modes and identify the topologically protected singularities in the phase bands. These applications are illustrated with one- and two-dimensional models that are readily accessible in cold-atom experiments. Our study opens a new framework for the characterization of Floquet topological phases.