  The topological characterization of nonequilibrium topological matter is
highly nontrivial because familiar approaches designed for equilibrium
topological phases may not apply. In the presence of crystal symmetry, Floquet
topological insulator states cannot be easily distinguished from normal
insulators by a set of symmetry eigenvalues at high symmetry points in the
Brillouin zone. This work advocates a physically motivated, easy-to-implement
approach to enhance the symmetry analysis to distinguish between a variety of
Floquet topological phases. Using a two-dimensional inversion-symmetric
periodically-driven system as an example, we show that the symmetry eigenvalues
for anomalous Floquet topological states, of both first-order and second-order,
are the same as for normal atomic insulators. However, the topological states
can be distinguished from one another and from normal insulators by inspecting
the occurrence of stable symmetry inversion points in their microscopic
dynamics. The analysis points to a simple picture for understanding how
topological boundary states can coexist with localized bulk states in anomalous
Floquet topological phases.
