Dynamical characterization of $Z_2$ Floquet topological phases via quantum quenches

The complete characterization of a generic $d$-dimensional Floquet topological phase is usually hard for the requirement of information about the micromotion throughout the entire driving period. In a recent work [L. Zhang et al., Phys. Rev. Lett. 125, 183001 (2020)], an experimentally feasible dynamical detection scheme was proposed to characterize the integer Floquet topological phases using quantum quenches. However, this theory is still far away from completion, especially for free-fermion Floquet topological phases, where the states can also be characterized by $Z_2$ invariants. Here we develop a full and unified dynamical characterization theory for the $Z_2$ Floquet topological phases of different dimensionality and tenfold-way symmetry classes by quenching the system from a trivial and static initial state to the Floquet topological regime through suddenly changing the parameters and turning on the periodic driving. By measuring the minimal information of Floquet bands via the stroboscopic time-averaged spin polarizations, we show that the topological spin texture patterns emerging on certain discrete momenta of Brillouin zone called the 0 or $\pi$ gap highest-order band-inversion surfaces provide a measurable dynamical $Z_2$ Floquet invariant, which uniquely determines the Floquet boundary modes in the corresponding quasienergy gap and characterizes the $Z_2$ Floquet topology. The applications of our theory are illustrated via one- and two-dimensional models that are accessible in current quantum simulation experiments. Our work provides a highly feasible way to detect the $Z_2$ Floquet topology and completes the dynamical characterization for the full tenfold classes of Floquet topological phases, which shall advance the research in theory and experiments.