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The complete characterization of a generic $d$-dimensional Floquettopological phase is usually hard for the requirement of information about themicromotion throughout the entire driving period. In a recent work [L. Zhang etal., Phys. Rev. Lett. 125, 183001 (2020)], an experimentally feasible dynamicaldetection scheme was proposed to characterize the integer Floquet topologicalphases using quantum quenches. However, this theory is still far away fromcompletion, especially for free-fermion Floquet topological phases, where thestates can also be characterized by $Z_{2}$ invariants. Here we develop thefirst full and unified dynamical characterization theory for the $Z_{2}$Floquet topological phases of different dimensionality and tenfold-way symmetryclasses by quenching the system from a trivial and static initial state to theFloquet topological regime through suddenly changing the parameters and turningon the periodic driving. By measuring the minimal information of Floquet bandsvia the stroboscopic time-averaged spin polarizations, we show that thetopological spin texture patterns emerging on certain discrete momenta ofBrillouin zone called the $0$ or $\pi$ gap highest-order band-inversionsurfaces provide a measurable dynamical $Z_{2}$ Floquet invariant, whichuniquely determines the Floquet boundary modes in the corresponding quasienergygap and characterizes the $Z_{2}$ Floquet topology. The applications of ourtheory are illustrated via one- and two-dimensional models that are accessiblein current quantum simulation experiments. Our work provides a highly feasibleway to detect the $Z_{2}$ Floquet topology and completes the dynamicalcharacterization for the full tenfold classes of Floquet topological phases,which shall advance the research in theory and experiments.

Dynamical characterization of $Z_{2}$ Floquet topological phases via quantum quenches