Floquet topological phases with time-reversal and space-inversion symmetries and dynamical detection of topological charges

For spinful systems with spin 1/2, it is generally believed that $\mathcal{P}$ and $\mathcal{T}$ invariant strong and second-order topologies exist in four band and eight band systems, respectively. Here, by using periodic driving, we find it is possible to have a strong topological insulator, a second-order topological insulator, and a hybrid-order topological insulator in a single four band system. Furthermore, we established a direct connection between topology and dynamics. More convenient experimental detection for these topological phases has also been proposed. This study provides the theoretical basis for a topological insulator that possess hybrid-order boundary states beyond the conventional regimes.