Higher-Order Floquet Topological Phases and Defects

We report the existence of higher-order Floquet topological phases. As a prototypical model, we study, both analytically and numerically, the two-dimensional π-flux square lattice with time-dependent nearest-neighbor hopping modulation. We show that with open boundary conditions, the quasienergy spectrum supports four-fold degenerate steady states at the center and/or edge of the Floquet zone localized at corners. The number of these Floquet corner states can be tuned by the amplitude and frequency of the drive. Under periodic boundary conditions and when the hopping modulation preserves diagonal mirror symmetry, the higher-order Floquet topology is revealed through a mirror-graded Floquet winding number. Moreover, we show that in a strip geometry, the eigenvalues of the Floquet Wannier operator reveal edge polarization. Finally, we show that inhomogeneous driving protocols can be used to “stir” domain walls defects to generate lower-dimensional Floquet topological vortex defects.