Higher-order Topological Phases in Dynamical Optical Lattices*

We propose a versatile framework to dynamically generate Floquet higher-order topological insulators by multi-step driving of topologically trivial Hamiltonians. Two analytically solvable examples are used to illustrate this procedure to yield Floquet quadrupole and octupole insulators with zero and/or $$\pi$$-corner modes protected by mirror symmetries. Furthermore, we introduce dynamical topological invariants from the full unitary return map and show its phase bands contain Weyl singularities whose topological charges form dynamical multipole moments in the Brillouin zone. Combining them with the topological index of Floquet Hamiltonian gives a pair of $$Z_2$$ invariant $$\nu_0$$ and $$\nu_{\pi}$$ which fully characterize the higher-order topology and predict the appearance of zero- and $$\pi$$-corner modes. Our work establishes a systematic route to construct and characterize Floquet higher-order topological phases.