Floquet gap-dependent topological classifications from color-decorated frequency lattices with space-time symmetries

We find a class of Floquet topological phases exhibiting gap-dependent topological classifications in quantum systems with a dynamical space-time symmetry and an antisymmetry. This is in contrast to all existing Floquet topological phases protected by static symmetries, where the topological classification across all quasienergy gaps is characterized by the same Abelian group. We demonstrate this gap-dependent classification phenomenon using the frequency-domain formulation of the time-dependent Hamiltonian. Moreover, we provide an interpretation of the resulting Floquet topological phases using a frequency lattice with a decoration represented by color degrees of freedom on the lattice vertices. These colors correspond to the coefficient N of the group extension of the system symmetry group G along the frequency lattice, given by N=Z×H1[A,M]. The distinct topological classifications that arise at different energy gaps in its quasienergy spectrum are described by the torsion product of the cohomology group H2[G,N] classifying the group extension.