Characterizing generalized Floquet topological states in hybrid space-time dimensions

In spatiotemporally modulated systems, topological states exist not only in energy gaps but also in momentum gaps. Such unconventional topological states impose challenges on topological physics. The underlying models also make the conventional Hamiltonian descriptions complicated. Here, we propose to describe such systems with space- and time-direction transfer matrices which substantially simplify the underlying theory and give direct information on the topological properties of the quasienergy and quasimomentum gaps. In particular, we find that the space- and time-direction reflection phases can serve as signatures for distinguishing various topological phases of the quasienergy and quasimomentum gaps. This approach directly reveals the topological properties of the band gap, avoiding the complexity in calculating bulk band topology in hybrid energy-moment space. By investigating two concrete models, we show that the method works well for both Hermitian and non-Hermitian systems. Furthermore, we uncover an unconventional topological state, called the anomalous Floquet quasimomentum gap, whose topological properties are invariant for different choices of the unit-cell center. This work advances the study of topological phenomena in hybrid space-time (energy-momentum) dimension that are attracting much interest due to the development of spatiotemporally modulated materials.