The mixing-spacetime symmetry in the Floquet-Bloch band theory

Abstract

We discover a class of spacetime symmetries unique to time-periodic systems, which we term "mixing symmetry" due to its combination of space and time coordinates in the symmetry transformation. We systematically enumerate the symmetry groups, and classify the corresponding Floquet-Bloch band theories by utilizing the winding number of quasi-energy. Moreover, we provide a comprehensive scheme for the experimental realization of these symmetries. The particle propagator exhibits an intriguing pattern that remains invariant even under transformations mixing space and time coordinates. We anticipate that this distinct feature can be observed in current cold atom experiments.