Topological properties of adiabatically varied Floquet systems

Energy or quasienergy (QE) band spectra depending on two parameters may have a nontrivial topological characterization by Chern integers. Band spectra of one-dimensional (1D) systems that are spanned by just one parameter, a Bloch phase, are topologically trivial. Recently, an ensemble of 1D Floquet (time-periodic) systems, double-kicked rotors (DKRs) that are classically nonintegrable and depend on an external parameter, has been studied. It was shown that a QE band spanned by both the Bloch phase and the external parameter is characterized by a Chern integer. The latter determines the change in the mean angular momentum of a state in the band when the external parameter is adiabatically varied by a natural period. We show here, under conditions much more general than in previous works, that the ensemble of DKRs for all values of the external parameter corresponds to a 1D double-kicked particle (DKP) having translational invariance in the position-momentum phase plane. This DKP can be characterized by a second Chern integer, which is shown to be connected with the integer above for the DKR ensemble. This connection is expressed by a Diophantine equation (DE), which we derive. The DE, involving the band degeneracies of the DKR ensemble and of the DKP system, determines the allowed values of the DKR-ensemble integer. In particular, this integer is generically nonzero, showing the general topological nontriviality of the DKR ensemble.