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 1D systems that are spanned by just one parameter, a Bloch phase, are topologically trivial. Recently, an ensemble of 1D Floquet systems, double kicked rotors (DKRs) depending 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, which determines the change in the mean angular momentum of a state in a 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 is fully described by a system having translational invariance on the phase plane. This system can be characterized by a \emph{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 number of QE bands and the degeneracy of the QE states of the phase-plane system, limits the values of the DKR-ensemble integer. In particular, this integer is generically nonzero, showing the general topological nontriviality of the DKR ensemble.