Floquet-Bloch theory and topology in periodically driven lattices

We propose a general framework to solve tight binding models in D dimensional lattices driven by ac electric fields. Our method is valid for arbitrary driving regimes and allows us to obtain effective Hamiltonians for different external field configurations. We establish an equivalence with time-independent lattices in D+1 dimensions and analyze their topological properties. Furthermore, we demonstrate that nonadiabaticity drives a transition from topological invariants defined in D+1 to D dimensions. Our results have potential applications in topological states of matter and nonadiabatic topological quantum computation, predicting novel outcomes for future experiments.We thank P. Delplace, A. G. Grushin, S. Kohler, P.C. E. Stamp, and M. Bu¨ttiker for useful discussions. A. G. L. acknowledges the JAE program (MICINN) and we both acknowledge Grant No. MAT2011-24331 and the ITN Grant No. 234970 (EU)