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We propose a general framework to solve tight binding models in D dimensionallattices driven by ac electric fields. Our method is valid for arbitrarydriving regimes and allows to obtain effective Hamiltonians for differentexternal fields configurations. We establish an equivalence with timeindependent lattices in D+1 dimensions, and analyze their topologicalproperties. Further, we demonstrate that non-adiabaticity drives a transitionfrom topological invariants defined in D+1 to D dimensions. Our approachprovides a theoretical framework to analyze ac driven systems, with potentialapplications in topological states of matter, and non-adiabatic topologicalquantum computation, predicting novel outcomes for future experiments.

Floquet-Bloch theory and topology in periodically driven lattices