Topological index for periodically driven time-reversal invariant 2D   systems
  We define a new $Z_2$-valued index to characterize the topological properties of periodically driven two dimensional crystals when the time-reversal symmetry is enforced. This index is associated with a spectral gap of the evolution operator over one period of time. When two such gaps are present, the Kane-Mele index of the eigenstates with eigenvalues between the gaps is recovered as the difference of the gap indices. This leads to an expression for the Kane-Mele invariant in terms of the Wess-Zumino amplitude. We illustrate the relation of the new index to the edge states in finite geometries by numerically solving an explicit model on the square lattice that is periodically driven in a time-reversal invariant way. 
David Carpentier and Pierre Delplace and Michel Fruchart and Krzysztof   Gaw\k{e}dzki
cond-mat.mes-hall cond-mat.quant-gas math-ph math.MP