Floquet topological states in shaking optical lattices

In this Rapid Communication we propose realistic schemes to realize topologically nontrivial Floquet states by shaking optical lattices, using both the one-dimensional lattice and two-dimensional honeycomb lattice as examples. The topological phase in the two-dimensional model exhibits quantum anomalous Hall effect. The transition between topological trivial and nontrivial states can be easily controlled by both shaking frequency and shaking amplitude. Our schemes have two major advantages. First, both the static Hamiltonian and the shaking scheme are sufficiently simple to implement. Secondly, it requires relatively small shaking amplitude and therefore heating can be minimized. These two advantages make our schemes much more practical.