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We analyze strong correlation effects and topological properties ofinteracting fermions with a Falicov-Kimball type interaction in circularlyshaken hexagonal optical lattices, which can be effectively described by theHaldane-Falicov-Kimball model, using the real-space Floquet dynamicalmean-field theory (DMFT). The Haldane model, a paradigmatic model of the Cherninsulator, is experimentally relevant, because it has been realized usingcircularly shaken hexagonal optical lattices. We show that in the presence ofstaggering a charge density wave emerges, which is affected by interactions andresonant tunneling. We demonstrate that interactions smear out the edge statesby introducing a finite life time of quasiparticles. Even though a generalmethod for calculating the topological invariant of a nonequilibrium steadystate is lacking, we extract the topological invariant using a Laughlin chargepump set-up. We find and attribute to the dissipations into the bath connectedto every lattice site, which is intrinsic to real-space Floquet DMFT methods,that the pumped charge is not an integer even for the non-interacting case atvery low reservoir temperatures. Furthermore, using the rate equation based onthe Floquet-Born-Markov approximation, we calculate the charge pump from therate equations for the non-interacting case to identify the role of thespectral properties of the bath. Starting from this approach we propose anexperimental protocol for measuring quantized charge pumping.

Charge density wave and charge pump of interacting fermions in circularly shaken hexagonal optical lattices