Bosonic fractional quantum Hall conductance in shaken honeycomb optical lattices without flat bands

We propose a scheme to realize bosonic fraction quantum Hall conductance in shaken honeycomb optical lattices. This scheme does not requires a very flat band, and the necessary long-range interaction relies on the s-wave scattering which is common in many ultracold-atom experiments. By filling the lattice at one-quarter with bosons under Feshbach resonance, two degenerate many-body ground states share one Chern number at 1, and exactly correspond to the fraction quantum Hall conductance at 1/2. Meanwhile, we prove that the fractional quantum Hall state can be prepared by adiabatically turning on the lattice shaking, and the fractional conductance is robust in the shaken lattice. This provides an easy way to initialize and prepare the fractional quantum Hall states in ultracold-atom platforms, and it paves a new way to investigate and simulate strong-correlated quantum matters with degenerate quantum gas.