Tunable intrinsic anomalous Hall effect in a shaken square optical lattice

We investigate the intrinsic anomalous linear and nonlinear Hall effect in a square optical lattice under the shaking. We find that the on-resonant shaking can induce a valley independent term which leads to the valley Hall effect. While the off-resonant shaking induces a valley-dependent term and the system shows an anomalous Hall effect. Furthermore, the nonlinear valley Hall effect also appears in this system, and the nonlinear valley Hall conductivities can be controlled by varying the shaking frequency and amplitude. These results will be useful for controlling the valley degree of freedom with ultracold atoms in an optical lattice.