Nonlinear Hall effect on a disordered lattice

The nonlinear Hall effect has recently attracted significant interest due to its potential as a promising spectral tool and for device applications. A theory of the nonlinear Hall effect on a disordered lattice is a crucial step towards explorations in realistic devices, but has not yet been addressed. We study the nonlinear Hall response on a lattice, which allows us to introduce strong disorder numerically. We reveal a disorder-induced Berry curvature that was not discovered in previous perturbation theories. The disorder-induced Berry curvature induces a fluctuation of the nonlinear Hall conductivity, which anomalously increases as the Fermi energy moves from the band edges to higher energies. More importantly, the fluctuation may explain those observations in recent experiments. We also find signatures of localization of the nonlinear Hall effect. This Letter shows a territory of the nonlinear Hall effect yet to be explored.