A quantum theory of the nonlinear Hall effect

The nonlinear Hall effect is an unconventional response, in which a voltage can be driven by two perpendicular currents in the Hall-bar measurement. Unprecedented in the family of the Hall effects, it can survive time-reversal symmetry but is sensitive to the breaking of discrete and crystal symmetries. It is quantum by nature because of its deep connection with the Berry curvature.
However, a full quantum description is still absent. Here we construct a quantum theory of the nonlinear Hall effect by using the diagrammatic technique. Quite different from nonlinear optics, 92 of the total 100 diagrams account for the disorder scattering, which plays a decisive role in the electronic transport. We present both qualitative and quantitative differences between the quantum theory and the semiclassical theories. As an application, we propose a pure electric experiment to detect the Berry curvature distribution near the band edge. We discuss the symmetry of the nonlinear-response diagrams and predict a pure disorder-induced nonlinear Hall effect for point groups C₃, C_3h, C_3v, D_3h, D₃ in 2D, and T, T_d, C_3h, D_3h in 3D. This work will be helpful for explorations of the topological physics beyond the linear regime.

arXiv:2004.09742