Symmetry and quantum kinetics of the nonlinear Hall effect

We argue that static nonlinear Hall conductivity can always be represented as a vector in two dimensions and as a pseudotensor in three dimensions independent of its microscopic origin. In a single-band model with a constant relaxation rate, this vector or tensor is proportional to the Berry curvature dipole I. Sodemann and L. Fu, Phys. Rev. Lett 115, 216806 (2015). Here, we develop a quantum Boltzmann formalism to second order in electric fields. We find that in addition to the Berry curvature dipole term, there exist additional disorder-mediated corrections to the nonlinear Hall tensor that have the same scaling in the impurity scattering rate. These can be thought of as the nonlinear counterparts to the side-jump and skew-scattering corrections to the Hall conductivity in the linear regime. We illustrate our formalism by computing the different contributions to the nonlinear Hall conductivity of two-dimensional tilted Dirac fermions.