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We argue that the static non-linear Hall conductivity can always berepresented as a vector in two-dimensions and as a pseudo-tensor inthree-dimensions independent of its microscopic origin. In a single band modelwith a constant relaxation rate this vector or tensor is proportional to theBerry curvature dipole \cite{Sodemann_2015}. Here, we develop a quantumBoltzmann formalism to second order in electric fields. We find that inaddition to the Berry Curvature Dipole term, there exist additional disordermediated corrections to the non-linear Hall tensor that have the same scalingin impurity scattering rate. These can be thought of as the non-linearcounterparts to the side-jump and skew-scattering corrections to the Hallconductivity in the linear regime. We illustrate our formalism by computing thedifferent contributions to the non-linear Hall conductivity of two-dimensionaltilted Dirac fermions.

Symmetry and Quantum Kinetics of the Non-linear Hall Effect