# Symmetry and Quantum Kinetics of the Non-linear Hall Effect

## TL;DR

This paper develops a quantum Boltzmann formalism to analyze the non-linear Hall effect, revealing additional disorder-mediated contributions beyond the Berry curvature dipole, with applications to tilted Dirac fermions.

## Contribution

It introduces a second-order quantum Boltzmann approach that uncovers disorder-mediated corrections to the non-linear Hall effect, extending previous Berry curvature dipole models.

## Key findings

- Disorder-mediated corrections scale with impurity scattering rate.
- Additional contributions analogous to side-jump and skew-scattering effects.
- Application to 2D tilted Dirac fermions demonstrates the formalism.

## Abstract

We argue that the static non-linear Hall conductivity can always be represented as a vector in two-dimensions and as a pseudo-tensor 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 \cite{Sodemann_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 non-linear Hall tensor that have the same scaling in impurity scattering rate. These can be thought of as the non-linear 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 non-linear Hall conductivity of two-dimensional tilted Dirac fermions.