# Disorder-induced nonlinear Hall effect with time-reversal symmetry

## TL;DR

This paper derives formulas for the nonlinear Hall effect considering disorder, applies them to a 2D Dirac model, and establishes a scaling law to distinguish disorder effects in experiments.

## Contribution

It introduces the first comprehensive theoretical framework for disorder-induced contributions to the nonlinear Hall effect and proposes a scaling law for experimental identification.

## Key findings

- Derived formulas for nonlinear Hall conductivity with disorder
- Applied formulas to tilted 2D Dirac model
- Established a general scaling law for experiments

## Abstract

The nonlinear Hall effect has opened the door towards deeper understanding of topological states of matter. It can be observed as the double-frequency Hall voltage response to an ac longitudinal current in the presence of time-reversal symmetry. Disorder plays indispensable roles in various linear Hall effects, such as the localization in the quantized Hall effects and the extrinsic mechanisms of the anomalous, spin, and valley Hall effects. Unlike in the linear Hall effects, disorder enters the nonlinear Hall effect even in the leading order. However, the disorder-induced contribution to the nonlinear Hall effect has not been addressed. Here, we derive the formulas of the nonlinear Hall conductivity in the presence of disorder scattering. We apply the formulas to calculate the nonlinear Hall response of the tilted 2D Dirac model, which is the symmetry-allowed minimal model for the nonlinear Hall effect and can serve as a building block in realistic band structures. More importantly, we construct the general scaling law of the nonlinear Hall effect, which may help in experiments to distinguish disorder-induced contributions to the nonlinear Hall effect. This work will be instructive for exploring unconventional responses upon breaking discrete or crystal symmetries in emergent physical systems and materials.