# Fundamental relation between longitudinal and transverse conductivities in the quantum Hall system

## TL;DR

This paper derives a fundamental relation between the longitudinal and transverse conductivities in quantum Hall systems, showing agreement with experimental data and providing analytic formulas within a simplified disorder model.

## Contribution

It establishes a proportional relation between the derivative of Hall conductivity and the square of longitudinal conductivity, extending understanding of conductivity behavior in quantum Hall systems.

## Key findings

- Derived analytic formulas for conductivities using a simple disorder model.
- Found a proportional relation between dσxy/dB and Bσxx^2.
- Confirmed the relation matches experimental results in GaAs/AlGaAs systems.

## Abstract

We investigate the relation between the diagonal ($\sigma_{xx}$) and off-diagonal ($\sigma_{xy}$) components of the conductivity tensor in the quantum Hall system. We calculate the conductivity components for a short-range impurity potential using the linear response theory, employing an approximation that simply replaces the self-energy by a constant value $-i \hbar /(2 \tau)$ with $\tau$ the scattering time. The approximation is equivalent to assuming that the broadening of a Landau level due to disorder is represented by a Lorentzian with the width $\Gamma = \hbar /(2 \tau)$. Analytic formulas are obtained for both $\sigma_{xx}$ and $\sigma_{xy}$ within the framework of this simple approximation at low temperatures. By examining the leading terms in $\sigma_{xx}$ and $\sigma_{xy}$, we find a proportional relation between $\mathrm{d}\sigma_{xy}/\mathrm{d}B$ and $B \sigma_{xx}^2$. The relation, after slight modification to account for the long-range nature of the impurity potential, is shown to be in quantitative agreement with experimental results obtained in the GaAs/AlGaAs two-dimensional electron system at the low magnetic-field regime where spin splitting is negligibly small.