Explanation for the Resistivity Law in Quantum Hall System

TL;DR

This paper proposes a theoretical explanation for the empirical resistivity law in quantum Hall systems, linking macroscopic longitudinal resistivity to local Hall resistivity fluctuations in a strong magnetic field.

Contribution

It introduces a model showing how slow decay of correlations or multi-scale fluctuations in local resistivity explain the empirical law in quantum Hall systems.

Findings

Macroscopic resistivity is weakly dependent on local longitudinal resistivity.

Resistivity proportional to fluctuations in local Hall resistivity.

Provides a theoretical basis for the empirical resistivity law.

Abstract

We consider a 2D electron system in a strong magnetic field, where the local Hall resistivity $\rho_{xy}(\vec r)$ is a function of position and $\rho_{xx}(\vec r)$ is small compared to $\rho_{xy}$. Particularly if the correlations fall off slowly with distance, or if fluctuations exist on several length scales, one finds that the macroscopic longitudinal resistivity $R_{xx}$ is only weakly dependent on $\rho_{xx}$ and is approximately proportional to the magnitude of fluctuations in $\rho_{xy}$. This may provide an explanation of the empirical law $R_{xx} \propto B \frac{dR_{xy}}{dB}$ where $R_{xy}$ is the Hall resistance, and $B$ is the magnetic field.