Explanation for the Resistivity Law in Quantum Hall System

Abstract

We consider a 2D electron system in a strong magnetic field, where the local Hall resistivity $ρ_{xy}(\vec r)$ is a function of position and $ρ_{xx}(\vec r)$ is small compared to $ρ_{xy}$. 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 $ρ_{xx}$ and is approximately proportional to the magnitude of fluctuations in $ρ_{xy}$. 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.