The Quantized Hall Insulator

Abstract

We model the insulator neighboring the 1/k quantum Hall phase by a random network of puddles of filling fraction 1/k. The puddles are coupled by weak tunnel barriers. Using Kirchoff's laws we prove that the macroscopic Hall resistivity is quantized at $k h/ e^2$ and independent of magnetic field and current bias -- in agreement with recent experimental observations. In addition, for k>1 this theory predicts non linear longitudinal response $V\sim I^α$ at zero temperature, and $V/I\sim T^{1-{1\over α}}$ at low bias. $α$ is determined using Renn and Arovas' theory for the single junction response, and is related to the Luttinger liquid spectra of the edge states straddling the typical tunnel barrier. The dependence of $V(I)$ on the magnetic field is related to the typical puddle size. Deviations of $V(I)$ from a pure power law are estimated using a series/parallel approximation for the two dimensional random nonlinear resistor network. We check the validity of this approximation by numerically solving for a finite square lattice network.