Quantized Hall insulator: Transverse and longitudinal transport

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 Kirchoffșs laws we prove that the macroscopic Hall resistivity is quantized at kh/(Formula presented) and independent of magnetic field and current bias, in agreement with recent experimental observations. In addition, for k>1 this theory predicts a nonlinear longitudinal response V∼(Formula presented) at zero temperature and V/I∼(Formula presented) at low bias. α is determined using Renn and Arovasșs theory for the single junction response [Phys. Rev. B 51, 16 832 (1995)] 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 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.