Integer quantum Hall transition in the presence of a long-range-correlated quenched disorder

Abstract

We theoretically study the effect of long-ranged inhomogeneities on the critical properties of the integer quantum Hall transition. For this purpose we employ the real-space renormalization-group (RG) approach to the network model of the transition. We start by testing the accuracy of the RG approach in the absence of inhomogeneities, and infer the correlation length exponent $\ensuremath{\nu}=2.39$ from a broad conductance distribution. We then incorporate macroscopic inhomogeneities into the RG procedure. Inhomogeneities are modeled by a smooth random potential with a correlator which falls off with distance as a power law, ${r}^{\ensuremath{-}\ensuremath{\alpha}}.$ Similar to the classical percolation, we observe an enhancement of $\ensuremath{\nu}$ with decreasing \ensuremath{\alpha}. Although the attainable system sizes are large, they do not allow one to unambiguously identify a cusp in the $\ensuremath{\nu}(\ensuremath{\alpha})$ dependence at ${\ensuremath{\alpha}}_{c}=2/\ensuremath{\nu},$ as might be expected from the extended Harris criterion. We argue that the fundamental obstacle for the numerical detection of a cusp in the quantum percolation is the implicit randomness in the Aharonov-Bohm phases of the wave functions. This randomness emulates the presence of a short-range disorder alongside the smooth potential.