Heavy-Tailed Hall Conductivity Fluctuations in Quantum Hall Transitions

TL;DR

This paper investigates the distribution of Hall conductivity in the integer quantum Hall effect, revealing heavy-tailed fluctuations near the transition point that challenge the assumption of self-averaging in small samples.

Contribution

It provides the first detailed analysis of the full distribution of Hall conductivity in a lattice model, highlighting heavy-tailed fluctuations and their implications for transport near criticality.

Findings

Heavy-tailed fluctuations with power-law decay in conductivity distribution.

Finite mean but divergent variance of conductivity near transition.

Breakdown of self-averaging in small, coherent samples at criticality.

Abstract

We study the full distribution of the zero-temperature Hall conductivity in a lattice model of the IQHE using the Kubo formula across disorder realizations. Near the localization-delocalization transition, the conductivity exhibits heavy-tailed fluctuations characterized by a power-law decay with exponent $\alpha \approx 2.3$--$2.5$, indicating a finite mean but a divergent variance. The heavy tail persists across a range of system sizes, correlation lengths of the disorder potential and fillings. Our results demonstrate a breakdown of self-averaging in transport in small, coherent samples near criticality, in agreement with findings in random matrix models of topological indices.