Heavy-Tailed Hall Conductivity Fluctuations in Quantum Hall Transitions

TL;DR

Problem: near IQHE plateau transitions, the full distribution of the zero-temperature Hall conductivity $\\sigma_{xy}$ exhibits heavy-tailed fluctuations. Approach: compute the distribution over about $10^5$ disorder realizations in a lattice model using the Kubo formula, across varying system sizes, disorder strengths, and correlation lengths. Findings: the distribution develops heavy tails with a power-law exponent $\\alpha\\approx 2.3$--$2.5$, yielding a finite mean but a divergent variance, and this behavior persists across system sizes and disorder parameters, indicating a breakdown of self-averaging in the critical regime. Significance: the results align with random-matrix theory predictions for topological indices and suggest heavy-tailed statistics are intrinsic to disorder-driven topological transitions, challenging ensemble averaging in mesoscopic samples.

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 $α\approx 2.3$--$2.5$, 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.

Figures (4)

• Figure 1: Ensemble mean and standard deviation of the Hall conductivity, $\sigma_{xy}(\nu)$, with $W=1.0$ and $\eta=1.25$. Solid thick lines show the ensemble mean, while solid thin lines denote the 95th percentile. The density of states is shown in gray. Large realization-to-realization fluctuations persist near transitions as $M$ increases.
• Figure 2: Distribution of Hall conductivity deviations, $P\!\left(|\sigma_{xy}-\langle\sigma_{xy}\rangle|\right)$, presented in log--log scale, at $\nu=2.5$ (a) and $\nu=2.0$ (b) with $\eta=0.75$. The dashed line is the fitted power law slope and the vertical solid line indicates where the tail begins.
• Figure 3: Fitted distribution width $\Delta\sigma_{0}$ (a) and power-law exponent $\alpha$ (b) from a distribution function of Hall conductivity. The dashed horizontal line is the power law $-2.5$ predicted by Berryberry2018geometricberry2020geometricberry2020quantum using random matrix models of topological indices.
• Figure S1: The widths of distributions defined as he $95$th percentile of the Hall conductivity $\sigma_{xy}(\nu)$. In each figure, we show the second moment of these widths between consecutive integer fillings.