Quartic multifractality and finite-size corrections at the spin quantum Hall transition

TL;DR

This paper investigates the multifractal spectrum at the spin quantum Hall transition, revealing it is essentially a quartic polynomial with precise coefficients, achieved through advanced simulations and novel finite-size correction analysis.

Contribution

We demonstrate that the multifractal spectrum at the spin quantum Hall transition is a quartic polynomial and introduce a new method for extracting it with high precision.

Findings

The multifractal spectrum $ au_q$ is a quartic polynomial in $q$.

The quartic curvature coefficient is determined as $oxed{ ext{(2.22±0.15)} imes 10^{-3}}$.

A novel finite-size correction analysis and a Kolmogorov-Smirnov based method improve precision.

Abstract

The spin quantum Hall (or class C) transition represents one of the few localization-delocalization transitions for which some of the critical exponents are known exactly. Not known, however, is the multifractal spectrum, $\tau_q$, which describes the system-size scaling of inverse participation ratios $P_q$, i.e., the $q$-moments of critical wavefunction amplitudes. We here report simulations based on the class C Chalker-Coddington network and demonstrate that $\tau_q$ is (essentially) a quartic polynomial in $q$. Analytical results fix all prefactors except the quartic curvature that we obtain as $\gamma=(2.22\pm{0.15})\cdot10^{-3}$. In order to achieve the necessary accuracy in the presence of sizable corrections to scaling, we have analyzed the evolution with system size of the entire $P_q$-distribution function. As it turns out, in a sizable window of $q$-values this distribution function exhibits a (single-parameter) scaling collapse already in the pre-asymptotic regime, where finite-size corrections are not negligible. This observation motivates us to propose a novel approach for extracting $\tau_q$ based on concepts borrowed from the Kolmogorov-Smirnov test of mathematical statistics. We believe that our work provides the conceptual means for high-precision investigations of multifractal spectra also near other localization-delocalization transitions of current interest, especially the integer (class A) quantum Hall effect.