Scaling Behavior of Level Statistics in Quantum Hall Regime

TL;DR

This study investigates the scaling properties of level statistics in quantum Hall systems, revealing how energy dependence of statistical measures relates to localization length and how finite-range disorder affects critical behavior.

Contribution

It provides a detailed numerical analysis of level statistics in quantum Hall regimes, including effects of finite-range disorder and extensions to higher Landau bands.

Findings

Critical exponent of localization length determined from level statistics.

Finite-range disorder improves critical behavior analysis.

Pathological behavior observed with short-range disorder in second Landau band.

Abstract

The scaling property of level statistics in the quantum Hall regime, i.e. 2D disordered electron systems subject to strong magnetic fields, is analyzed numerically in the light of the random matrix theory. The energy dependences of the effective level repulsion parameter, the two level correlation, the GUE-GOE crossover parameter, and the rigidity (or $\Delta_3$-statistics) of the level distributions are investigated for different system sizes by unfolding the original data and by dividing the unfolded spectrum into small regions. It is shown that the critical exponent of the localization length as a function of energy can be determined through the energy dependence of the level statistics. The analyses are carried out not only for the lowest Landau band (LB) but also for the second lowest LB. Furthermore the effect of finite range of disordered potential is studied. The short-ranged potential case in the second lowest LB is found to be pathological as in other studies of critical behavior, and it is confirmed that this pathological behavior is improved in the case of disordered potential with finite ranges.