Log-normal Distribution of Level Curvatures in the Localized Regime: Analytical Verification

TL;DR

This paper analytically and numerically investigates the distribution of level curvatures in one-dimensional disordered rings, revealing a log-normal behavior in the extreme localization limit and highlighting differences in positive moments.

Contribution

It provides an analytical verification of the log-normal distribution of level curvatures in the localized regime, supported by numerical analysis of moments.

Findings

Negative moments can be evaluated analytically in the localization limit.

The curvature distribution exhibits a log-normal asymptotic behavior.

Positive moments differ from other log-normal quantities.

Abstract

We study numerically and analytically the moments of the dimensionless level curvature for one-dimensional disordered rings of the circumference L pierced by a magnetic flux. The negative moments of the curvature distribution can be evaluated analytically in the extreme localization limit. The ensuing small curvature asymptotic of the corresponding distribution has a "log-normal" behavior. Numerically studied positive moments show differences from other log-normally distributed quantities.