Spectral Statistics in the Lowest Landau Band

TL;DR

This paper investigates spectral statistics in the lowest Landau band of a disordered 2D system under strong magnetic field, confirming some theoretical predictions and discovering a new large-scale behavior in spectral fluctuations.

Contribution

It verifies predicted relations between spectral statistics exponents and identifies a new power-law regime at large spectral scales.

Findings

Confirmed the relation $eta = 1 - rac{1}{ u d}$ between spectral statistics exponents.

Found a new spectral fluctuation regime with a large power-law exponent $oxed{ ext{1.38}}$ for $ar N > 60$.

Observed that existing models do not fully describe the entire spacings distribution in this system.

Abstract

We study the spectral statistics in the center of the lowest Landau band of a 2D disordered system with smooth potential and strong transverse magnetic field. Due to the finite size of the system, the energy range in which there are extended states is finite as well. The behavior in this range can be viewed as the analogue of the Anderson metal-insulator transition for the case of the Hall system. Accordingly, we verify recent predictions regarding the exponent of the asymptotic power law of $\Sigma^2 (\bar N)$, $\gamma$, and that of the stretched exponential dominating the large $s$ behavior of the spacings distribution, $\alpha$. Both the relations, $\alpha = 1- \gamma$, and $\gamma = 1 - {1\over{\nu d}}$ where $\nu$ is the critical exponent of the localization length and $d$ is the dimension, are found to hold within the accuracy of our computations. However, we find that none of several possible models of the entire spacings distribution correctly describes our situation. Finally, for very large $\bar N$, $\bar N > 60$, we find a new regime in which $\Sigma^2 (\bar N)$ behaves as a power law with an unexpectedly large power, $\gamma_1 = 1.38 \pm 0.02$.