Critical statistics in a power-law random banded matrix ensemble

TL;DR

This paper studies the spectral properties of a power-law random banded matrix ensemble, revealing unique critical statistics at the localization-delocalization transition point, with eigenstates being multifractal.

Contribution

It provides numerical evidence that spectral statistics at the critical point differ from semi-Poisson, highlighting unique features of systems with a localization-delocalization transition.

Findings

Spectral statistics at criticality differ from semi-Poisson.

Eigenstates are multifractal at the transition.

The model exhibits a localization-delocalization transition at ermu=1.

Abstract

We investigate the statistical properties of the eigenvalues and eigenvectors in a random matrix ensemble with $H_{ij}\sim |i-j|^{-\mu}$. It is known that this model shows a localization-delocalization transition (LDT) as a function of the parameter $\mu$. The model is critical at $\mu=1$ and the eigenstates are multifractals. Based on numerical simulations we demonstrate that the spectral statistics at criticality differs from semi-Poisson statistics which is expected to be a general feature of systems exhibiting a LDT or `weak chaos'.