Statistical properties of power-law random banded unitary matrices in the delocalization-localization transition regime

TL;DR

This paper investigates the statistical properties of power-law random banded unitary matrices (PRBUM) in the transition between delocalized and localized states, revealing new features and their relation to quantum criticality.

Contribution

It provides a detailed numerical analysis of PRBUM, highlighting novel statistical features and their implications for understanding quantum phase transitions.

Findings

PRBUM can serve as a unitary analog of power-law random Hermitian matrices for Anderson transition.

Eigenvector fractal dimension depends uniquely on ensemble parameters, differing from Hermitian models.

Nearest neighbor spacing distribution varies with time-reversal symmetry, showing semi-Poisson and anomalous level repulsion.

Abstract

Power-law random banded unitary matrices (PRBUM), whose matrix elements decay in a power-law fashion, were recently proposed to model the critical statistics of the Floquet eigenstates of periodically driven quantum systems. In this work, we numerically study in detail the statistical properties of PRBUM ensembles in the delocalization-localization transition regime. In particular, implications of the delocalization-localization transition for the fractal dimension of the eigenvectors, for the distribution function of the eigenvector components, and for the nearest neighbor spacing statistics of the eigenphases are examined. On the one hand, our results further indicate that a PRBUM ensemble can serve as a unitary analog of the power-law random Hermitian matrix model for Anderson transition. On the other hand, some statistical features unseen before are found from PRBUM. For example, the dependence of the fractal dimension of the eigenvectors of PRBUM upon one ensemble parameter displays features that are quite different from that for the power-law random Hermitian matrix model. Furthermore, in the time-reversal symmetric case the nearest neighbor spacing distribution of PRBUM eigenphases is found to obey a semi-Poisson distribution for a broad range, but display an anomalous level repulsion in the absence of time-reversal symmetry.