PT-Symmetric $SU(2)$-like Random Matrix Ensembles: Invariant Distributions and Spectral Fluctuations

TL;DR

This paper studies a new class of 2x2 PT-symmetric random matrices, analyzing their spectral properties and level repulsion, revealing connections to quantum chaos and extending random matrix theory.

Contribution

It introduces a novel ensemble of PT-symmetric 2x2 matrices with specific invariant distributions and explores their spectral fluctuation properties.

Findings

Probability densities follow power laws with domain-dependent exponents.

Level repulsion parameter $ u$ is at least 2, matching Gaussian Unitary Ensemble.

Systematic framework for PT-symmetric random matrix spectral analysis.

Abstract

We consider an ensemble of $2\times 2$ normal matrices with complex entries representing operators in the quantum mechanics of 2 - level parity-time reversal (PT) symmetric systems. The randomness of the ensemble is endowed by obtaining probability distributions based on symmetry and statistical independence. The probability densities turn out to be power law with exponents that depend on the boundedness of the domain. For small spacings, $\sigma$, the probability density varies as $\sigma^{\nu}$, $\nu \geq 2$. The degree of level repulsion is a parameter of great interest as it makes a connection to quantum chaos; the lower bound of $\nu$ for our ensemble coincides with the Gaussian Unitary Ensemble. We believe that the systematic development presented here paves the way for further generalizations in the field of random matrix theory for PT-symmetric quantum systems.