Crossover between the Gaussian orthogonal ensemble, the Gaussian unitary ensemble, and Poissonian statistics

TL;DR

This paper develops a unified formula for level spacing distributions describing all crossovers among Poisson, GOE, and GUE statistics, and applies it to magnetoexcitons to analyze quantum chaos and symmetry breaking.

Contribution

It introduces a general formula for arbitrary crossovers between Poisson, GOE, and GUE statistics, extending previous specific cases in random matrix theory.

Findings

The formula accurately describes known special cases of crossovers.

Application to magnetoexcitons reveals transitions between different statistical regimes.

The approach distinguishes regular and chaotic behavior in quantum systems.

Abstract

Until now only for specific crossovers between Poissonian statistics (P), the statistics of a Gaussian orthogonal ensemble (GOE), or the statistics of a Gaussian unitary ensemble (GUE) analytical formulas for the level spacing distribution function have been derived within random matrix theory. We investigate arbitrary crossovers in the triangle between all three statistics. To this aim we propose an according formula for the level spacing distribution function depending on two parameters. Comparing the behavior of our formula for the special cases of P$\rightarrow$GUE, P$\rightarrow$GOE, and GOE$\rightarrow$GUE with the results from random matrix theory, we prove that these crossovers are described reasonably. Recent investigations by F.~Schweiner~\emph{et al.} [Phys. Rev. E~\textbf{95}, 062205 (2017)] have shown that the Hamiltonian of magnetoexcitons in cubic semiconductors can exhibit all three statistics in dependence on the system parameters. Evaluating the numerical results for magnetoexcitons in dependence on the excitation energy and on a parameter connected with the cubic valence band structure and comparing the results with the formula proposed allows us to distinguish between regular and chaotic behavior as well as between existent or broken antiunitary symmetries. Increasing one of the two parameters, transitions between different crossovers, e.g., from the P$\rightarrow$GOE to the P$\rightarrow$GUE-crossover, are observed and discussed.