Generalized random matrix conjecture for chaotic systems

TL;DR

This paper extends the random matrix conjecture for quantum chaotic systems to finite energies, proposing a specific statistical equivalence with the Circular Unitary Ensemble based on classical and quantum timescales.

Contribution

It introduces a semiclassical framework that generalizes the random matrix conjecture to finite energies for chaotic systems without time-reversal symmetry.

Findings

Spectrum near large finite energy matches Circular Unitary Ensemble statistics.

Effective matrix dimension depends on Heisenberg time and classical characteristic time.

Conjecture applies to both quantum chaotic systems and maps.

Abstract

The eigenvalues of quantum chaotic systems have been conjectured to follow, in the large energy limit, the statistical distribution of eigenvalues of random ensembles of matrices of size $N\rightarrow\infty$. Here we provide semiclassical arguments that extend the validity of this correspondence to finite energies. We conjecture that the spectrum of a generic fully chaotic system without time-reversal symmetry has, around some large but finite energy $E$, the same statistical properties as the Circular Unitary Ensemble of random matrices of dimension $N_{\rm eff} = \tH / \sqrt{24 d_1}$, where $\tH$ is Heisenberg time and $\sqrt{d_1}$ is a characteristic classical time, both evaluated at energy $E$. A corresponding conjecture is also made for chaotic maps.