Random matrix theory of integrability-to-chaos transition

TL;DR

This paper introduces a random matrix framework to understand the transition from integrability to chaos in quantum systems, revealing universal features in matrix element distributions.

Contribution

It formulates a simple random matrix ensemble that accurately models level spacing distributions during the transition, uncovering universal power-law behaviors.

Findings

The ensemble reproduces level spacing distributions across various systems.

Distributions of matrix elements are dominated by simple power laws.

Universal features are observed in diverse physical systems.

Abstract

The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.