No-resonance conditions, random matrices, and quantum chaotic models

TL;DR

This paper examines no-resonance conditions in quantum many-body chaotic systems and random matrices, showing that sums of eigenvalues follow Poisson statistics and exploring implications for quantum equilibration theory.

Contribution

It demonstrates that sums of eigenvalues in quantum chaotic and random matrix models follow Poisson statistics, challenging assumptions about no-resonance conditions.

Findings

Eigenvalue sums exhibit Poisson statistics

No-resonance conditions can be violated or nearly violated

Bounds in quantum equilibration are generalized for these cases

Abstract

In this article we investigate no-resonance conditions for quantum many body chaotic systems and random matrix models. No-resonance conditions are properties of the spectrum of a model, usually employed as a theoretical tool in the analysis of late time dynamics. The first order no-resonance condition holds when a spectrum is non-degenerate, while higher order no-resonance conditions imply sums of an equal number of energies are non-degenerate outside of permutations of the indices. This resonance condition is usually assumed to hold for quantum chaotic models. In this work we use several tests from random matrix theory to demonstrate that the statistics of sums of eigenvalues, that are of interest to due to the no-resonance conditions, have Poisson statistics, and lack level repulsion. This result is produced for both a quantum chaotic Hamiltonian as well as the Gaussian Unitary Ensemble and Gaussian Orthogonal Ensemble. This implies some models may have violations of the no-resonance condition or "near" violations. We finish the paper by generalizing important bounds in quantum equilibration theory to cases where the no-resonance conditions are violated, and to the case of random matrix models.