Integrable matrix theory: Level statistics

TL;DR

This paper investigates the level statistics of integrable matrices, revealing conditions under which they exhibit Poisson or level repulsion statistics, and demonstrates their ergodic and stationary nature in the large matrix limit.

Contribution

It provides a basis-independent construction of integrable matrices and characterizes their level statistics, including the conditions for Poisson behavior and the measure-zero exceptions.

Findings

Poisson statistics emerge when the number of integrals scales at least as log N.

Level repulsion occurs at small n or specific parameter values.

Ensembles are shown to be stationary and ergodic with respect to level statistics.

Abstract

We study level statistics in ensembles of integrable $N\times N$ matrices linear in a real parameter $x$. The matrix $H(x)$ is considered integrable if it has a prescribed number $n>1$ of linearly independent commuting partners $H^i(x)$ (integrals of motion) $\left[H(x),H^i(x)\right] = 0$, $\left[H^i(x), H^j(x)\right]$ = 0, for all $x$. In a recent work, we developed a basis-independent construction of $H(x)$ for any $n$ from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the $N\to\infty$ limit provided $n$ scales at least as $\log{N}$; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values $x=x_0$ or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at $ \mathcal{O}(N^{-0.5})$ deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest neighbor level statistics.