Integrable matrix theory: 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.Comment: 18 pages, 26 figures, discussion on number variance added; published versio

Rotationally invariant ensembles of integrable matrices

“…(8)(9), while ansatz type-M ≥ 1 are given by Eqs. (38)(39)(40)(41)(42) along with Eqs. (43)(44)(45)(46).…”

“…The final step is to determine ansatz T through Eqs. (38)(39)(40). The choice of x 0 , |γ , Λ and H 2 defines the ansatz type-M commuting family, while the choice of H 1 specifies a matrix within the family.…”

“…Eq. (26) is particularly significant because it allows one to study the level statistics of the ensemble of N × N type-1 integrable matrices H 1 N , which according to numerical simulations generally turn out to be Poisson [40].…”

“…Moreover, it turns out that certain choices drastically affect the level statistics, e.g. those where d k and ε k are correlated [21,40].…”

“…Here we first derive such a formulation and then obtain an appropriate probability distribution of random integrable matrices with a given number of integrals of motion. In a follow-up work [40] we will study level statistics of these ensembles as well as spectral statistics of individual integrable matrices, see Fig. 1 for an example.…”