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.

18 pages, 26 figures, discussion on number variance added; published version

References in corpus (4)

• Localization of interacting fermions at high temperature
• The Link between Integrability, Level Crossings, and Exact Solution in Quantum Models
• Saturation of number variance in embedded random matrix ensembles
• Rotationally invariant ensembles of integrable matrices

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