# Random-matrix behavior of quantum nonintegrable many-body systems with Dyson's three symmetries

## TL;DR

This paper introduces a one-dimensional spin model that exhibits Dyson's three symmetry classes, demonstrating that its spectral statistics align with random matrix theory predictions and revealing universal ratios in matrix element fluctuations.

## Contribution

The study constructs a nonintegrable spin model covering all three Dyson symmetry classes and verifies universal spectral and matrix element properties predicted by random matrix theory.

## Key findings

- Spectral statistics match random matrix predictions for each symmetry class.
- Universal ratios of matrix element fluctuations depend only on symmetries.
- Long-time dynamics reveal symmetry-dependent universal behaviors.

## Abstract

We propose a one-dimensional nonintegrable spin model with local interactions that covers Dyson's three symmetry classes (classes A, AI, and AII) depending on the values of parameters. We show that the nearest-neighbor spacing distribution in each of these classes agrees with that of random matrices with the corresponding symmetry. By investigating the ratios between the standard deviations of diagonal and off-diagonal matrix elements, we numerically find that they become universal, depending only on symmetries of the Hamiltonian and an observable, as predicted by random matrix theory. These universal ratios are evaluated from long-time dynamics of small isolated quantum systems.