Probing symmetries of quantum many-body systems through gap ratio statistics

TL;DR

This paper extends the analysis of gap ratio statistics to systems with symmetries, providing analytical formulas and demonstrating their effectiveness in diagnosing symmetries in complex quantum many-body systems.

Contribution

It introduces analytical surmises for gap ratio distributions in block-structured random matrices and applies them to identify symmetries in various quantum many-body models.

Findings

Analytical formulas match numerical simulations well.

Method effectively detects symmetries in complex quantum systems.

Applicable to a wide range of many-body physics models.

Abstract

The statistics of gap ratios between consecutive energy levels is a widely used tool, in particular in the context of many-body physics, to distinguish between chaotic and integrable systems, described respectively by Gaussian ensembles of random matrices and Poisson statistics. In this work we extend the study of the gap ratio distribution P(r) to the case where discrete symmetries are present. This is important, since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place. Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks. We check our formulae against simulations from large random matrices, showing excellent agreement. We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically-driven spin systems. In all these models the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces. We finally discuss the relevance of our analysis for existing results in the literature, as well as its practical usefulness, and point out possible future applications and extensions.