Level statistics detect generalized symmetries

TL;DR

This paper demonstrates that level statistics can detect generalized symmetries, including non-invertible, nonlocal, and q-deformed symmetries, revealing their influence on spectral properties and phase protection.

Contribution

It provides multiple examples showing how level statistics identify generalized symmetries beyond conventional ones, including non-invertible and q-deformed symmetries, and introduces a new q-deformed inversion symmetry.

Findings

Level statistics detect generalized symmetries in various models.

Resolving generalized symmetries is necessary to observe level repulsion.

Discovered a q-deformed inversion symmetry related to SPT phases.

Abstract

Level statistics are a useful probe for detecting symmetries and distinguishing integrable and non-integrable systems. I show by way of several examples that level statistics detect the presence of generalized symmetries that go beyond conventional lattice symmetries and internal symmetries. I consider non-invertible symmetries through the example of Kramers-Wannier duality at an Ising critical point, symmetries with nonlocal generators through the example of a spin-$1$ anisotropic Heisenberg chain, and $q$-deformed symmetries through an example closely related to recent work on $q$-deformed SPT phases. In each case, conventional level statistics detect the generalized symmetries, and these symmetries must be resolved before seeing characteristic level repulsion in non-integrable systems. For the $q$-deformed symmetry, I discovered via level statistics a $q$-deformed generalization of inversion that is interesting in its own right and that may protect $q$-deformed SPT phases.