# Eigenstate phases with finite on-site non-Abelian symmetry

## TL;DR

This paper investigates eigenstate phases in disordered spin chains with non-Abelian symmetry, developing a formalism for invariant Hamiltonians and identifying thermal and many-body localized phases with symmetry breaking.

## Contribution

It introduces a general group-theoretic framework for constructing symmetric Hamiltonians and applies it to study eigenstate phases with $S_3$ symmetry, revealing distinct thermal and MBL phases.

## Key findings

- Identification of thermal and MBL phases with $S_3$ symmetry
- Detection of spontaneous symmetry breaking from $S_3$ to $ ext{Z}_3$
- Evidence of an intermediate phase with full $S_3$ symmetry

## Abstract

We study the eigenstate phases of disordered spin chains with on-site finite non-Abelian symmetry. We develop a general formalism based on standard group theory to construct local spin Hamiltonians invariant under any on-site symmetry. We then specialize to the case of the simplest non-Abelian group, $S_3$, and numerically study a particular two parameter spin-1 Hamiltonian. We observe a thermal phase and a many-body localized phase with a spontaneous symmetry breaking (SSB) from $S_3$ to $\mathbb{Z}_3$ in our model Hamiltonian. We diagnose these phases using full entanglement distributions and level statistics. We also use a spin-glass diagnostic specialized to detect spontaneous breaking of the $S_3$ symmetry down to $\mathbb{Z}_3$. Our observed phases are consistent with the possibilities outlined by Potter and Vasseur [Phys. Rev. B 94, 224206 (2016)], namely thermal/ ergodic and spin-glass many-body localized (MBL) phases. We also speculate about the nature of an intermediate region between the thermal and MBL+SSB regions where full $S_3$ symmetry exists.