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An important challenge in the field of many-body quantum dynamics is toidentify non-ergodic states of matter beyond many-body localization (MBL).Strongly disordered spin chains with non-Abelian symmetry and chains ofnon-Abelian anyons are natural candidates, as they are incompatible withstandard MBL. In such chains, real space renormalization group methods predicta partially localized, non-ergodic regime known as a quantum critical glass (acritical variant of MBL). This regime features a tree-like hierarchy ofintegrals of motion and symmetric eigenstates with entanglement entropy thatscales as a logarithmically enhanced area law. We argue that such tentativenon-ergodic states are perturbatively unstable using an analytic computation ofthe scaling of off-diagonal matrix elements and accessible level spacing oflocal perturbations. Our results indicate that strongly disordered chains withnon-Abelian symmetry display either spontaneous symmetry breaking or ergodicthermal behavior at long times. We identify the relevant length and time scalesfor thermalization: even if such chains eventually thermalize, they can exhibitnon-ergodic dynamics up to parametrically long time scales with a non-analyticdependence on disorder strength.

Perturbative instability of non-ergodic phases in non-Abelian quantum chains