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

Disordered Systems and Neural Networks;Statistical Mechanics;Strongly Correlated Electrons