Symmetry constraints on many-body localization

We derive general constraints on the existence of many-body localized (MBL) phases in the presence of global symmetries, and show that MBL is not possible with symmetry groups that protect multiplets (e.g. all non-Abelian symmetry groups). Based on simple representation theoretic considerations, we derive general Mermin-Wagner-type principles governing the possible alternative fates of non-equilibrium dynamics in isolated, strongly disordered quantum systems. Our results rule out the existence of MBL symmetry protected topological phases with non-Abelian symmetry groups, as well as time-reversal symmetry protected electronic topological insulators, and in fact all fermion topological insulators and superconductors in the 10-fold way classification. Moreover, extending our arguments to systems with intrinsic topological order, we rule out MBL phases with non-Abelian anyons as well as certain classes of symmetry enriched topological orders.