The effect of $SU(2)$ symmetry on many-body localization and thermalization

The many-body localized (MBL) phase is characterized by a complete set of quasi-local integrals of motion and area-law entanglement of excited eigenstates. We study the effect of non-Abelian continuous symmetries on MBL, considering the case of $SU(2)$ symmetric disordered spin chains. The $SU(2)$ symmetry imposes strong constraints on the entanglement structure of the eigenstates, precluding conventional MBL. We construct a fixed-point Hamiltonian, which realizes a non-ergodic (but non-MBL) phase characterized by eigenstates having logarithmic scaling of entanglement with the system size, as well as an incomplete set of quasi-local integrals of motion. We study the response of such a phase to local symmetric perturbations, finding that even weak perturbations induce multi-spin resonances. We conclude that the non-ergodic phase is generally unstable and that $SU(2)$ symmetry implies thermalization. The approach introduced in this work can be used to study dynamics in disordered systems with non-Abelian symmetries, and provides a starting point for searching non-ergodic phases beyond conventional MBL.