A Unified Symmetry Classification of Many-Body Localized Phases

Anderson localization admits a complete symmetry classification given by the Altland-Zirnbauer (AZ) tenfold scheme, whereas an analogous framework for interacting many-body localization (MBL) has remained elusive. Here we develop a symmetry-based classification of static MBL phases formulated at the level of local integrals of motion (LIOMs). We show that a symmetry is compatible with stable MBL if and only if its action can be consistently represented within a quasi-local LIOM algebra, without enforcing extensive degeneracies or nonlocal operator mixing. This criterion sharply distinguishes symmetry classes: onsite Abelian symmetries are compatible with stable MBL and can host distinct symmetry-protected topological MBL phases, whereas continuous non-Abelian symmetries generically preclude stable MBL. By systematically combining AZ symmetries with additional onsite symmetries, we construct a complete classification table of MBL phases, identify stable, fragile, and unstable classes, and provide representative lattice realizations. Our results establish a unified and physically transparent framework for understanding symmetry constraints on MBL.