A Unified Symmetry Classification of Many-Body Localized Phases

TL;DR

This work extends the Altland–Zirnbauer symmetry framework to interacting disordered systems by formulating MBL in terms of a quasi-local LIOM algebra and analyzing how global symmetries act covariantly on these LIOMs. The authors derive a criterion for when a symmetry is compatible with stable MBL, showing that onsite Abelian symmetries typically permit stable MBL and can host SPT-MBL phases, while continuous non-Abelian symmetries generically destroy localization. They construct a comprehensive symmetry classification that combines AZ data with onsite symmetries (e.g., U(1), Z_n, SU(2)) and reveal that several AZ classes collapse to a smaller set of MBL classes (A, AI, AII, AIII, D, C) in the interacting, LIOM framework. The resulting table clarifies which symmetry actions on the LIOM algebra support stable, fragile, or unstable MBL phases and unifies previously disparate observations under a physically transparent, symmetry-guided paradigm. This framework provides a foundation for understanding localization in higher dimensions, driven/open systems, and topology–localization interplay beyond static 1D models, with concrete lattice realizations illustrating the classification.

Abstract

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.