Nested Shadows of Anyons:A Framework for Identifying Topological Phases

Abstract: The 1-form symmetries in two-dimensional topological systems are ``shadowed'' as global symmetries in their one-dimensional quantum transfer matrices. In this work, we introduce a distinct shadow effect arising from the pair-creation of anyons, which manifests as a local symmetry of the quantum transfer matrix. The interplay between these two shadow effects provides a powerful framework for characterizing topological phases without extensive numerical simulations. Specifically, we derive the phase diagram of the filtered toric code state and precisely identify phase boundaries using the nested shadows of anyons. Additionally, we reveal that a class of topological states host gapless edge modes protected by 1-form symmetry rather than global symmetry. Finally, we apply our approach to the three-dimensional toric code and X-cube states, uncovering a nontrivial path in phase space that connects them through a subdimensional critical point, which is highly challenging to detect numerically due to the complexity of simulating three-dimensional systems.