Nonlinear Symmetry-Fragmentation of Nonabelian Anyons In Symmetry-Enriched Topological Phases: A String-Net Model Realization

Abstract: Symmetry-enriched topological (SET) phases combine intrinsic topological order with global symmetries, giving rise to novel symmetry phenomena. While SET phases with Abelian anyons are relatively well understood, those involving non-Abelian anyons remain elusive. This obscurity stems from the multi-dimensional internal gauge spaces intrinsic to non-Abelian anyons -- a feature first made explicit in [1,2] and further explored and formalized in our recent works [3-8]. These internal spaces can transform in highly nontrivial ways under global symmetries. In this work, we employ an exactly solvable model -- the multifusion Hu-Geer-Wu string-net model introduced in a companion paper [9] -- to reveal how the internal gauge spaces of non-Abelian anyons transform under symmetries. We uncover a universal mechanism, global symmetry fragmentation (GSF), whereby symmetry-invariant anyons exhibit internal Hilbert space decompositions into eigensubspaces labeled by generally fractional symmetry charges. Meanwhile, symmetry-permuted anyons hybridize and fragment their internal spaces in accordance with their symmetry behavior. These fragmented structures realize genuinely nonlinear symmetry representations -- to be termed coherent representations -- that transcend conventional linear and projective classifications, reflecting the categorical nature of symmetries in topological phases. Our results identify nonlinear fragmentation as a hallmark of non-Abelian SETs and suggest new routes for symmetry-enabled control in topological quantum computation.