Nonlinear symmetry-fragmentation of nonabelian anyons in symmetry-enriched topological phases: a string-net model realization

A bstract 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 nonabelian anyons remain elusive. This obscurity stems from the multi-dimensional internal gauge spaces intrinsic to nonabelian anyons — a feature first made explicit in 1 and further explored in our recent works 2–7. 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 8 — to reveal how the internal gauge spaces of nonabelian anyons transform under symmetries. We uncover a universal phenomenon, 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 that transcend conventional linear and projective classifications. Our results identify nonlinear global symmetry fragmentation as a hallmark of SETs and may shed new light on symmetry-enabled control in topological quantum computation.