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

TL;DR

This paper introduces the concept of global symmetry fragmentation in nonabelian anyons within symmetry-enriched topological phases, revealing nonlinear symmetry representations through an exactly solvable string-net model.

Contribution

It uncovers a universal mechanism called global symmetry fragmentation that leads to nonlinear symmetry representations in nonabelian SET phases, expanding understanding beyond traditional classifications.

Findings

Discovery of symmetry-invariant anyons with fractional symmetry charges

Identification of symmetry-permuted anyons with hybridized internal spaces

Proposal of nonlinear, coherent symmetry representations in topological phases

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 nonabelian anyons remain elusive. This obscurity stems from the multidimensional internal gauge spaces intrinsic to nonabelian 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 nonabelian 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 nonabelian SETs and suggest new routes for symmetry-enabled control in topological quantum computation.