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

TL;DR

This work addresses how nonabelian anyons transform under global symmetries in symmetry-enriched topological (SET) phases. Using an exactly solvable enlarged string-net model, it introduces global symmetry fragmentation (GSF) and demonstrates that symmetry-invariant nonabelian anyons decompose into fractional-charge eigensubspaces while symmetry-permuted anyons further fragment their internal spaces, yielding genuinely nonlinear (coherent) representations. In the concrete case of the $D(S_3)$ phase with EM-exchange symmetry ($\mathbb{Z}_2$), the authors read off symmetry actions from half-braiding data and show explicit fragmentation patterns, such as $H$ splitting into charges $2/3$ and $1/6$ and $C\oplus F$ forming 2D irreps with charges $0$ and $1/2$, with several fragments displaying nonlinear representations. The results suggest that nonlinear symmetry fragmentation is a general feature of SETs with nonabelian anyons and opens routes for symmetry‑assisted control in topological quantum computation, including potential connections to universal QC in such 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.

Figures (4)

• Figure 1: HGW string-net model lattice. Each plaquette has a tail (wavy line).
• Figure 2: Anyon $a$ crossing a domain wall (thick line) is transformed to be anyon $\mathcal{G}(a)$.
• Figure 3: Composing symmetry transformations is equivalent to anyon crossing two domain walls (thick lines). Edge orientation is upward.
• Figure 4: The fattened lattice of one plaquette.