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arXiv:2607.24740

Anyon Condensation In Symmetry-Enriched Topological Phases: $G$-Grading of Multifusion Categories

Nianrui Fu, Siyuan Wang, Yu Zhao, Yidun Wan

Published Jul 27, 2026• 0 citations
arXivHTMLPDF

Abstract

Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a $G$-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension $E$ of $G$ by $N$: an $E$-graded fusion category induces a $G$-SET input by passing to the quotient symmetry $G$, and the resulting input naturally carries a compatible $N$-grading that implements the further condensation inside the SET phase while preserving $G$. We illustrate the construction using three quantum-double examples: the trivial extension $\mathbb{Z}_2\times\mathbb{Z}_2$, the non-Abelian semidirect product $S_3$, and the nontrivial central extension $\mathbb{Z}_4$. The $\mathbb{Z}_4$ example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.

DOI
10.48550/arxiv.2607.24740

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Anyon Condensation In Symmetry-Enriched Topological Phases: GGG-Grading of Multifusion Categories
Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a GGG-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension EEE of GGG by NNN: an EEE-graded fusion category induces a GGG-SET input by passing to the quotient symmetry GGG, and the resulting input naturally carries a compatible NNN-grading that implements the further condensation inside the SET phase while preserving GGG. We illustrate the construction using three quantum-double examples: the trivial extension Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​, the non-Abelian semidirect product S3S_3S3​, and the nontrivial central extension Z4\mathbb{Z}_4Z4​. The Z4\mathbb{Z}_4Z4​ example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.
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.
We construct exactly solvable models for a wide class of symmetry-enriched topological (SET) phases. Our construction applies to two-dimensional (2D) bosonic SET phases with finite unitary on-site symmetry group GGG and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: If we couple them to a dynamical lattice gauge field with gauge group GGG, the resulting gauge theories are equivalent to string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2{\mathbb{Z}}_{2}Z2​ symmetry which exchanges the eee and mmm type anyons. We further illustrate our construction with a number of additional examples.
A bstract We present a systematic framework for constructing exactly-solvable lattice models of symmetry-enriched topological (SET) phases based on an enlarged version of the string-net model. We also gauge the global symmetries of our SET models to obtain string-net models of pure topological phases. Without invoking externally imposed onsite symmetry actions, our approach promotes the string-net model of a pure topological order, specified by an input unitary fusion category F \mathcal{F} F F , to an SET model, specified by a multifusion category together with a set of isomorphisms. Two complementary construction strategies are developed in the main text: (i) promotion via outer automorphisms of F \mathcal{F} F F and (ii) promotion via the Frobenius algebras of F \mathcal{F} F F . The global symmetries derived via these two strategies are intrinsic to topological phases and are thus termed blood symmetries, as opposed to adopted symmetries, which can be arbitrarily imposed on topological phases. We propose the concept of symmetry-gauging family of topological phases, which are related by gauging their blood symmetries. With our approach, we construct the first explicit lattice realization of a nonabelian-symmetry-enriched topological phase — the S 3 symmetry-enriched ℤ 2 × ℤ 2 quantum-double phase. The approach further reveals the role of local excitations in SET phases and establishes their symmetry constraints.
The interplay between symmetry and topological properties plays a very important role in modern physics. In the past decade, the concept of symmetry-enriched topological (SET) phases was proposed and their classifications have been systematically studied for bosonic systems. Very recently, the concept of SET phases has been generalized into fermionic systems and their corresponding classification schemes are also proposed. Nevertheless, how to realize all these fermionic SET (fSET) phases in lattice models remains to be a difficult open problem. In this paper, we first construct exactly solvable models for nonanomalous nonchiral 2 + 1 D fSET phases, namely, the symmetry-enriched fermionic string-net models, which are described by commuting-projector Hamiltonians whose ground states are the fixed-point wave functions of each fSET phase. Mathematically, we provide a partial definition to G -graded super fusion category (when the total symmetry G f is a direct product of physical symmetry G and fermion-parity symmetry Z 2 f ), which is the input data of a symmetry-enriched fermionic string-net model. Next, we construct exactly solvable models for nonchiral 2 + 1 D fSET phases with ’t Hooft anomaly, especially the H 3 ( G , Z 2 ) fermionic ’t Hooft anomaly which is different from the well-known bosonic H 4 [ G , U ( 1 ) T ] anomaly. In our construction, this H 3 ( G , Z 2 ) fermionic ’t Hooft anomaly is characterized by a violation of fermion-parity conservation in some of the surface F moves (a type of renormalization moves for the ground state wave functions of surface fSET phases), and also by a new fermionic obstruction Θ in the surface pentagon equation. We demonstrate this construction in a concrete example that the surface topological order is a Z 4 gauge theory embedded into a fermion system and the total symmetry G f = Z 2 f × Z 2 × Z 4 . We further conjecture that the H 2 ( G , Z 2 ) fermionic ’t Hooft anomaly is characterized by a possible violation of nonterminating q -type (called σ -type) string in the fusion rules of surface models.
We propose a definition of a non-invertible symmetry enriched topological order (NI-SETO), and we implement our definition for string net models. We do so in two ways, using full inclusions of unitary fusion categories (UFCs), as well as anyon condensation. In both cases, the NI-SETO is a relative center of UFCs. All NI-SETOs can be realized in either model, where we can use enriched UFCs to get chiral examples on the boundary of a 3D Walker-Wang model representing the anomaly. We describe several examples of NI-SETOs and compute the qualitative symmetry action on anyons and symmetry defects using tube algebra techniques.
In topological phases in 2+12+12+1 dimensions, anyons fall into representations of quantum group symmetries. As proposed in our work [Hung and Wan, Int. J. Mod. Phys. B 28, 1450172 (2014)], the physics of a symmetry enriched phase can be extracted by the mathematics of (hidden) quantum group symmetry breaking of a ``parent phase.'' This offers a unified framework and classification of the symmetry enriched (topological) phases, including symmetry protected trivial phases as well. In this paper, we extend our investigation to the case where the ``parent'' phases are non-Abelian topological phases. We show explicitly how one can obtain the topological data and symmetry transformations of the symmetry enriched phases from that of the ``parent'' non-Abelian phase. Two examples are computed: (1) the $\text{Ising}\ifmmode\times\else\texttimes\fi{}\overline{\text{Ising}}$ phase breaks into the Z2{\mathbb{Z}}_{2}Z2​ toric code with Z2{\mathbb{Z}}_{2}Z2​ global symmetry; (2) the SU(2)8\mathrm{SU}{(2)}_{8}SU(2)8​ phase breaks into the chiral Fibonacci $\ifmmode\times\else\texttimes\fi{}$ Fibonacci phase with a Z2{\mathbb{Z}}_{2}Z2​ symmetry, a first non-Abelian example of symmetry enriched topological phase beyond the gauge-theory construction.