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arXiv:2408.04006 · SciPost Physics

Generalizations of Kitaev’s honeycomb model from braided fusion categories

Luisa Eck, Paul Fendley

Published Jun 2, 2025• 9 citations
arXivHTMLPDF

Abstract

Fusion surface models, as introduced by Inamura and Ohmori, extend the concept of anyon chains to 2+1 dimensions, taking fusion 2-categories as their input. In this work, we construct and analyze fusion surface models on the honeycomb lattice built from braided fusion 1-categories. These models preserve mutually commuting plaquette operators and anomalous 1-form symmetries. Their Hamiltonian is chosen to mimic the structure of Kitaev’s honeycomb model, which is unitarily equivalent to the Ising fusion surface model. In the anisotropic limit, where one coupling constant is dominant, the fusion surface models reduce to Levin-Wen string-nets. In the isotropic limit, they are described by weakly coupled anyon chains and are likely to realize chiral topological order. We focus on three specific examples: (i) Kitaev’s honeycomb model with a perturbation breaking time-reversal symmetry that realizes chiral Ising topological order, (ii) a \mathbb{Z}_N ℤ N generalization proposed by Barkeshli et al., which potentially realizes chiral parafermion topological order, and (iii) a novel Fibonacci honeycomb model featuring a non-invertible 1-form symmetry.

DOI
10.21468/scipostphys.18.6.170

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Floquet topological order (FTO) extends topology beyond equilibrium, giving rise to phenomena, such as the dynamical transmutation of anyons. Here, we realize $\mathbb{Z}_N$ FTO in a generalized Kitaev honeycomb model with clock degrees of freedom using a three-step periodic drive. The driven system admits an emergent description in terms of constrained parafermions coupled to conserved $\mathbb{Z}_N$ gauge fields. At the boundary, the drive generates chiral parafermionic Floquet edge modes. We resolve their propagation through correlation functions, characterize their fractional quantum-information transport through the radical chiral unitary index, and use their dynamics to probe exchange statistics. In the bulk, we explore the dynamical anyon transmutation with a non-equilibrium loop order parameter. At a Floquet sweet spot, where the drive becomes Clifford, these phenomena can be accessed using stabilizer simulations. Short-time matrix-product-state simulations away from this point show that the chiral edge response persists at accessible times. Our results establish $\mathbb{Z}_N$ Kitaev models as a setting for exploring intrinsically non-equilibrium topological phenomena on qudit-based quantum processors.

cond-mat.str-elCross-listedarXiv:2610.07156•Oct 7, 2026•0 citations
Fusion surface models, as introduced by Inamura and Ohmori, extend the concept of anyon chains to 2+1 dimensions, taking fusion 2-categories as their input. In this work, we construct and analyze fusion surface models on the honeycomb lattice built from braided fusion 1-categories. These models preserve mutually commuting plaquette operators and anomalous 1-form symmetries. Their Hamiltonian is chosen to mimic the structure of Kitaev’s honeycomb model, which is unitarily equivalent to the Ising fusion surface model. In the anisotropic limit, where one coupling constant is dominant, the fusion surface models reduce to Levin-Wen string-nets. In the isotropic limit, they are described by weakly coupled anyon chains and are likely to realize chiral topological order. We focus on three specific examples: (i) Kitaev’s honeycomb model with a perturbation breaking time-reversal symmetry that realizes chiral Ising topological order, (ii) a \mathbb{Z}_N ℤ N generalization proposed by Barkeshli et al., which potentially realizes chiral parafermion topological order, and (iii) a novel Fibonacci honeycomb model featuring a non-invertible 1-form symmetry.
Fusion surface models generalize the concept of anyon chains to 2+1 dimensions, utilizing fusion 2-categories as their input. We investigate bond-algebraic dualities in these systems and show that distinct module tensor categories \mathcal{M} ℳ over the same braided fusion category \mathcal{B} ℬ give rise to dual lattice models. This extends the 1+1d result that dualities in anyon chains are classified by module categories over fusion categories. We analyze two concrete examples: (i) a \text{Rep}(S_3) Rep ( S 3 ) model with a constrained Hilbert space, dual to the spin- \tfrac{1}{2} 1 2 XXZ model on the honeycomb lattice, and (ii) a bilayer Kitaev honeycomb model, dual to a spin- \tfrac{1}{2} 1 2 model with XXZ and Ising interactions. Unlike regular \mathcal{M}=\mathcal{B} ℳ = ℬ fusion surface models, which conserve only 1-form symmetries, models constructed from \mathcal{M} ≠ \mathcal{B} ℳ ≠ ℬ can exhibit both 1-form and 0-form symmetries, including non-invertible ones.
We construct (2+1)-dimensional lattice systems, which we call fusion surface models. These models have finite non-invertible symmetries described by general fusion 2-categories. Our method can be applied to build microscopic models with, for example, anomalous or non-anomalous one-form symmetries, 2-group symmetries, or non-invertible one-form symmetries that capture non-abelian anyon statistics. The construction of these models generalizes the construction of the 1+1d anyon chains formalized by Aasen, Fendley, and Mong. Along with the fusion surface models, we also obtain the corresponding three-dimensional classical statistical models, which are 3d analogues of the 2d Aasen-Fendley-Mong height models. In the construction, the “symmetry TFTs” for fusion 2-category symmetries play an important role.
We construct large families of mixed-state topological orders in (3+1)D, including intrinsically mixed-state topological orders. We realize these theories by relating the data parameterizing braided fusion 2-categories to the strong higher-form symmetries and their anomalies that characterize mixed-state topological order. We focus here on bosonic theories, which include theories which admit an emergent fermion in the spectrum of line operators. Our general framework is exemplified on the lattice, where we present concrete models and identify how the data of braided fusion 2-categories are physically encoded. We then formalize this data and show that every braided fusion 2-category arises from a general construction (decoherence followed by strong and weak gauging) as the category of topological operators of a (3+1)D mixed state, with intrinsically mixed-state topological orders distinguished by the presence of strong higher-form symmetries whose anomalies cannot be realized in any (3+1)D gapped ground state. Notably, we show that (3+1)D admits a new mechanism with no (2+1)D counterpart: intrinsically mixed-state order whose topological operators braid non-degenerately, but which carries a gravitational anomaly.
This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category C\mathcal{C}C on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided C∗\rm C^*C∗-tensor category of superselection sectors SSS\mathsf{SSS}SSS. By constructing explicit isomorphisms between the fusion spaces of SSS\mathsf{SSS}SSS and those of the Drinfeld center Z(C)Z(\mathcal{C})Z(C), we show that these two categories have isomorphic FFF- and RRR-symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center,  SSSf≃Z(C).\,\mathsf{SSS}_f \simeq Z(\mathcal{C}).SSSf​≃Z(C). This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum dimensions.
We explore a large NNN generalization of the Kitaev model on the honeycomb lattice with a simple nearest-neighbor interacting Hamiltonian. In particular, we focus on the Z4{\mathbb{Z}}_{4}Z4​ case with isotropic couplings, which is characterized by an exact Z4{\mathbb{Z}}_{4}Z4​ 1-form symmetry. Guided by symmetry considerations and an analytical study in the single-chain limit, on infinitely long cylinders, we find the model is gapped with an extremely short correlation length. Combined with the Z4{\mathbb{Z}}_{4}Z4​ 1-form symmetry, this suggests the model is topologically ordered. To pin down the nature of this phase, we further study the model on both finite and infinitely long strips, where we consistently find a c=1c=1c=1 conformal field theory (CFT) description, suggesting the existence of chiral edge modes described by a free boson CFT. Further evidence is found by studying the dimer correlators on infinitely long strips. We find the dimer correlation functions show a power-law decay with the exponent close to 2 on the boundary of the strip, while decaying much faster in the bulk. Combined with the topological entanglement entropy extracted from cylinder geometry, we identify the phase as a chiral spin liquid with a $\mathrm{U}{(1)}_{\ensuremath{-}8}$ chiral topological order. A unified perspective for all ZN{\mathbb{Z}}_{N}ZN​ type Kitaev models is also discussed.
Chiral Parafermions in Floquet ZN\mathbb{Z}_NZN​ Topological Order
Floquet topological order (FTO) extends topology beyond equilibrium, giving rise to phenomena, such as the dynamical transmutation of anyons. Here, we realize ZN\mathbb{Z}_NZN​ FTO in a generalized Kitaev honeycomb model with clock degrees of freedom using a three-step periodic drive. The driven system admits an emergent description in terms of constrained parafermions coupled to conserved ZN\mathbb{Z}_NZN​ gauge fields. At the boundary, the drive generates chiral parafermionic Floquet edge modes. We resolve their propagation through correlation functions, characterize their fractional quantum-information transport through the radical chiral unitary index, and use their dynamics to probe exchange statistics. In the bulk, we explore the dynamical anyon transmutation with a non-equilibrium loop order parameter. At a Floquet sweet spot, where the drive becomes Clifford, these phenomena can be accessed using stabilizer simulations. Short-time matrix-product-state simulations away from this point show that the chiral edge response persists at accessible times. Our results establish ZN\mathbb{Z}_NZN​ Kitaev models as a setting for exploring intrinsically non-equilibrium topological phenomena on qudit-based quantum processors.