Dualities between 2+1d fusion surface models from braided fusion categories

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 $\mathcal{M}$ over the same braided fusion category $\mathcal{B}$ 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 $\text{Rep}(S_3)$ model with a constrained Hilbert space, dual to the spin-$\tfrac{1}{2}$ XXZ model on the honeycomb lattice, and (ii) a bilayer Kitaev honeycomb model, dual to a spin-$\tfrac{1}{2}$ model with XXZ and Ising interactions. Unlike regular $\mathcal{M}=\mathcal{B}$ fusion surface models, which conserve only 1-form symmetries, models constructed from $\mathcal{M} \neq \mathcal{B}$ can exhibit both 1-form and 0-form symmetries, including non-invertible ones.

💡 Research Summary

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The paper develops a systematic framework for dualities in 2+1‑dimensional fusion surface models, extending the well‑known classification of dualities in 1+1‑dimensional anyon chains. The key idea is to start from a braided fusion category 𝔅, which supplies a set of 1‑form categorical symmetries, and then choose a module tensor category 𝔐 over 𝔅. Different choices of 𝔐 generate lattice Hamiltonians that share the same bond algebra (the algebra generated by local terms) because the bond algebra depends only on 𝔅. Consequently, models built from distinct 𝔐 are dual to each other: they have identical energy spectra (up to possible differences in degeneracies and symmetry sectors) but may differ in their Hilbert space structure and symmetry content.

When 𝔐 = 𝔅, the resulting model is a conventional Levin‑Wen‑type string‑net with only 1‑form symmetries given by the simple objects of 𝔅. In this case the symmetry category is precisely 𝔅, and any non‑abelian object yields a non‑invertible symmetry line. If 𝔐 ≠ 𝔅, the symmetry category changes to the Morita‑dual fusion category 𝔅*𝔐 = End_𝔅(𝔐). This new category provides 0‑form (global) symmetries, while the original 1‑form symmetries from 𝔅 remain. Both categories share the same Drinfeld center Z(𝔅), guaranteeing that the dual models have identical topological data (anyons, braiding, etc.). The paper shows that the duality transformation can be written as a matrix‑product operator (MPO) built from a module functor X ∈ Fun(𝔐,𝔑); the MPO entries are determined solely by the ⊳‑F symbols of the module action.

Two concrete examples illustrate the construction.

1. Rep(S₃) model vs. spin‑½ XXZ chain: Taking 𝔅 = Rep(S₃) and 𝔐 = Rep(S₃) yields a constrained Hilbert space that maps onto a Rydberg‑blockade ladder. Choosing 𝔐 = Vec (the trivial module) gives the standard spin‑½ XXZ chain on the honeycomb lattice. The local Hamiltonian terms realize the so(3)₂ BMW algebra with a Jones‑Wenzl projector. The duality MPO is obtained from the restriction functor Res_{S₃}^{ℤ₁}, and the two models share the same spectrum, though the Rydberg ladder exhibits a ℤ₂ symmetry and a non‑invertible 1‑form symmetry, while the XXZ chain possesses only the Morita‑dual ℤ₂ global symmetry.
2. Bilayer Kitaev honeycomb vs. XXZ‑Ising model: Here 𝔅 is again a non‑abelian braided category, while 𝔐 = Rep(ℤ₂) ⊗ Rep(ℤ₂) introduces two independent ℤ₂ degrees of freedom per site. The original bilayer Kitaev model (with inter‑layer coupling) is dual to a spin‑½ model that combines XXZ interactions (in‑plane) with Ising couplings (between layers). The dual model enjoys both a ℤ₂ 0‑form symmetry (from 𝔅*𝔐) and the original 1‑form non‑invertible symmetry from 𝔅, leading to a richer symmetry‑enriched topological phase diagram.

The authors also discuss the symmetry 2‑category of the dual models. For 𝔐 ≠ 𝔅, the 0‑form symmetries correspond to objects of 𝔅*𝔐, while the 1‑form symmetries correspond to objects of 𝔅. Non‑invertible symmetries arise from non‑abelian objects of 𝔅, obeying fusion rules such as D(1) × D(1) = 1 + D(1) + D(2). Because 𝔅 and 𝔅*𝔐 have the same Drinfeld center, the dual models share the same anyon content, ensuring that the duality does not change the underlying topological order.

The paper concludes by emphasizing that the framework opens a pathway to construct tractable 2+1‑dimensional lattice models with both categorical 1‑form and conventional 0‑form symmetries, including non‑invertible ones. Potential future directions include exploring more exotic braided categories, designing experimental platforms (e.g., Rydberg atom arrays or superconducting qubit lattices) to realize these models, and classifying mixed‑symmetry phases where 0‑form and 1‑form symmetries coexist. This work thus bridges the gap between abstract categorical dualities and concrete spin‑model realizations in three‑dimensional quantum many‑body physics.