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

Luisa Eck

TL;DR

This work develops a systematic bond-algebraic approach to dualities in 2+1d fusion surface models built from braided fusion categories, by introducing module tensor categories $ ext{M}$ over a fixed category $ ext{B}$. Distinct choices of $ ext{M}$ yield dual lattice models that share the same bond algebra, extending 1+1d dualities classified by module categories to higher dimensions. It provides two concrete pairs of dual models—the Rep$(S_3)$-based XXZ honeycomb and its constrained dual, and a Kitaev bilayer dual to an XXZ-Ising model—analyzing their phase diagrams and symmetry content, including 0-form and 1-form (non-invertible) symmetries. The paper also develops the symmetry fusion 2-category framework, showing how Morita duality of fusion 2-categories governs the full symmetry content of both regular and dual models, and discusses implications for topological order, symmetry enrichment, and potential future directions such as non-invertible 0-form symmetries and circuit implementations.

Abstract

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.

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

TL;DR

This work develops a systematic bond-algebraic approach to dualities in 2+1d fusion surface models built from braided fusion categories, by introducing module tensor categories over a fixed category . Distinct choices of yield dual lattice models that share the same bond algebra, extending 1+1d dualities classified by module categories to higher dimensions. It provides two concrete pairs of dual models—the Rep-based XXZ honeycomb and its constrained dual, and a Kitaev bilayer dual to an XXZ-Ising model—analyzing their phase diagrams and symmetry content, including 0-form and 1-form (non-invertible) symmetries. The paper also develops the symmetry fusion 2-category framework, showing how Morita duality of fusion 2-categories governs the full symmetry content of both regular and dual models, and discusses implications for topological order, symmetry enrichment, and potential future directions such as non-invertible 0-form symmetries and circuit implementations.

Abstract

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 over the same braided fusion category 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 model with a constrained Hilbert space, dual to the spin- XXZ model on the honeycomb lattice, and (ii) a bilayer Kitaev honeycomb model, dual to a spin- model with XXZ and Ising interactions. Unlike regular fusion surface models, which conserve only 1-form symmetries, models constructed from can exhibit both 1-form and 0-form symmetries, including non-invertible ones.
Paper Structure (14 sections, 56 equations, 2 figures)

This paper contains 14 sections, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review: Dual 1+1d anyon chains
  3. Construction of dual 2+1d fusion surface models from braided fusion categories
  4. Example: XXZ honeycomb model and $\text{Rep}(S_3)$ fusion surface model
  5. XXZ model from $\mathcal{M} = \text{Vec}$
  6. Fusion surface model from $\mathcal{M} = \text{Rep}(S_3)$
  7. Example: Kitaev bilayer and XXZ-Ising honeycomb model
  8. Kitaev honeycomb bilayer model from $\mathcal{M} = \text{Ising} \boxtimes \overline{\text{Ising}}$
  9. XXZ-Ising model from $\mathcal{M} = \text{Ising}$
  10. Symmetry fusion 2-category of dual models
  11. Mathematical background
  12. Symmetries of the regular ($\mathcal{M}$ = $\mathcal{B}$) fusion surface model
  13. Symmetries of fusion surface models with $\mathcal{M}$ $\neq$ $\mathcal{B}$
  14. Conclusions and outlook

Figures (2)

  • Figure 1: (i) Concurrences $C_x$, $C_y$ of the XXZ honeycomb model \ref{['eq:hamxxz']} and (ii) second derivative of the ground state energy, computed with infinite DMRG on a $L_y=4$ cylinder with bond dimension $D=800$. The plots are for fixed $J_y$ = $J_z$ = $(J_x-1)/2$ and $\Delta$ = $1$.
  • Figure 2: (i) Ferromagnetic order parameter $|\langle X_1 \rangle|$ of the XXZ-Ising model \ref{['eq:hamIsingXXZ']} and (ii) second derivative of the ground state energy, computed with infinite DMRG on a $L_y$ = $2$ cylinder with bond dimension $D$ = $200$. The plots are for fixed $J_y$ = $J_z$ = $J_x$ = $1$.