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Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories

Kansei Inamura, Kantaro Ohmori

TL;DR

This work develops a higher-categorical generalization of lattice symmetry constructions by introducing fusion surface models: $2+1$d quantum lattices whose finite non-invertible symmetries are encoded by fusion 2-categories. Grounded in the Douglas–Reutter TFT (and its symmetry TFT viewpoint) and its 4d state-sum, the authors build 3d height models and obtain 2+1d fusion surface models via an anisotropic limit, with a precisely defined state space, commuting-projector-like Hamiltonians, and a robust graphical calculus. The framework accommodates anomalous and non-anomalous one-form symmetries, non-invertible 1-form symmetries, 2-groups, and even channeled cases such as Kitaev-type realizations, while enabling non-chiral topological phases through separable algebras in a fusion 2-category. By providing explicit input data, symmetry actions, and unitarity conditions, the paper offers a practical route to realize and study higher-categorical symmetries and their associated topological orders in lattice models, with clear connections to known constructions like Levin–Wen and to examples like anomalous $ ext{Z}_2$ 1-form symmetry and the Kitaev honeycomb model without a magnetic field.

Abstract

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.

Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories

TL;DR

This work develops a higher-categorical generalization of lattice symmetry constructions by introducing fusion surface models: d quantum lattices whose finite non-invertible symmetries are encoded by fusion 2-categories. Grounded in the Douglas–Reutter TFT (and its symmetry TFT viewpoint) and its 4d state-sum, the authors build 3d height models and obtain 2+1d fusion surface models via an anisotropic limit, with a precisely defined state space, commuting-projector-like Hamiltonians, and a robust graphical calculus. The framework accommodates anomalous and non-anomalous one-form symmetries, non-invertible 1-form symmetries, 2-groups, and even channeled cases such as Kitaev-type realizations, while enabling non-chiral topological phases through separable algebras in a fusion 2-category. By providing explicit input data, symmetry actions, and unitarity conditions, the paper offers a practical route to realize and study higher-categorical symmetries and their associated topological orders in lattice models, with clear connections to known constructions like Levin–Wen and to examples like anomalous 1-form symmetry and the Kitaev honeycomb model without a magnetic field.

Abstract

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.
Paper Structure (40 sections, 86 equations, 28 figures, 1 table)

This paper contains 40 sections, 86 equations, 28 figures, 1 table.

Table of Contents

  1. Introduction and Summary
  2. Motivation
  3. Review of the Aasen-Fendley-Mong model
  4. Fusion category.
  5. Examples.
  6. Symmetry TFT construction.
  7. AFM height model.
  8. Quantum anyon chain.
  9. Examples.
  10. Generalization to 2+1 dimensions
  11. Fusion 2-category.
  12. Symmetry TFT construction.
  13. Fusion surface models
  14. Input data.
  15. State space.
  16. ...and 25 more sections

Figures (28)

  • Figure 1: A timelike insertion of a symmetry defect realizes a twisted boundary condition, while a spacelike insertion does a symmetry operation on a state.
  • Figure 7: The $F$-move in a fusion category.
  • Figure 14: The AFM height model realized as the TVBW model on a slab with one topological boundary and one non-topological boundary. The non-topological boundary is obtained by decorating the Dirichlet boundary with a defect network as shown in Figure \ref{['fig:AFMdecorate']}.
  • Figure 15: The decorated boundary in Figure \ref{['fig:TV']}. The transfer matrix $T$ of the AFM height model is defined by the region indicated in the figure.
  • Figure 16: The dynamical variables in the AFM height model.
  • ...and 23 more figures