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

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.

61 pages. Minor changes, references updated

References in corpus (21)

• Generalized Global Symmetries
• Interacting anyons in topological quantum liquids: The golden chain
• Duality and defects in rational conformal field theory
• Universal Non-Invertible Symmetries
• Topological defects for the free boson CFT
• Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond
• Topological Defects on the Lattice: Dualities and Degeneracies
• Branes and Non-Invertible Symmetries
• Fusion 2-categories and a state-sum invariant for 4-manifolds
• Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
• Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality
• Decomposition, condensation defects, and fusion
• Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond
• Discrete torsion defects
• Higher representations for extended operators
• Anomalies of $(1+1)D$ categorical symmetries
• On Enriching the Levin-Wen model with Symmetry
• Neutrino Masses from Generalized Symmetry Breaking
• A toy model for categorical charges
• Building 1D lattice models with $G$-graded fusion category
• An introduction to decomposition

Cited by in corpus (26)

• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Remarks on Geometric Engineering, Symmetry TFTs and Anomalies
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Lattice T-duality from non-invertible symmetries in quantum spin chains
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• (SPT-)LSM theorems from projective non-invertible symmetries
• Topological defects
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Entanglement and the density matrix renormalisation group in the generalised Landau paradigm
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Fusion 3-Categories for Duality Defects
• Generalizations of Kitaev's honeycomb model from braided fusion categories
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Duality-preserving deformation of 3+1d lattice ${\mathbb{Z}}_2$ gauge theory with exact gapped ground states
• Twisted gauging and topological sectors in (2+1)d abelian lattice gauge theories
• Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Dualities between 2+1d fusion surface models from braided fusion categories
• Les Houches Lecture Notes on Tensor Networks
• Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization
• Chiral Edge States Emerging on Anyon-Net