Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory

Abstract

We introduce weak Hopf symmetry as a tool to explore (1+1)-dimensional topological phases with non-invertible symmetries. Drawing inspiration from Symmetry Topological Field Theory (SymTFT), we construct a lattice model featuring two boundary conditions: one that encodes topological symmetry and another that governs non-topological dynamics. This cluster ladder model generalizes the well-known cluster state model. We demonstrate that the model exhibits weak Hopf symmetry, incorporating both the weak Hopf algebra and its dual. On a closed manifold, the symmetry reduces to cocommutative subalgebras of the weak Hopf algebra. Additionally, we introduce weak Hopf tensor network states to provide an exact solution for the model. As every fusion category corresponds to the representation category of some weak Hopf algebra, fusion category symmetry naturally corresponds to a subalgebra of the dual weak Hopf algebra. Consequently,the cluster ladder model offers a lattice realization of arbitrary fusion category symmetries.

v2: 68pages, typos are corrected

References in corpus (41)

• Generalized Global Symmetries
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Interacting anyons in topological quantum liquids: The golden chain
• Generalized Symmetries in Condensed Matter
• Symmetries and Strings of Adjoint QCD${}_2$
• Orbifold groupoids
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• A short introduction to Fibonacci anyon models
• Topological defects for the free boson CFT
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• Lecture Notes on Generalized Symmetries and Applications
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• On lattice models of gapped phases with fusion category symmetries
• Dualities in one-dimensional quantum lattice models: topological sectors
• Cluster state as a non-invertible symmetry protected topological phase
• An Introduction to Higher-Form Symmetries
• Higher categorical symmetries and gauging in two-dimensional spin systems
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Topological field theories and symmetry protected topological phases with fusion category symmetries
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Topological holography, quantum criticality, and boundary states
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Fermionization of fusion category symmetries in 1+1 dimensions
• Generalized Cluster States Based on Finite Groups
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation
• Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras
• Boundary and domain wall theories of 2d generalized quantum double model
• On weak Hopf symmetry and weak Hopf quantum double model
• Invertible bimodule categories and generalized Schur orthogonality
• On Topology of the Moduli Space of Gapped Hamiltonians for Topological Phases
• Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
• Generalizations of Kitaev's honeycomb model from braided fusion categories
• Topological and nontopological degeneracies in generalized string-net models
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Non-invertible symmetries in finite-group gauge theory
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs

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