Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model

Abstract

We investigate the multifusion generalization of string-net ground states and lattice Hamiltonians, delving into its associated weak Hopf symmetry. For the multifusion string-net, the gauge symmetry manifests as a general weak Hopf algebra, leading to a reducible vacuum string label; the charge symmetry, serving as a quantum double of gauge symmetry, constitutes a connected weak Hopf algebra. This implies that the associated topological phase retains its characterization by a unitary modular tensor category (UMTC). The bulk charge symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit construction of the weak Hopf tube algebra structure and thoroughly discuss its properties. The gapped boundary and domain wall models are extensively discussed, with these $1d$ phases characterized by unitary multifusion categories (UMFCs). We delve into the gauge and charge symmetries of these $1d$ phases, as well as the construction of the boundary and domain wall tube algebras. Additionally, we illustrate that the domain wall tube algebra can be regarded as a cross product of two boundary tube algebras. As an application of our model, we elucidate how to interpret the defective string-net as a restricted multifusion string-net.

arXiv commentv1: 64 pages

References in corpus (42)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Topological quantum memory
• String-net condensation: A physical mechanism for topological phases
• Quantum Error Correction for Quantum Memories
• Models for gapped boundaries and domain walls
• Condensate induced transitions between topologically ordered phases
• Quantized Thermal Transport in the Fractional Quantum Hall Effect
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Anyon condensation and its applications
• Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
• Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
• Generalizations and limitations of string-net models
• Full Dyon Excitation Spectrum in Generalized Levin-Wen Models
• Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory
• Boundary Degeneracy of Topological Order
• Generalized string-net models: A thorough exposition
• A hierarchy of topological tensor network states
• Ground State Degeneracy in the Levin-Wen Model for Topological Phases
• Broken quantum symmetry and confinement phases in planar physics
• Generalized string-nets for unitary fusion categories without tetrahedral symmetry
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries
• Electric-magnetic duality of lattice systems with topological order
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Kitaev lattice models as a Hopf algebra gauge theory
• The center functor is fully faithful
• Mapping between Morita equivalent string-net states with a constant depth quantum circuit
• Kitaev models based on unitary quantum groupoids
• On Enriching the Levin-Wen model with Symmetry
• Computing data for Levin-Wen with defects
• Uniqueness of unitary structure for unitarizable fusion categories
• Computing associators of endomorphism fusion categories
• Boundary and domain wall theories of 2d generalized quantum double model
• Invertible bimodule categories and generalized Schur orthogonality
• On weak Hopf symmetry and weak Hopf quantum double model
• State Sum Invariants of Three Manifolds from Spherical Multi-fusion Categories
• Topological and nontopological degeneracies in generalized string-net models
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory

Cited by in corpus (6)

• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems