Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models

Abstract

For a class of two-dimensional Euclidean lattice field theories admitting topological lines encoded into a spherical fusion category, we explore aspects of their realisations as boundary theories of a three-dimensional topological quantum field theory. After providing a general framework for explicitly constructing such realisations, we specialise to non-abelian generalisations of the Ising model and consider two operations: gauging an arbitrary subsymmetry and performing Fourier transforms of the local weights encoding the dynamics of the theory. These are carried out both in a traditional way and in terms of the three-dimensional topological quantum field theory. Whenever the whole symmetry is gauged, combining both operations recovers the non-abelian Kramers-Wannier duals à la Freed and Teleman of the generalised Ising models. Moreover, we discuss the interplay between renormalisation group fixed points of gapped symmetric phases and these generalised Kramers-Wannier dualities.

References in corpus (50)

• Fault-tolerant quantum computation by anyons
• Generalized Global Symmetries
• String-net condensation: A physical mechanism for topological phases
• Models for gapped boundaries and domain walls
• Topological Defect Lines and Renormalization Group Flows in Two Dimensions
• Interacting anyons in topological quantum liquids: The golden chain
• TFT construction of RCFT correlators I: Partition functions
• Duality and defects in rational conformal field theory
• Generalised twisted partition functions
• Boundary conformal field theory approach to the two-dimensional critical Ising model with a defect line
• On finite symmetries and their gauging in two dimensions
• Kramers-Wannier duality from conformal defects
• On gauging finite subgroups
• PEPS as ground states: degeneracy and topology
• Symmetries and Strings of Adjoint QCD${}_2$
• Orbifold groupoids
• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• Topological Defects on the Lattice I: The Ising model
• Non-Invertible Higher-Categorical Symmetries
• Universal Non-Invertible Symmetries
• Anyons and matrix product operator algebras
• Bicategories for boundary conditions and for surface defects in 3-d TFT
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Explicit tensor network representation for the ground states of string-net models
• TFT construction of RCFT correlators IV: Structure constants and correlation functions
• Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
• TFT construction of RCFT correlators II: Unoriented world sheets
• The Ponzano-Regge model
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• Wave Function and Strange Correlator of Short Range Entangled states
• Defect Lines in the Ising Model and Boundary States on Orbifolds
• Anyonic quantum spin chains: Spin-1 generalizations and topological stability
• Anyonic Chains, Topological Defects, and Conformal Field Theory
• Boundary-bulk relation in topological orders
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Monoidal 2-structure of Bimodule Categories
• Mapping topological to conformal field theories through strange correlators
• On lattice models of gapped phases with fusion category symmetries
• Dualities in one-dimensional quantum lattice models: topological sectors
• Generalized Charges, Part I: Invertible Symmetries and Higher Representations
• Matrix product operator symmetries and intertwiners in string-nets with domain walls
• Characterizing Topological Order with Matrix Product Operators
• The dual of pure non-Abelian lattice gauge theory as a spin foam model
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
• Topological aspects of the critical three-state Potts model
• On the stability of topological order in tensor network states
• Electric-Magnetic duality in twisted quantum double model of topological orders
• The structure of fixed-point tensor network states characterizes patterns of long-range entanglement
• Manifestly T-Duality Symmetric Matrix Models

Cited by in corpus (3)

• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Les Houches Lecture Notes on Tensor Networks
• A non-semisimple non-invertible symmetry