Models for gapped boundaries and domain walls

2011/04/26 by Alexei Kitaev, Liang Kong · 1 voice · 43 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Topological Materials and Phenomena #cond-mat.str-el #math.CT #math.QA

paper · pdf · doi:10.1007/s00220-012-1500-5

published as Commun. Math. Phys. 313 (2012) 351-373 · 21 pages, a typo is corrected (thanks for Juven Wang)

arxiv published 2011/04/26 · openalex publication_date 2012/06/06 · arxiv created 2012/09/28 · arxiv updated 2012/10/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31

Abstract

We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category \calC as in the Levin-Wen model, whereas the boundary is associated with a module category over \calC. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

Citations

• String-net condensation: A physical mechanism for topological phases
• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Nonabelions in the fractional quantum hall effect

Cited by

• Emergent Symmetry and Phase Transitions on the Domain Wall of ℤ₂ Topological Orders
• Unified Framework for Quantum Code Embedding
• On Frustration-Free Quantum Spin Models
• Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs
• Dipoles and Anyonic Directional Confinement via Twisted Toric Codes
• An Algebraic Theory of Gapped Domain Wall Partons
• Holography for bulk-boundary local topological order
• Tunable anyonic permeability across ℤ₂ spin liquid junctions
• Discrete quantum systems from topological field theory
• Topological Mixed States: Phases of Matter from Axiomatic Approaches
• Symmetry Theories, Wigner's Function, Compactification, and Holography
• Generalized Hall Conductivities in Local Commuting Projector Models: Generalized Symmetries and Protected Surface Modes
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to p-String Condensation
• Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries
• Fast decoders for qudit topological codes
• Tensor networks with a twist: Anyon-permuting domain walls and defects in projected entangled pair states
• Non-invertible symmetry enriched string net topological orders
• Gapped Boundaries of Kitaev's Quantum Double Models: A Lattice Realization of Anyon Condensation from Lagrangian Algebras
• Construction of bosonic symmetry-protected-trivial states and their topological invariants viaG×SO(∞)nonlinearσmodels
• Generalizations and limitations of string-net models
• 3D Gravity Does Not Average Like Narain at Genus One: Rigidity of Virasoro Topological Boundaries
• Layer Construction of 3D Topological States and String Braiding Statistics
• Boundary degeneracy of topological order
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Theory of twist liquids: Gauging an anyonic symmetry
• An Invariant of Topologically Ordered States Under Local Unitary Transformations
• Anomalous matrix product operator symmetries and 1D mixed-state phases
• Prethermal Strong Zero Modes and Topological Qubits
• One dimensional gapped quantum phases and enriched fusion categories
• Topological color code and symmetry-protected topological phases
• Electric–magnetic duality of lattice systems with topological order
• Anomaly indicators and bulk-boundary correspondences for three-dimensional interacting topological crystalline phases with mirror and continuous symmetries
• Braiding statistics and congruent invariance of twist defects in bosonic bilayer fractional quantum Hall states
• Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code
• Model wave functions for interfaces between lattice Laughlin states
• Choreographed entanglement dances: Topological states of quantum matter
• Twofold twist defect chains at criticality
• Ising anyons in frustration-free Majorana-dimer models
• Constructing topological models by symmetrization: A projected entangled pair states study
• Fusion basis for lattice gauge theory and loop quantum gravity
• An Effective String Theory Toolbox for Quantum Hall Interfaces III: Open Worldsheets, Endpoint Conditions, and Branes
• Generalized Symmetries of Non-SUSY and Discrete Torsion String Backgrounds
• Higher-Form Anomalies Imply Intrinsic Long-Range Entanglement
• Protected gates for topological quantum field theories

Discussions

• In topologically ordered phases of matter, point-like defects can have some peculiar properties compared to higher-dimensional defects (such as boundaries or domain walls). For one thing, it is known [stackexchange, 3 points]

Related

• Classification of Gapped Domain Walls of Topological Orders in 2+1 dimensions: A Levin-Wen Model Realization
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Domain Wall Solutions
• Classifying 2D topological phases: mapping ground states to string-nets
• Gapped Boundary Phases of Topological Insulators via Weak Coupling