Models for gapped boundaries and domain walls

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.

21 pages, a typo is corrected (thanks for Juven Wang)

References in corpus (5)

• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Duality and defects in rational conformal field theory
• Topological boundary conditions in abelian Chern-Simons theory
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Invertible defects and isomorphisms of rational CFTs

Cited by in corpus (315)

• Zoo of quantum-topological phases of matter
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Topological Defect Lines and Renormalization Group Flows in Two Dimensions
• Genons, twist defects, and projective non-Abelian braiding statistics
• Symmetries and Strings of Adjoint QCD${}_2$
• Higher Gauging and Non-invertible Condensation Defects
• Orbifold groupoids
• Protected edge modes without symmetry
• Quantum memories at finite temperature
• Choreographed entangle dances: topological states of quantum matter
• Theory of defects in Abelian topological states
• On the classification of topological orders
• Anyon condensation and its applications
• Three-dimensional topological lattice models with surface anyons
• Bicategories for boundary conditions and for surface defects in 3-d TFT
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• A theory of 2+1D bosonic topological orders
• Unfolding the color code
• Gauging quantum states: from global to local symmetries in many-body systems
• Symmetry fractionalization and twist defects
• Classification of Topological Defects in Abelian Topological States
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Hierarchy of topological order from finite-depth unitaries, measurement and feedforward
• Symmetry enriched string-nets: Exactly solvable models for SET phases
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• Diagrammatics for Bose condensation in anyon theories
• A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries
• Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Digital simulation of non-Abelian anyons with 68 programmable superconducting qubits
• Poking holes and cutting corners to achieve Clifford gates with the surface code
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• Generalizations and limitations of string-net models
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• Long-range entanglement from measuring symmetry-protected topological phases
• Boson Condensation in Topologically Ordered Quantum Liquids
• Full Dyon Excitation Spectrum in Generalized Levin-Wen Models
• Boundary Degeneracy of Topological Order
• Generalized string-net models: A thorough exposition
• Higher central charges and topological boundaries in 2+1-dimensional TQFTs
• Twisted Fracton Models in Three Dimensions
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Topological color code and symmetry-protected topological phases
• Cage-Net Fracton Models
• Boundary-bulk relation in topological orders
• Efficient Decoders for Qudit Topological Codes
• Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies
• Fusion basis for lattice gauge theory and loop quantum gravity
• Topological Order from Measurements and Feed-Forward on a Trapped Ion Quantum Computer
• Remarks on Boundaries, Anomalies, and Noninvertible Symmetries
• Prethermal Strong Zero Modes and Topological Qubits
• Topological Defect Networks for Fractons of all Types
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Layer construction of 3D topological states and string braiding statistics
• Advantages of versatile neural-network decoding for topological codes
• Ground State Degeneracy of Topological Phases on Open Surfaces
• Dualities in one-dimensional quantum lattice models: topological sectors
• Cheshire charge in (3+1)-D topological phases
• Synthetic Topological Degeneracy by Anyon Condensation
• An invariant of topologically ordered states under local unitary transformations
• Gapped boundaries, group cohomology and fault-tolerant logical gates
• Classification of topological phases with finite internal symmetries in all dimensions
• Anyon condensation and the color code
• Higher categorical symmetries and gauging in two-dimensional spin systems
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Generalized ADE Classification of Gapped Domain Walls
• Entanglement phases as holographic duals of anyon condensates
• Interactions along an Entanglement Cut in 2+1D Abelian Topological Phases
• The boundaries and twist defects of the color code and their applications to topological quantum computation
• Matrix product operator symmetries and intertwiners in string-nets with domain walls
• Logical blocks for fault-tolerant topological quantum computation
• Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries
• Isometric Tensor Network representation of string-net liquids
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Ground-state degeneracy for abelian anyons in the presence of gapped boundaries
• Protected gates for topological quantum field theories
• Coherent Transmutation of Electrons into Fractionalized Anyons
• Unconventional Fusion and Braiding of Topological Defects in a Lattice Model
• Electric-magnetic duality of lattice systems with topological order
• Braiding and Gapped Boundaries in Fracton Topological Phases
• Orbifolds of n-dimensional defect TQFTs
• Universal Quantum Computation with Gapped Boundaries
• Gapless edges of 2d topological orders and enriched monoidal categories
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• A systematic construction of gapped non-liquid states
• Quantum gravity kinematics from extended TQFTs
• Topological Entanglement Entropy with a Twist
• Duality Defects in $E_8$
• Pauli topological subsystem codes from Abelian anyon theories
• A Solution to the 1+1D Gauged Chiral Fermion Problem
• Sequential Quantum Circuits as Maps between Gapped Phases
• Generalized Kitaev Models and Slave Genons
• Topological Defect Lines in Two Dimensional Fermionic CFTs
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• A mathematical theory of gapless edges of 2d topological orders. Part II
• Position-Dependent Excitations and UV/IR Mixing in the ${\mathbb{Z}}_N$ Rank-2 Toric Code and its Low-Energy Effective Field Theory
• Lorentzian Dynamics and Factorization Beyond Rationality
• A Non-Perturbative Definition of the Standard Models
• Exact emergent higher-form symmetries in bosonic lattice models
• Ising Anyons in Frustration-Free Majorana-Dimer Models
• Higher-group symmetry in finite gauge theory and stabilizer codes
• On the Holographic Dual of a Topological Symmetry Operator
• Locality-Preserving Logical Operators in Topological Stabiliser Codes
• Classifying phases protected by matrix product operator symmetries using matrix product states
• Categories of quantum liquids I
• Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond
• Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions
• A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Gapped boundary theories in three dimensions
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• Kitaev lattice models as a Hopf algebra gauge theory
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• On Defects Between Gapped Boundaries in Two-Dimensional Topological Phases of Matter
• The Morita theory of quantum graph isomorphisms
• Twisted Quantum Double Model of Topological Orders with Boundaries
• Symmetry Enriched Phases via Pseudo Anyon Condensation
• Microscopic Study of the Halperin - Laughlin Interface through Matrix Product States
• Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1D
• The center functor is fully faithful
• Unification of bosonic and fermionic theories of spin liquids on the kagome lattice
• Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the $ADE$ Classification
• Model States for a Class of Chiral Topological Order Interfaces
• One dimensional gapped quantum phases and enriched fusion categories
• Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
• Efficient Machine Learning Representations of Surface Code with Boundaries, Defects, Domain Walls and Twists
• Fermion Condensation and Gapped Domain Walls in Topological Orders
• Lattice model constructions for gapless domain walls between topological phases
• Defects in the 3-dimensional toric code model form a braided fusion 2-category
• A topological phase transition on the edge of the 2d ${\mathbb{Z}}_2$ topological order
• A universal tripartite entanglement signature of ungappable edge states
• Entanglement bootstrap approach for gapped domain walls
• Topological aspects of the critical three-state Potts model
• On the Brauer groups of symmetries of abelian Dijkgraaf-Witten theories
• Topological Interface between Pfaffian and anti-Pfaffian Order in $\nu =5/2$ Quantum Hall Effect
• Braiding Statistics and Congruent Invariance of Twist Defects in Bosonic Bilayer Fractional Quantum Hall States
• Boundary Hamiltonian theory for gapped topological orders
• Globally symmetric topological phase: from anyonic symmetry to twist defect
• Mapping between Morita equivalent string-net states with a constant depth quantum circuit
• A Unified Framework of Topological Phases with Symmetry
• Kitaev models based on unitary quantum groupoids
• Charge $2e/3$ superconductivity and topological degeneracies without localized zero modes in bilayer fractional quantum Hall states
• Tensor Networks with a Twist: Anyon-permuting domain walls and defects in PEPS
• Non-invertible symmetry-protected topological order in a group-based cluster state
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Boundary theory of the X-cube model in the continuum
• Entropic Barriers for Two-Dimensional Quantum Memories
• Frustrated topological symmetry breaking: geometrical frustration and anyon condensation
• Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases
• Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes
• A Defect Verlinde Formula
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup
• Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy
• Robust teleportation of a surface code and cascade of topological quantum phase transitions
• Composing topological domain walls and anyon mobility
• Domain wall topological entanglement entropy
• Modular transformations through sequences of topological charge projections
• Robustness of Topological Order in the Toric Code with Open Boundaries
• Computing data for Levin-Wen with defects
• Quantum robustness and phase transitions of the 3D Toric Code in a field
• Anyon Condensation: Coherent states, Symmetry Enriched Topological Phases, Goldstone Theorem, and Dynamical Rearrangement of Symmetry
• Seeing topological entanglement through the information convex
• Folding approach to topological orders enriched by mirror symmetry
• Characterizing topological order by the information convex
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• Pointed Drinfeld center functor
• Fusing Binary Interface Defects in Topological Phases: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ case
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Towards Non-Invertible Anomalies from Generalized Ising Models
• Gapped boundaries of (3+1)d topological orders
• Entanglement Entropy of Generalized Moore-Read Fractional Quantum Hall State Interfaces
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point
• From 3D topological quantum field theories to 4D models with defects
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Domain walls in topological phases and the Brauer-Picard ring for $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$
• Theory of weak symmetry breaking of translations in ${\mathbb{Z}}_2$ topologically ordered states and its relation to topological superconductivity from an exact lattice ${\mathbb{Z}}_2$ charge-flux attachment
• Non-Liquid Cellular States
• From orbifolding conformal field theories to gauging topological phases
• Gauging (3+1)-dimensional topological phases: an approach from surface theories
• Gapped domain walls between 2+1D topologically ordered states
• On tensor network representations of the (3+1)d toric code
• Boundary and domain wall theories of 2d generalized quantum double model
• Exactly solvable models for U(1) symmetry-enriched topological phases
• Ground state entanglement constrains low-energy excitations
• Graph gauge theory of mobile non-Abelian anyons in a qubit stabilizer code
• Constructing topological models by symmetrization: A PEPS study
• Families of Gapped Interfaces Between Fractional Quantum Hall States
• A remark on gapped domain walls between topological phases
• Invertible bimodule categories and generalized Schur orthogonality
• Noninvertible Symmetry-Resolved Affleck-Ludwig-Cardy Formula and Entanglement Entropy from the Boundary Tube Algebra
• Generalized Color Codes Supporting Non-Abelian Anyons
• On weak Hopf symmetry and weak Hopf quantum double model
• Topological defect network representations of fracton stabilizer codes
• Majorana Edge States for Z2 Topological Orders of the Wen-plaquette Model and the Toric-code Model
• Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
• Classification of symmetry-protected topological many-body localized phases in one dimension
• Quantum Current and Holographic Categorical Symmetry
• Lifting topological codes: Three-dimensional subsystem codes from two-dimensional anyon models
• Boundary Modes in the Chamon Model
• Boundary SymTFT
• A note on permutation twist defects in topological bilayer phases
• Entanglement and the density matrix renormalisation group in the generalised Landau paradigm
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• Screw dislocations in the X-cube fracton model
• Extend The Levin-Wen Model To Two-dimensional Topological Orders With Gapped Boundary Junctions
• Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
• Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons
• Two-dimensional algebra in lattice gauge theory
• Categories of quantum liquids II
• Fermion Parity Flips and Majorana Bound States at twist defects in Superconducting Fractional Topological Phases
• Low-overhead quantum computing with the color code
• Model wavefunctions for interfaces between lattice Laughlin states
• Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae
• Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model
• Low-overhead magic state distillation with color codes
• String operators for Cheshire strings in topological phases
• Locality of Edge States and Entanglement Spectrum from Strong Subadditivity
• Fractional Hall Conductivity and Spin-c Structure in Solvable Lattice Hamiltonians
• Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking
• Interplay between short and long-range entanglement in symmetry protected phases
• No Strings Attached: Boundaries and Defects in the Cubic Code
• Topological symmetry breaking: Domain walls and partial instability of chiral edges
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
• Model wavefunctions for an interface between lattice Laughlin and Moore-Read states
• Bulk-to-boundary anyon fusion from microscopic models
• On Quantum Entanglement in Topological Phases on a Torus
• Crossing with the circle in Dijkgraaf-Witten theory and applications to topological phases of matter
• Renormalization of Ising cage-net model and generalized foliation
• Correspondence between bulk entanglement and boundary excitation spectra in 2d gapped topological phases
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTs
• Topological and nontopological degeneracies in generalized string-net models
• Computing Defects Associated to Bounded Domain Wall Structures: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ Case
• Solvable Lattice Hamiltonians with Fractional Hall Conductivity
• Anomalies and symmetry fractionalization in reflection-symmetric topological order
• Topological interfaces in Chern-Simons theory and $Ad{S}_3/CF{T}_2$
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Gauge symmetry of excited states in projected entangled-pair state simulations
• Entanglement Entropy of Topological Orders with Boundaries
• A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation
• Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase
• Anyon exclusions statistics on surfaces with gapped boundaries
• Ungappable edge theories with finite dimensional Hilbert spaces
• From effective Hamiltonian to anomaly inflow in topological orders with boundaries
• Topological physics in quantum critical systems
• General scheme for preparation of different topological states on the cluster states
• Gauging Defects in Quantum Spin Systems
• Edge Theories for Anyon Condensation Phase Transitions
• Conformal geometry from entanglement
• Non-invertible symmetries in finite-group gauge theory
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Boundary topological entanglement entropy in two and three dimensions
• A graph-based formalism for surface codes and twists
• Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory
• Generalized Symmetries of Non-SUSY and Discrete Torsion String Backgrounds
• Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• Self-correction from higher-form symmetry protection on a boundary
• Soft symmetries of topological orders
• Fractional domain wall statistics in spin chains with anomalous symmetries
• Dihedral twist liquid models from emergent Majorana fermions
• Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to $p$-String Condensation
• Twofold twist defect chains at criticality
• The Symmetry Enriched Center Functor is Fully Faithful
• Systematic construction of stabilizer codes via gauging abelian boundary symmetries
• Color codes with twists: construction and universal gate set implementation
• A 3D lattice defect and efficient computations in topological MBQC
• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• Multi-flavor string-net models
• Dualities between 2+1d fusion surface models from braided fusion categories
• Revisiting the symmetry-resolved entanglement for non-invertible symmetries in $1+1$d conformal field theories
• An algebraic quantum field theoretic approach to toric code with gapped boundary
• Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs
• The Kitaev honeycomb model on surfaces of genus $g\ge 2$
• Lattice models from CFT on surfaces with holes I: Torus partition function via two lattice cells
• Numerical identification and gapped boundaries of abelian fermionic topological order
• Remote detectability from entanglement bootstrap I: Kirby's torus trick
• 2-Drinfel'd double symmetry of the 4d Kitaev model
• Fermionic defects of topological phases and logical gates
• The boundary phase transitions of the 2+1D ${\mathbb{Z}}_N$ topological order via topological Wick rotation
• Topological Phase Transitions Induced by Varying Topology and Boundaries in the Toric Code
• Local topological order and boundary algebras
• Tube Category, Tensor Renormalization and Topological Holography
• Generalized Statistics on Lattices
• Analytic framework for self-dual criticality in ${\mathbb{Z}}_k$ gauge theory with matter
• Preparing topologically ordered states by Hamiltonian interpolation
• Parent Lindbladians for Matrix Product Density Operators
• Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle
• Rényi-like entanglement probe of the chiral central charge
• Classification of Gapped Domain Walls of Topological Orders in 2+1 dimensions: A Levin-Wen Model Realization
• Les Houches Lecture Notes on Tensor Networks
• Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization
• Topological Degeneracy Induced Flat Bands in two-Dimensional Holed Systems
• Extended string-net models with all anyons at finite temperature
• Fusion of Low-Entanglement Excitations in 2D Toric Code
• Condensation Completion and Defects in 2+1D Topological Orders
• Knots and entanglement
• Unveiling topological order through multipartite entanglement
• Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries
• A geometrical description of untwisted 3d Dijkgraaf-Witten TQFT with defects
• Tunable anyonic permeability across ${\mathbb{Z}}_2$ spin liquid junctions
• Emergent Symmetry and Phase Transitions on the Domain Wall of ${\mathbb{Z}}_2$ Topological Orders
• Unified Framework for Quantum Code Embedding
• Universal fault tolerant quantum computation in 2D without getting tied in knots
• Categorical computation