Boundary Degeneracy of Topological Order

Abstract

We introduce the concept of boundary degeneracy of topologically ordered states on a compact orientable spatial manifold with boundaries, and emphasize that the boundary degeneracy provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions. By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries. By Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which encodes more than the fusion algebra of fractionalized quasiparticles. We apply our theory to Kitaev's toric code and Levin-Wen string-net models. We predict that the $Z_2$ toric code and $Z_2$ double-semion model (more generally, the $Z_k$ gauge theory and the $U(1{)}_k\times U(1{)}_{-k}$ non-chiral fractional quantum Hall state at even integer $k$) can be numerically and experimentally distinguished, by measuring their boundary degeneracy on an annulus or a cylinder.

15 pages, 4 figures. v3: the expanded version, add new tables for clarification, with some new corrections

References in corpus (8)

• Genons, twist defects, and projective non-Abelian braiding statistics
• Protected edge modes without symmetry
• Topological boundary conditions in abelian Chern-Simons theory
• Theory of defects in Abelian topological states
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Mutual Chern-Simons theory for Z_2 topological order
• Ground State Degeneracy of Topological Phases on Open Surfaces
• Ground-state degeneracy for abelian anyons in the presence of gapped boundaries

Cited by in corpus (72)

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• Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids
• Boson Condensation in Topologically Ordered Quantum Liquids
• Higher central charges and topological boundaries in 2+1-dimensional TQFTs
• Superconducting analogue of the parafermion fractional quantum Hall states
• Boundary-bulk relation in topological orders
• Symmetric Mass Generation
• Cheshire charge in (3+1)-D topological phases
• Theory of Disordered $\nu =5/2$ Quantum Thermal Hall State: Emergent Symmetry and Phase Diagram
• Generalized ADE Classification of Gapped Domain Walls
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• A K matrix Construction of Symmetry Enriched Phases of Matter
• Interactions along an Entanglement Cut in 2+1D Abelian Topological Phases
• Non-Perturbative Regularization of 1+1D Anomaly-Free Chiral Fermions and Bosons: On the equivalence of anomaly matching conditions and boundary gapping rules
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• A systematic construction of gapped non-liquid states
• Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory
• Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model
• `Unhinging' the surfaces of higher-order topological insulators and superconductors
• Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers
• A Solution to the 1+1D Gauged Chiral Fermion Problem
• A Non-Perturbative Definition of the Standard Models
• A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• On Defects Between Gapped Boundaries in Two-Dimensional Topological Phases of Matter
• Twisted Quantum Double Model of Topological Orders with Boundaries
• Entanglement bootstrap approach for gapped domain walls
• Topological Interface between Pfaffian and anti-Pfaffian Order in $\nu =5/2$ Quantum Hall Effect
• Boundary Hamiltonian theory for gapped topological orders
• Boundary theory of the X-cube model in the continuum
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy
• Topological Phases on Non-orientable Surfaces: Twisting by Parity Symmetry
• Robustness of Topological Order in the Toric Code with Open Boundaries
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• Entanglement in tripartitions of topological orders: a diagrammatic approach
• Entanglement Entropy of Generalized Moore-Read Fractional Quantum Hall State Interfaces
• Gapped boundaries of (3+1)d topological orders
• Gapped domain walls between 2+1D topologically ordered states
• Non-Liquid Cellular States
• Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy II -- Cutting through the boundary
• Families of Gapped Interfaces Between Fractional Quantum Hall States
• Boundary and domain wall theories of 2d generalized quantum double model
• Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
• Exactly soluble model of boundary degeneracy
• Boundary Modes in the Chamon Model
• Interfaces and the extended Hilbert space of Chern-Simons theory
• Topological skyrmion phases of matter
• Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
• Model wavefunctions for interfaces between lattice Laughlin states
• Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking
• CT or P Problem and Symmetric Gapped Fermion Solution
• Topological symmetry breaking: Domain walls and partial instability of chiral edges
• Topological interfaces in Chern-Simons theory and $Ad{S}_3/CF{T}_2$
• Correspondence between bulk entanglement and boundary excitation spectra in 2d gapped topological phases
• Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence
• Entanglement Entropy of Topological Orders with Boundaries
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Fractional quantum Hall states with gapped boundaries in an extreme lattice limit
• Topological Phase Transitions Induced by Varying Topology and Boundaries in the Toric Code
• Numerical identification and gapped boundaries of abelian fermionic topological order
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Abelian Topological Order on Lattice Enriched with Electromagnetic Background
• Interacting 1D Chiral Fermions with Pairing: Transition from Integrable to Chaotic
• Quantized Transport of $\nu =2/3$ Fractional Quantum Hall Edge with Disordered Superconducting Proximity