Boundary-bulk relation in topological orders

Abstract

In this paper, we study the relation between an anomaly-free $n+$1D topological order, which are often called $n+$1D topological order in physics literature, and its $n$D gapped boundary phases. We argue that the $n+$1D bulk anomaly-free topological order for a given $n$D gapped boundary phase is unique. This uniqueness defines the notion of the "bulk" for a given gapped boundary phase. In this paper, we show that the $n+$1D "bulk" phase is given by the "center" of the $n$D boundary phase. In other words, the geometric notion of the "bulk" corresponds precisely to the algebraic notion of the "center". We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the "bulk" satisfies the same universal property as that of the "center" of an algebra in mathematics, i.e. "bulk = center". The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.

14 pages, 12 figures, This paper gives a concise explanation of one of the main results in arXiv:1502.01690. We have tried to make it easier for physicists to read

References in corpus (12)

• Non-Abelian Anyons and Topological Quantum Computation
• Condensate induced transitions between topologically ordered phases
• Topological boundary conditions in abelian Chern-Simons theory
• Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
• Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
• Cardy algebras and sewing constraints, I
• Gapless edges of 2d topological orders and enriched monoidal categories
• Drinfeld center of enriched monoidal categories
• Poissonian tunneling through an extended impurity in the quantum Hall effect
• A remark on gapped domain walls between topological phases
• The monoidal centre as a limit
• Extended TQFT arising from enriched multi-fusion categories

Cited by in corpus (61)

• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• On the classification of topological orders
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Generalized string-net models: A thorough exposition
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Dualities in one-dimensional quantum lattice models: topological sectors
• A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Classification of topological phases with finite internal symmetries in all dimensions
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Braiding and Gapped Boundaries in Fracton Topological Phases
• Gapless edges of 2d topological orders and enriched monoidal categories
• A systematic construction of gapped non-liquid states
• Minimal nondegenerate extensions
• Topological non-linear $\sigma$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems
• A mathematical theory of gapless edges of 2d topological orders. Part II
• Categories of quantum liquids I
• Classifying phases protected by matrix product operator symmetries using matrix product states
• Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
• One dimensional gapped quantum phases and enriched fusion categories
• Defects in the 3-dimensional toric code model form a braided fusion 2-category
• Quantum dynamical characterization and simulation of topological phases with high-order band inversion surfaces
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Non-invertible symmetry-protected topological order in a group-based cluster state
• String Condensations in 3+1D and Lagrangian Algebras
• Gapped Boundary Theory of the Twisted Gauge Theory Model of Three-Dimensional Topological Orders
• Gapped boundaries of (3+1)d topological orders
• Towards Non-Invertible Anomalies from Generalized Ising Models
• (3+1)D topological orders with only a ${\mathbb{Z}}_2$-charged particle
• Gapped domain walls between 2+1D topologically ordered states
• Boundary and domain wall theories of 2d generalized quantum double model
• Topological Orders in (4+1)-Dimensions
• Noninvertible Symmetry-Resolved Affleck-Ludwig-Cardy Formula and Entanglement Entropy from the Boundary Tube Algebra
• Enriched string-net models and their excitations
• On weak Hopf symmetry and weak Hopf quantum double model
• Quantum Current and Holographic Categorical Symmetry
• Entanglement Area Law for Shallow and Deep Quantum Neural Network States
• Categories of quantum liquids II
• String operators for Cheshire strings in topological phases
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant
• Enriched monoidal categories I: centers
• U(1) Gauging, Continuous TQFTs, and Higher Symmetry Structures
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• The Symmetry Enriched Center Functor is Fully Faithful
• 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
• 2-Drinfel'd double symmetry of the 4d Kitaev model
• The boundary phase transitions of the 2+1D ${\mathbb{Z}}_N$ topological order via topological Wick rotation
• Tube Category, Tensor Renormalization and Topological Holography
• Systematic construction of stabilizer codes via gauging abelian boundary symmetries
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Simplicity condition and boundary-bulk duality
• Fusion of Low-Entanglement Excitations in 2D Toric Code
• A non-semisimple non-invertible symmetry
• Measuring the Unique Identifiers of Topological Order Based on Boundary-Bulk Duality and Anyon Condensation
• Categorical computation