Symmetry as a shadow of topological order and a derivation of topological holographic principle

Abstract

Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out that the algebra of local symmetric operators contains a special class of extended operators -- transparent patch operators, which reveal the selection sectors and hence the corresponding symmetry. The algebra of those transparent patch operators in $n$-dimensional space gives rise to a non-degenerate braided fusion $n$-category, which happens to describe a topological order in one higher dimension (for finite symmetry). Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, non-invertible (higher) symmetries (also known as algebraic higher symmetries), and non-invertible gravitational anomalies. Thus, topological order in one higher dimension, replacing group, provides a unified and systematic description of the above generalized symmetries. This is referred to symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: \emph{boundary uniquely determines the bulk}, or more precisely, the algebra of local boundary operators uniquely determines the bulk topological order. As an application of the Symm/TO correspondence, we show the equivalence between ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry with mixed anomaly and ${\mathbb{Z}}_4$ symmetry, as well as between many other symmetries, in 1-dimensional space.

35 pages, 21 figures; PRB version

References in corpus (20)

• Generalized Global Symmetries
• Duality and defects in rational conformal field theory
• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• Hilbert Space Fragmentation and Commutant Algebras
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
• Classification of topological phases with finite internal symmetries in all dimensions
• Pauli stabilizer models of twisted quantum doubles
• Microscopic definitions of anyon data
• On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups
• Topological field theories and symmetry protected topological phases with fusion category symmetries
• Topological Higher Gauge Theory - from BF to BFCG theory
• A mathematical theory of gapless edges of 2d topological orders. Part II
• Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
• One dimensional gapped quantum phases and enriched fusion categories
• Symmetry protected topological phases beyond groups: The q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension

Cited by in corpus (52)

• Unifying Constructions of Non-Invertible Symmetries
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Lecture Notes on Generalized Symmetries and Applications
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Subsystem Non-Invertible Symmetry Operators and Defects
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Exact emergent higher-form symmetries in bosonic lattice models
• Topological holography, quantum criticality, and boundary states
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• Anomalies of Generalized Symmetries from Solitonic Defects
• Fermionization of fusion category symmetries in 1+1 dimensions
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems
• Numerical Methods for Detecting Symmetries and Commutant Algebras
• Towards Non-Invertible Anomalies from Generalized Ising Models
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point
• Symmetries as Ground States of Local Superoperators: Hydrodynamic Implications
• Quantum Current and Holographic Categorical Symmetry
• Non-invertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae
• Category of SET orders
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
• Classical origins of Landau-incompatible transitions
• Additivity, Haag duality, and non-invertible symmetries
• Noninvertible symmetry and topological holography for modulated SPT in one dimension
• Enriched monoidal categories I: centers
• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• Quantum cellular automata and categorical dualities of spin chains
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Generalized Landau Paradigm for quantum phases and phase transitions
• Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models
• Fractional domain wall statistics in spin chains with anomalous symmetries
• Local topological order and boundary algebras
• Tube Category, Tensor Renormalization and Topological Holography
• The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology
• Generalized Statistics on Lattices
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• The boundary phase transitions of the 2+1D ${\mathbb{Z}}_N$ topological order via topological Wick rotation
• Exploring two dimensional ${\mathbb{Z}}_2$ invariant phases with time reversal symmetry and their transitions with topological operations
• Non-onsite symmetry breaking: topological phase coexistence and criticality
• A non-semisimple non-invertible symmetry
• Entropic order parameters and topological holography
• Anomalies of global symmetries on the lattice
• Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle
• Emergent Symmetry and Phase Transitions on the Domain Wall of ${\mathbb{Z}}_2$ Topological Orders
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• Boundary critical phenomena in the quantum Ashkin-Teller model