Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions

Abstract

For a zero-temperature Landau symmetry breaking transition in $n$-dimensional space that completely breaks a finite symmetry $G$, the critical point at the transition has the symmetry $G$. In this paper, we show that the critical point also has a dual symmetry - a $(n-1)$-symmetry described by a higher group when $G$ is Abelian or an algebraic $(n-1)$-symmetry beyond higher group when $G$ is non-Abelian. In fact, any $G$-symmetric system can be viewed as a boundary of $G$-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk $G$-gauge theory gives rise to the symmetry and the dual symmetry respectively. So any $G$-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3+1D $Z_2$ gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a $Z_2$ 0-symmetry and its dual - a $Z_2$ 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a $Z_2$ 1-symmetry and its dual - another $Z_2$ 1-symmetry.

arXiv comment27 pages, 7 figures, 1 table

References in corpus (38)

• Fault-tolerant quantum computation by anyons
• Generalized Global Symmetries
• String-net condensation: A physical mechanism for topological phases
• AdS/CFT Correspondence and Symmetry Breaking
• Entanglement renormalization
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Tensor renormalization group approach to 2D classical lattice models
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Quantum orders in an exact soluble model
• Topological Defect Lines and Renormalization Group Flows in Two Dimensions
• Duality and defects in rational conformal field theory
• A symmetry principle for Topological Quantum Order
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• Constraints on symmetry from holography
• An Invitation to Higher Gauge Theory
• Fermions, strings, and gauge fields in lattice spin models
• Symmetry and Topological Order
• Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems
• Topological Transition on the Conformal Manifold
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
• Boundary-bulk relation in topological orders
• Adjoint QCD${}_4$, Deconfined Critical Phenomena, Symmetry-Enriched Topological Quantum Field Theory, and Higher Symmetry-Extension
• Structure of 2D Topological Stabilizer Codes
• Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions
• Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries
• A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
• Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry
• Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models
• Topological phases from higher gauge symmetry in 3+1D
• Invertible defects and isomorphisms of rational CFTs
• Topological Higher Gauge Theory - from BF to BFCG theory
• Topological non-linear $\sigma$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems
• A topological phase transition on the edge of the 2d ${\mathbb{Z}}_2$ topological order
• Constraints on order and disorder parameters in quantum spin chains

Cited by in corpus (138)

• Generalized Symmetries in Condensed Matter
• Non-Invertible Duality Defects in 3+1 Dimensions
• Orbifold groupoids
• Non-Invertible Global Symmetries and Completeness of the Spectrum
• On the classification of topological orders
• Quantum criticality under decoherence or weak measurement
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Non-invertible Time-reversal Symmetry
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Non-invertible 1-form symmetry and Casimir scaling in 2d Yang-Mills theory
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Lecture Notes on Generalized Symmetries and Applications
• Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond $N$-ality
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Remarks on Boundaries, Anomalies, and Noninvertible Symmetries
• On lattice models of gapped phases with fusion category symmetries
• Dualities in one-dimensional quantum lattice models: topological sectors
• Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Cluster state as a non-invertible symmetry protected topological phase
• Generalized Charges, Part I: Invertible Symmetries and Higher Representations
• Measuring Rényi entanglement entropy with high efficiency and precision in quantum Monte Carlo simulations
• Bootstrapping Non-Invertible Symmetries
• Scaling of disorder operator at deconfined quantum criticality
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Universal Features of Higher-Form Symmetries at Phase Transitions
• Subsystem Non-Invertible Symmetry Operators and Defects
• Higher-form symmetry breaking at Ising transitions
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Scaling of disorder operator at $(2+1)d$ U(1) quantum criticality
• Duality Defects in $E_8$
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Exact emergent higher-form symmetries in bosonic lattice models
• Categories of quantum liquids I
• Topological holography, quantum criticality, and boundary states
• Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
• One dimensional gapped quantum phases and enriched fusion categories
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Lattice model constructions for gapless domain walls between topological phases
• Bosonic Rational Conformal Field Theories in Small Genera, Chiral Fermionization, and Symmetry/Subalgebra Duality
• Unlocking the general relationship between energy and entanglement spectra via the wormhole effect
• Remarks on Geometric Engineering, Symmetry TFTs and Anomalies
• Generating Lattice Non-invertible Symmetries
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Field theories with higher-group symmetry from composite currents
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• 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
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance
• Emergent Symmetry in Quantum Phase Transitions: From Deconfined Quantum Critical Point to Gapless Quantum Spin Liquid
• (SPT-)LSM theorems from projective non-invertible symmetries
• Topological defects
• Towards Non-Invertible Anomalies from Generalized Ising Models
• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement
• When the moduli space is an orbifold: Spontaneous breaking of continuous non-invertible symmetries
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point
• Classical model emerges in quantum entanglement: Quantum Monte Carlo study for an Ising-Heisenberg bilayer
• Symmetries as Ground States of Local Superoperators: Hydrodynamic Implications
• On tensor network representations of the (3+1)d toric code
• Sampling reduced density matrix to extract fine levels of entanglement spectrum and restore entanglement Hamiltonian
• Low energy effective field theories of fermion liquids and mixed $U(1)\times {\mathbb{R}}^d$ anomaly
• Noninvertible Symmetry-Resolved Affleck-Ludwig-Cardy Formula and Entanglement Entropy from the Boundary Tube Algebra
• Demonstrating the wormhole mechanism of the entanglement spectrum via a perturbed boundary
• Quantum Current and Holographic Categorical Symmetry
• Definition and Classification of Fermi Surface Anomalies
• Boundary SymTFT
• Reweight-annealing method for evaluating the partition function via quantum Monte Carlo calculations
• Non-invertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry
• Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model
• Relevant long-range interaction of the entanglement Hamiltonian emerges from a short-range gapped system
• Category of SET orders
• Fusion 3-Categories for Duality Defects
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• Symmetry fractionalization and duality defects in Maxwell theory
• Higher Gauge Theory
• Generalizations of Kitaev's honeycomb model from braided fusion categories
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
• Additivity, Haag duality, and non-invertible symmetries
• Decoherence through Ancilla Anyon Reservoirs
• Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Deriving the non-perturbative gravitational dual of quantum Liouville theory from BCFT operator algebra
• Measuring the Boundary Gapless State and Criticality via Disorder Operator
• Topologically ordered time crystals
• Self-$G$-ality in 1+1 dimensions
• 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
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Noninvertible symmetry and topological holography for modulated SPT in one dimension
• A Multicritical Point with Infinite Fractal Symmetries
• Non-invertible symmetries in finite-group gauge theory
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Soft symmetries of topological orders
• Generalized Landau Paradigm for quantum phases and phase transitions
• Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models
• Generalized Modulated Symmetries in ${\mathbb{Z}}_2$ Topological Ordered Phases
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki 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
• The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology
• Generalized Statistics on Lattices
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Universal Deformations
• Self-duality protected multi-criticality in deconfined quantum phase transitions
• Worldline deconfinement and emergent long-range interaction in the entanglement Hamiltonian and in the entanglement spectrum
• Dynamic critical exponents as an emergent property at interacting topological quantum critical points
• Emergent non-Hermitian conservation laws at exceptional points
• A non-semisimple non-invertible symmetry
• Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle
• Entropic order parameters and topological holography
• Threefold Way for Typical Entanglement
• Non-onsite symmetry breaking: topological phase coexistence and criticality
• Analogs of deconfined quantum criticality for non-invertible symmetry breaking in 1d
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• Exponential $U(1)$ Symmetry-Breaking Phase as a Disorder-Free Quantum Glass
• Equivalence class of Emergent Single Weyl fermion lattice models in 3 dimensions: gapless superconductors and superfluids versus chiral fermions
• Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains
• Note on searching for critical lattice models as entropy critical points from strange correlator
• Holographic duality between bulk topological order and boundary mixed-state order