Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries

Abstract

We discuss (1+1)d gapless phases with non-invertible global symmetries, also referred to as categorical symmetries. This includes gapless phases showing properties analogous to gapped symmetry protected topological (SPT) phases, known as gapless SPT (or gSPT) phases; and gapless phases showing properties analogous to gapped spontaneous symmetry broken (SSB) phases, that we refer to as gapless SSB (or gSSB) phases. We fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram describing possible deformations connecting them. This phase diagram is partially ordered and defines a so-called Hasse diagram. Based on these deformations, we identify gapless phases exhibiting symmetry protected criticality, that we refer to as intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases. This includes the first examples of igSPT and igSSB phases with non-invertible symmetries. Central to this analysis is the Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. On a mathematical note, gSPT phases are classified by functors between fusion categories, generalizing the fact that gapped SPT phases are classified by fiber functors; and gSSB phases are classified by functors from fusion to multi-fusion categories. Finally, our framework can be applied to understand gauging of trivially acting non-invertible symmetries, including possible patterns of decomposition arising due to such gaugings.

arXiv comment54 pages, v2: added a generalized superconductivity interpretation in terms of condensed, confined and deconfined charges, v3: added complete list of gapless phases for Rep(D8) and derivations, v4: published in SciPost Physics

References in corpus (153)

• Fault-tolerant quantum computation by anyons
• String-net condensation: A physical mechanism for topological phases
• Condensate induced transitions between topologically ordered phases
• Detection of Symmetry Protected Topological Phases in 1D
• A classification of symmetry enriched topological phases with exactly solvable models
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Non-Invertible Duality Defects in 3+1 Dimensions
• Symmetries and Strings of Adjoint QCD${}_2$
• Higher Gauging and Non-invertible Condensation Defects
• Orbifold groupoids
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• Non-Invertible Higher-Categorical Symmetries
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Non-Invertible Global Symmetries and Completeness of the Spectrum
• Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
• Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
• Universal Non-Invertible Symmetries
• Finite Group Modular Data
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• Gapless topological phases and symmetry-enriched quantum criticality
• Intrinsically Gapless Topological Phases
• Gapless Symmetry Protected Topological Order
• Asymptotic density of states in 2d CFTs with non-invertible symmetries
• Non-Invertible Symmetries from Holography and Branes
• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases
• Diagrammatics for Bose condensation in anyon theories
• Non-Invertible Symmetries of $\mathcal{N}=4$ SYM and Twisted Compactification
• The Branes Behind Generalized Symmetry Operators
• Non-invertible Time-reversal Symmetry
• Branes and Non-Invertible Symmetries
• Topological interactions in broken gauge theories
• Unifying Constructions of Non-Invertible Symmetries
• Exploring Non-Invertible Symmetries in Free Theories
• On Continuous 2-Category Symmetries and Yang-Mills Theory
• Non-invertible Symmetries of Class $\mathcal{S}$ Theories
• Non-Invertible Symmetry Webs
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Lecture Notes on Generalized Symmetries and Applications
• Non-invertible Symmetries and Higher Representation Theory I
• Remarks on Boundaries, Anomalies, and Noninvertible Symmetries
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Non-Invertible Duality Transformation Between SPT and SSB Phases
• Non-Invertible Defects in 5d, Boundaries and Holography
• Obstructions to Gapped Phases from Non-Invertible Symmetries
• Decomposition, condensation defects, and fusion
• Continuous Generalized Symmetries in Three Dimensions
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Generalized Charges, Part I: Invertible Symmetries and Higher Representations
• Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains
• On anomalies and gauging of U(1) non-invertible symmetries in 4d QED
• Classification of topological phases with finite internal symmetries in all dimensions
• An Introduction to Higher-Form Symmetries
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Higher categorical symmetries and gauging in two-dimensional spin systems
• Bootstrapping Non-Invertible Symmetries
• Non-invertible Symmetries and Higher Representation Theory II
• What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
• Bulk-boundary correspondence for intrinsically-gapless SPTs from group cohomology
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Subsystem Non-Invertible Symmetry Operators and Defects
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Anomalies and gauging of U(1) symmetries
• Topological symmetry in quantum field theory
• Introduction to Generalized Global Symmetries in QFT and Particle Physics
• Solitonic symmetry beyond homotopy: Invertibility from bordism and noninvertibility from topological quantum field theory
• Topological Defect Lines in Two Dimensional Fermionic CFTs
• A mathematical theory of gapless edges of 2d topological orders. Part II
• Gauging Noninvertible Defects: A 2-Categorical Perspective
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Exact emergent higher-form symmetries in bosonic lattice models
• On the Holographic Dual of a Topological Symmetry Operator
• Categories of quantum liquids I
• Decorated Defect Construction of Gapless-SPT States
• Topological holography, quantum criticality, and boundary states
• Universal entanglement spectrum in gapless symmetry protected topological states
• Notes on gauging noninvertible symmetries, part 1: Multiplicity-free cases
• SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds
• Comments on Non-invertible Symmetries in Argyres-Douglas Theories
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• One dimensional gapped quantum phases and enriched fusion categories
• Anomalies of $(1+1)D$ categorical symmetries
• Remarks on Geometric Engineering, Symmetry TFTs and Anomalies
• Non-invertible symmetries along 4d RG flows
• Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Anomalies of non-invertible self-duality symmetries: fractionalization and gauging
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Non-invertible symmetry-protected topological order in a group-based cluster state
• ICTP Lectures on (Non-)Invertible Generalized Symmetries
• Non-invertible symmetries and boundaries in four dimensions
• Parity and Spin CFT with boundaries and defects
• The holography of non-invertible self-duality symmetries
• Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup
• Topological operators, noninvertible symmetries and decomposition
• Decomposition, Trivially-Acting Symmetries, and Topological Operators
• Non-Invertible Chiral Symmetry and Exponential Hierarchies
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Orbifolds by 2-groups and decomposition
• Emergent generalized symmetry and maximal symmetry-topological-order
• Neutrino Masses from Generalized Symmetry Breaking
• Topological defects
• "Zoology" of non-invertible duality defects: the view from class $\mathcal{S}$
• Representation theory for categorical symmetries
• Non-Invertible Symmetry in Calabi-Yau Conformal Field Theories
• Symmetry TFTs for Non-Invertible Defects
• Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients
• Branes and symmetries for $\mathcal{N}=3$ S-folds
• Lectures on Generalized Symmetries
• Categorical Landau Paradigm for Gapped Phases
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories
• Four-manifolds and Symmetry Categories of 2d CFTs
• Category of SET orders
• Symmetry TFTs and Anomalies of Non-Invertible Symmetries
• Lattice Realizations of Topological Defects in the critical (1+1)-d Three-State Potts Model
• Classification of 1+1D gapless symmetry protected phases via topological holography
• Monopoles, Scattering, and Generalized Symmetries
• Three-dimensional orbifolds by 2-groups
• Non-invertible Global Symmetries in the Standard Model
• Higher Structure of Chiral Symmetry
• When are Duality Defects Group-Theoretical?
• A Goldstone theorem for continuous non-invertible symmetries
• On the 6d Origin of Non-invertible Symmetries in 4d
• From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors
• Symmetry TFT for Subsystem Symmetry
• A SymTFT for Continuous Symmetries
• Top Down Approach to Topological Duality Defects
• Dilaton shifts, probability measures, and decomposition
• An introduction to decomposition
• Para-fusion Category and Topological Defect Lines in ${\mathbb{Z}}_N$-parafermionic CFTs
• 2+1D symmetry-topological-order from local symmetric operators in 1+1D
• Non-invertible Symmetries in 2D from Type IIB String Theory
• SymTrees and Multi-Sector QFTs
• On the Fate of Stringy Non-Invertible Symmetries
• Coupling a Cosmic String to a TQFT
• Classification of modular data up to rank 12
• R7-Branes as Charge Conjugation Operators
• Non-Invertible Gauss Law and Axions
• Aspects of Categorical Symmetries from Branes: SymTFTs and Generalized Charges
• Lattice realization of the axial $U(1)$ noninvertible symmetry
• Spontaneously Broken $(-1)$-Form U(1) Symmetries
• Field theories with defects and the centre functor
• SymTFT out of equilibrium: from time crystals to braided drives and Floquet codes
• Non-Invertible Duality Interfaces in Field Theories with Exotic Symmetries
• Emergent non-invertible symmetries in $\mathcal{N}=4$ Super-Yang-Mills theory
• Duality defects in $D_n$-type Niemeier lattice CFTs
• Swampland Constraints on the SymTFT of Supergravity
• Invertibility of Condensation Defects and Symmetries of 2 + 1d QFTs
• Axion Domain Walls, Small Instantons, and Non-Invertible Symmetry Breaking
• Non-invertible duality defect and non-commutative fusion algebra
• Quantum Duality in Electromagnetism and the Fine-Structure Constant
• ${\mathbb{Z}}_N$ Duality and Parafermions Revisited

Cited by in corpus (13)

• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Gauge theory and mixed state criticality
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• von Neumann Subfactors and Non-invertible Symmetries
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Entropic order parameters and topological holography
• Topological Manipulations On $\mathbb{R}$ Symmetries Of Abelian Gauge Theory
• Decomposition in 2d non-invertible gaugings
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Generalized symmetry enriched criticality in (3+1)d
• Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains