The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries

Abstract

We provide a generalization of the Symmetry Topological Field Theory (SymTFT) framework to characterize phase transitions and gapless phases with categorical symmetries. The central tool is the club sandwich, which extends the SymTFT setup to include an interface between two topological orders: there is a symmetry boundary, which is gapped, and a physical boundary that may be gapless, but in addition, there is also a gapped interface in the middle. The club sandwich generalizes so-called Kennedy-Tasaki (KT) transformations. Building on the results in [1, 2] on gapped phases with categorical symmetries, we construct gapless theories describing phase transitions with non-invertible symmetries by applying suitable KT transformations on known phase transitions provided by the critical Ising model and the 3-state Potts model. We also describe in detail the order parameters in these gapless theories characterizing the phase transitions, which are generally mixtures of conventional and string-type order parameters mixed together by the action of categorical symmetries. Additionally, removing the physical boundary from the club sandwiches results in club quiches, which characterize all possible gapped boundary phases with (possibly non-invertible) symmetries that can arise on the boundary of a bulk gapped phase. We also provide a mathematical characterization of gapped boundary phases with symmetries as pivotal tensor functors whose targets are pivotal multi-fusion categories.

58 pages, v2: typos fixed, v3: published version

References in corpus (72)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Non-Invertible Duality Defects in 3+1 Dimensions
• Higher Gauging and Non-invertible Condensation Defects
• Orbifold groupoids
• Non-Invertible Higher-Categorical Symmetries
• Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
• Universal Non-Invertible Symmetries
• Asymptotic density of states in 2d CFTs with non-invertible symmetries
• Non-Invertible Symmetries from Holography and Branes
• 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
• 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 Duality Transformation Between SPT and SSB Phases
• Non-Invertible Defects in 5d, Boundaries and Holography
• Obstructions to Gapped Phases from Non-Invertible Symmetries
• On topological phases of spin chains
• 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
• On anomalies and gauging of U(1) non-invertible symmetries in 4d QED
• An Introduction to Higher-Form Symmetries
• Higher categorical symmetries and gauging in two-dimensional spin systems
• Non-invertible Symmetries and Higher Representation Theory II
• Bootstrapping Non-Invertible Symmetries
• What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Subsystem Non-Invertible Symmetry Operators and Defects
• 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
• 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
• Categories of quantum liquids I
• Classifying phases protected by matrix product operator symmetries using matrix product states
• SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds
• Comments on Non-invertible Symmetries in Argyres-Douglas Theories
• One dimensional gapped quantum phases and enriched fusion categories
• Non-invertible symmetries along 4d RG flows
• Anomalies of non-invertible self-duality symmetries: fractionalization and gauging
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Non-invertible symmetries and boundaries in four dimensions
• Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters
• Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states
• Symmetry protected topological phases beyond groups: The q-deformed Affleck-Kennedy-Lieb-Tasaki model
• The holography of non-invertible self-duality symmetries
• Parity and Spin CFT with boundaries and defects
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Emergent generalized symmetry and maximal symmetry-topological-order
• Topological defects
• Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Four-manifolds and Symmetry Categories of 2d CFTs
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• Category of SET orders
• When are Duality Defects Group-Theoretical?
• 2+1D symmetry-topological-order from local symmetric operators in 1+1D
• Para-fusion Category and Topological Defect Lines in ${\mathbb{Z}}_N$-parafermionic CFTs

Cited by in corpus (20)

• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Boundary SymTFT
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Gauge theory and mixed state criticality
• 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
• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• (-1)-Form Symmetries and Anomaly Shifting from SymTFT
• 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
• Entropic order parameters and topological holography
• Analogs of deconfined quantum criticality for non-invertible symmetry breaking in 1d
• Exploring two dimensional ${\mathbb{Z}}_2$ invariant phases with time reversal symmetry and their transitions with topological operations
• Critical theories connecting gapped phases with ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry from the duality web
• Generalized symmetry enriched criticality in (3+1)d
• Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains