Gapped Phases with Non-Invertible Symmetries: (1+1)d

Abstract

We propose a general framework to characterize gapped infra-red (IR) phases of theories with non-invertible (or categorical) symmetries. In this paper we focus on (1+1)d gapped phases with fusion category symmetries. The approach that we propose uses the Symmetry Topological Field Theory (SymTFT) as a key input: associated to a field theory in d spacetime dimensions, the SymTFT lives in one dimension higher and admits a gapped boundary, which realizes the categorical symmetries. It also admits a second, physical, boundary, which is generically not gapped. Upon interval compactification of the SymTFT by colliding the gapped and physical boundaries, we regain the original theory. In this paper, we realize gapped symmetric phases by choosing the physical boundary to be a gapped boundary condition as well. This set-up provides computational power to determine the number of vacua, the symmetry breaking pattern, and the action of the symmetry on the vacua. The SymTFT also manifestly encodes the order parameters for these gapped phases, thus providing a generalized, categorical Landau paradigm for (1+1)d gapped phases. We find that for non-invertible symmetries the order parameters involve multiplets containing both untwisted and twisted sector local operators, and hence can be interpreted as mixtures of conventional and string order parameters. We also observe that spontaneous breaking of non-invertible symmetries can lead to vacua that are physically distinguishable: unlike the standard symmetries described by groups, non-invertible symmetries can have different actions on different vacua of an irreducible gapped phase. This leads to the presence of relative Euler terms between physically distinct vacua. We also provide a mathematical description of symmetric gapped phases as 2-functors from delooping of fusion category characterizing the symmetry to Euler completion of 2-vector spaces.

141 pages, v2: corrected an omission in the analysis of TY(Z_N) gapped phases reported by A. Antinucci, references added, v3: some charges corrected in section 7.1, v4,5: referee comments addressed and references added

References in corpus (80)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Condensate induced transitions between topologically ordered phases
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Non-Invertible Duality Defects in 3+1 Dimensions
• String order and symmetries in quantum spin lattices
• Symmetries and Strings of Adjoint QCD${}_2$
• 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
• Construction of two-dimensional topological field theories with non-invertible symmetries
• Non-invertible Time-reversal Symmetry
• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• 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
• Obstructions to Gapped Phases from Non-Invertible Symmetries
• On lattice models of gapped phases with fusion category symmetries
• Non-Invertible Defects in 5d, Boundaries and Holography
• Non-Invertible Duality Transformation Between SPT and SSB Phases
• 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
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Bootstrapping Non-Invertible Symmetries
• Non-invertible Symmetries and Higher Representation Theory II
• What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Subsystem Non-Invertible Symmetry Operators and Defects
• Topological field theories and symmetry protected topological phases with fusion category symmetries
• 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
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Gauging Noninvertible Defects: A 2-Categorical Perspective
• 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
• Fermionization of fusion category symmetries in 1+1 dimensions
• One dimensional gapped quantum phases and enriched fusion categories
• Non-invertible symmetries along 4d RG flows
• Topological Quantum Computation with Gapped Boundaries
• Anomalies of non-invertible self-duality symmetries: fractionalization and gauging
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Non-invertible symmetries and boundaries in four dimensions
• Lattice Models for Phases and Transitions with Non-Invertible Symmetries
• 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
• 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
• Representation theory for categorical symmetries
• Computing associators of endomorphism fusion categories
• Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients
• Four-manifolds and Symmetry Categories of 2d CFTs
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• Monopoles, Scattering, and Generalized Symmetries
• When are Duality Defects Group-Theoretical?
• Higher Structure of Chiral Symmetry
• Para-fusion Category and Topological Defect Lines in ${\mathbb{Z}}_N$-parafermionic CFTs

Cited by in corpus (31)

• Topological holography, quantum criticality, and boundary states
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• Gauging non-invertible symmetries on the lattice
• SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Boundary SymTFT
• Entanglement and the density matrix renormalisation group in the generalised Landau paradigm
• Non-Invertible T-duality at Any Radius via Non-Compact SymTFT
• Twisted Partition Functions as Order Parameters
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Topological physics in quantum critical systems
• 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
• von Neumann Subfactors and Non-invertible Symmetries
• Generalized Landau Paradigm for quantum phases and phase transitions
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Soft symmetries of topological orders
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Analogs of deconfined quantum criticality for non-invertible symmetry breaking in 1d
• $N$-ality symmetry and SPT phases in (1+1)d
• Exploring two dimensional ${\mathbb{Z}}_2$ invariant phases with time reversal symmetry and their transitions with topological operations
• Intertwined order of generalized global symmetries
• Non-onsite symmetry breaking: topological phase coexistence and criticality
• Entropic order parameters and topological holography
• Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains
• Symmetric entanglers for non-invertible SPT phases
• Decomposition in 2d non-invertible gaugings
• Higher-order topological phases protected by noninvertible and subsystem symmetries