Cluster state as a non-invertible symmetry protected topological phase

Abstract

We show that the standard 1+1d ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ cluster model has a non-invertible global symmetry, described by the fusion category Rep(D${}_8$). Therefore, the cluster state is not only a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry protected topological (SPT) phase, but also a non-invertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D${}_8$) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these non-invertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.

19 pages, 1 figure, 3 tables; v2: minor changes; v3: minor changes

References in corpus (44)

• Classification of topological quantum matter with symmetries
• Generalized Global Symmetries
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Topological phases of fermions in one dimension
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Classifying quantum phases using Matrix Product States and PEPS
• Symmetry Protected Topological phases of Quantum Matter
• Braiding statistics approach to symmetry-protected topological phases
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Cluster-state quantum computation
• Generalized Symmetries in Condensed Matter
• Boundary conformal field theory approach to the two-dimensional critical Ising model with a defect line
• On finite symmetries and their gauging in two dimensions
• On gauging finite subgroups
• Kramers-Wannier-like duality defects in (3+1)d gauge theories
• Non-Invertible Duality Defects in 3+1 Dimensions
• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• Topological Defects on the Lattice I: The Ising model
• Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions
• Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
• Setting the quantum integrand of M-theory
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• Topological Transition on the Conformal Manifold
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Topological order in 1D Cluster state protected by symmetry
• 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
• On lattice models of gapped phases with fusion category symmetries
• Bosonic Anomalies, Induced Fractional Quantum Numbers and Degenerate Zero Modes: the anomalous edge physics of Symmetry-Protected Topological States
• Twisted boundary condition and Lieb-Schultz-Mattis ingappability for discrete symmetries
• Topological field theories and symmetry protected topological phases with fusion category symmetries
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Anomalous Symmetries End at the Boundary
• The spin-1/2 XXZ Heisenberg chain, the quantum algebra U_q[sl(2)], and duality transformations for minimal models
• $SL(2,\mathbb{Z})$ action on QFTs with ${\mathbb{Z}}_2$ symmetry and the Brown-Kervaire invariants
• Notes on gauging noninvertible symmetries, part 1: Multiplicity-free cases
• 't Hooft anomalies and boundaries
• Mixed global anomalies and boundary conformal field theories
• Kennedy-Tasaki transformation and non-invertible symmetry in lattice models beyond one dimension
• Constraints on order and disorder parameters in quantum spin chains

Cited by in corpus (50)

• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Generating Lattice Non-invertible Symmetries
• Lattice T-duality from non-invertible symmetries in quantum spin chains
• Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems
• Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance
• (SPT-)LSM theorems from projective non-invertible symmetries
• Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation
• Gauging non-invertible symmetries on the lattice
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Boundary SymTFT
• Non-invertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry
• Symmetry Transmutation and Anomaly Matching
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Category of SET orders
• Generalized Kramers-Wannier Duality from Bilinear Phase Map
• Additivity, Haag duality, and non-invertible symmetries
• Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective
• LSM and CPT
• Stabilizer Rényi Entropy Encodes Fusion Rules of Topological Defects and Boundaries
• Global symmetries of quantum lattice models under non-invertible dualities
• Quantum Cluster State Model with Haagerup Fusion Category Symmetry
• Duality-preserving deformation of 3+1d lattice ${\mathbb{Z}}_2$ gauge theory with exact gapped ground states
• 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
• Self-$G$-ality in 1+1 dimensions
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Many-body quantum catalysts for transforming between phases of matter
• Generalized Landau Paradigm for quantum phases and phase transitions
• The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization
• Gauging Modulated Symmetries via Multiple Gauge Symmetry Operators and Adaptive Quantum Circuits
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• $N$-ality symmetry and SPT phases in (1+1)d
• Intertwined order of generalized global symmetries
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Analogs of deconfined quantum criticality for non-invertible symmetry breaking in 1d
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• From gauging to duality in one-dimensional quantum lattice models
• Les Houches Lecture Notes on Tensor Networks
• Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Symmetric entanglers for non-invertible SPT phases
• Entanglement Spectrum Resolved by Loop Symmetries
• Non-invertible translation from Lieb-Schultz-Mattis anomaly
• Revealing effects of local dimension on variable-range interacting model by connecting Lieb-Robinson bounds and multipartite entanglement
• Quantum many-body scarring from Kramers-Wannier duality
• Generalized symmetry enriched criticality in (3+1)d