Symmetry protection of topological order in one-dimensional quantum spin systems

Abstract

We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-$S$ Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\pi$-rotations about $x,y$ and $z$ axes; (ii) time-reversal symmetry $S^{x,y,z}\to -S^{x,y,z}$; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B \textbf{81}, 064439 (2010)]. On the other hand, an even-$S$ Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.

9 pages, 6 figures, extended version: several new examples and numerical results added. Journal reference added

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• Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension
• Electronic structure and magnetic properties of the strong-rung spin-1 ladder compound Rb${}_3$Ni${}_2$(NO${}_3$)${}_7$
• Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid
• Twice Hidden String Order and Competing Phases in the Spin-1/2 Kitaev-Gamma Ladder
• The Asymmetric Valence-Bond-Solid States in Quantum Spin Chains: The Difference Between Odd and Even Spins
• Dissipative Symmetry-Protected Topological Order
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• Superuniversality from disorder at two-dimensional topological phase transitions
• The bilinear-biquadratic model on the complete graph
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Critical behavior of the extended Hubbard model with bond dimerization
• Efficient variational quantum circuit structure for correlated topological phases
• Topological Phase Transition in the Extended Cluster Compass Ladder
• Quantum Berezinskii-Kosterltz-Thouless Transition for Topological Insulator
• Ground-State Phase Diagram of the Bond-Alternating $S=2$ Quantum Spin Chain with the $XXZ$ and On-Site Anisotropies -- Symmetry Protected Topological Phase versus Trivial Phase
• Spin parity effects in monoaxial chiral ferromagnetic chain
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements
• Antiferromagnetic Ordering in MnF(salen)
• Topological transitions to Weyl states in bulk Bi${}_2$Se${}_3$: Effect of hydrostatic pressure and doping
• Spin-1 Haldane chains of superconductor-semiconductor hybrids
• Converting topological insulators into topological metals within the tetradymite family
• Reentrance of Topological Phase in Spin-1 Frustrated Heisenberg Chain
• Entanglement in non-critical inhomogeneous quantum chains
• Generalized Kramers-Wannier Duality from Bilinear Phase Map
• Pivoting through the chiral-clock family
• Gappability Index for Quantum Many-Body Systems
• Anomaly inflow, dualities, and quantum simulation of abelian lattice gauge theories induced by measurements
• Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model
• Robust symmetry-protected metrology with the Haldane phase
• Magnetization plateau of the $S=2$ antiferromagnetic Heisenberg chain with anisotropies
• Valence bond fluctuations in the Kitaev spin model
• Negativity in the Generalized Valence Bond Solid State
• Resilience of hidden order to symmetry-preserving disorder
• Bulk Entanglement Spectrum in Gapped Spin Ladders
• Entanglement properties of the two-dimensional SU(3) AKLT state
• High Chern numbers and topological flat bands in high-field polarized Kitaev magnets on the star lattice
• Geometric Entanglement and Quantum Phase Transition in Generalized Cluster-XY models
• Composite symmetry protected topological order and effective models
• Higher Berry curvature from matrix product states
• Classical topological paramagnetism
• Fractional corner magnetization of collinear antiferromagnets
• An Interplay of Topology and Quantized Geometric Phase for two Different Symmetry-Class Hamiltonians
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Breakdown of superdiffusion in perturbed quantum integrable spin chains and ladders
• Reentrant Topological Phase Transition in a Bridging Model between Kitaev and Haldane Chains
• Attractor metadynamics in terms of target points in slow-fast systems: adiabatic vs. symmetry protected flow in a recurrent neural network
• Adaptive variational ground state preparation for spin-1 models on qubit-based architectures
• Anomalous domain wall condensation in a modified Ising chain
• Topological Field Theories of Three-Dimensional Rotation Symmetric Insulators: Coupling Curvature and Electromagnetism
• Bulk and boundary entanglement transitions in the projective gauge-Higgs model
• Quantum antiferromagnetic Heisenberg half-odd integer spin model as the entanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
• Symmetry-protected topological phase transitions and robust chiral order on a tunable zigzag lattice
• Phase diagram of spin-$1$ chains with Dzyaloshinskii-Moriya interaction
• Interacting topological phases in thin films of topological mirror Kondo insulators
• Role of anomalous symmetry in 0-$\pi$ qubits
• 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
• Topological phases of unitary dynamics: Classification in Clifford category
• Gauge theory and mixed state criticality
• Interaction-induced phase transitions at topological quantum criticality of an extended Su-Schrieffer-Heeger model
• Symmetry protected topological phases in spin-1 ladders and their phase transitions
• Exotic Haldane Superfluid Phase of Soft-Core Bosons in Optical Lattices
• Existence of two distinct valence bond solid states in the dimerized frustrated ferromagnetic $J_1$-$J_1^{\prime }$-$J_2$ chain
• The resource theory of tensor networks
• Self-$G$-ality in 1+1 dimensions
• Transfer learning from Hermitian to non-Hermitian quantum many-body physics
• Discrete symmetries and Lieb-Schultz-Mattis theorem
• Detection of gapped phases of a 1D spin chain with onsite and spatial symmetries
• Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions
• Interacting Majorana modes at surfaces of noncentrosymmetric superconductors
• Inexorable Edge Kondo Breakdown in Topological Kondo Insulators
• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• A bosonic topological phase in a paired superfluid
• Fate of symmetry protected coherence in open quantum system
• Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems
• Classical and quantum spin liquids
• Luther-Emery liquid and dominant singlet superconductivity in the hole-doped Haldane spin-1 chain
• Quantum Inception Score
• Inferring Hidden Symmetries of Exotic Magnets from Detecting Explicit Order Parameters
• Embedding semiclassical periodic orbits into chaotic many-body Hamiltonians
• Topological Phases in Half-Integer Higher Spin $J_1$-$J_2$ Heisenberg Chains
• Haldane and Dimer phases in a frustrated spin chain: an exact groundstate and associated topological phase transition
• Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction
• Preparing matrix product states via fusion: constraints and extensions
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Competing superconducting instabilities in the one-dimensional p-band degenerate cold fermionic system
• Bell-type correlation at quantum phase transitions in spin-1 chain
• Complete characterization of spin chains with two Ising symmetries
• Advancing Hybrid Quantum-Classical Algorithms via Mean-Operators
• Quasi-fractionalization of spin in a cluster-based Haldane state supported by chirality of a triangular spin tube
• Quantum phase transition in skewed ladders: an entanglement entropy and fidelity study
• Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model
• Duality and topology
• Quantum order by disorder in Frustrated Spin Nanotubes
• Bound-States Dynamics in One-Dimensional Multi-Species Fermionic Systems
• Engineering Hierarchical Symmetries
• Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Refined Phase Diagram for a Spin-1 System Exhibiting a Haldane Phase
• The Ground State of the S=1 Antiferromagnetic Heisenberg Chain is Topologically Nontrivial if Gapped
• Competition between dimerization and vector chirality in the spin-$3/2$ $J_1$-$J_2$ Heisenberg chain with uniaxial single-ion anisotropy
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• Lowest-energy states in parity-transformation eigenspaces of SO(N) spin chain
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Ground-State Phase Diagram of S=2 Heisenberg Chains with Alternating Single-Site Anisotropy
• Generalized Landau Paradigm for quantum phases and phase transitions
• Topological quantum phase transition in strongly correlated Kondo insulators in 1D
• Field theory analysis of S=1 antiferromagnetic bond-alternating chains in the dimer phase
• Quantum simulation of superexchange magnetism in linear ion crystals
• Counter-intuitive yet efficient regimes for measurement based quantum computation on symmetry protected spin chains
• Spontaneous time-reversal symmetry breaking without magnetism in a $S=1$ chain
• Topological Dipole Insulator
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Nonlocal order parameter of pair superfluids
• Parity-dependent phase diagrams in spin-cluster two-leg ladders
• Tunable quantum spin chain with three-body interactions
• Superradiance, charge density waves and lattice gauge theory in a generalized Rabi-Hubbard chain
• Frustrated mixed-spin ladders: Evidence for a bond order wave phase between rung-singlet and Haldane phases
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• One-dimensional symmetric phases protected by frieze symmetries
• Haldane phases and phase diagrams of the S = 3/2, 1 bilinear-biquadratic Heisenberg model on the orthogonal dimer chain
• A Simple and General Equation for Matrix Product Unitary Generation
• Translational Symmetry Broken Magnetization Plateau of the $S=2$ Antiferromagnetic Chain with Anisotropies
• $Z_2\times Z_2$ symmetry and $Z_4$ Berry phase of bosonic ladder
• Topological and Error-Correcting Properties for Symmetry-Protected Topological Order
• Redundant string symmetry-based error correction: Demonstrations on quantum devices
• Promising regimes for the observation of topological degeneracy in spin chains
• Quantum phase transitions to topological Haldane phases in spin-one chains studied by linked-cluster expansions
• Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
• Local dynamics and detection of topology in spin-1 chains
• Eleven Competing Phases in the Heisenberg-Gamma (J$\Gamma$) Ladder
• Mass ratio of elementary excitations in frustrated antiferromagnetic chains with dimerization
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation
• Threefold Way for Typical Entanglement
• Kondo spin liquid in Kondo necklace model: Classical disordered phase versus symmetry-protected topological state
• Entanglement-enabled symmetry-breaking orders
• S=2 Quantum Spin Chain with the Biquadratic Exchange Interaction
• Reconfigurable dissipative entanglement between many spin ensembles: from robust quantum sensing to many-body state engineering
• Intertwined order of generalized global symmetries
• Angular-time evolution for the Affleck-Kennedy-Lieb-Tasaki chain and its edge-state dynamics
• Impurity screening by defects in (1+1)$d$ quantum critical systems
• SPT indices emerging from translation invariance in two dimensional quantum spin systems
• Entropic order parameters and topological holography
• Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities
• Spectral signatures of nonstabilizerness and criticality in infinite matrix product states
• Topological Phases of Spin-1/2 Ferromagnetic--Antiferromagnetic Alternating Heisenberg Chains with Alternating Next-Nearest-Neighbour Interaction
• Quantum phase with spontaneous translational symmetry breaking in an extended diamond chain
• Bulk-edge correspondence in the Haldane phase of the bilinear-biquadratic spin-1 Hamiltonian
• Interrelations among frustration-free models via Witten's conjugation
• Symmetries and field tensor network states
• Entanglement renormalization and symmetry fractionalization
• Condensation Completion and Defects in 2+1D Topological Orders
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• $N$-ality symmetry and SPT phases in (1+1)d
• Topological Phase Transition in the Two-Leg Hubbard Model: Emergence of the Haldane Phase via Diagonal Hopping and Strong Interactions
• Efficient Berry Phase Calculation via Adaptive Variational Quantum Computing Approach
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Instability of the Haldane Phase: Roles of Charge Fluctuations and Hund's Coupling
• Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems
• Equivalence class of Emergent Single Weyl fermion lattice models in 3 dimensions: gapless superconductors and superfluids versus chiral fermions
• Hidden subsystem symmetry protected states in competing topological orders
• Bosonic Haldane insulator in the presence of local disorder: A quantum Monte Carlo study
• Dynamically generated quadrupole polarization using Floquet adiabatic evolution
• Parafermionic representation of Potts-based cluster chain
• Many-body Keldysh Crossover in the DC-driven Haldane Spin Chain
• Evidence for valence-bond pairing in a one-dimensional two-orbital system
• Magnetically tuned topological phase in graphene nanoribbon heterojunctions
• Tensorization of neural networks for improved privacy and interpretability
• Construction of Entangled Many-body States via the Higgs Mechanism
• Triangular lattice models of the Kalmeyer-Laughlin spin liquid from coupled wires
• Detection of 2D SPT Order with Partial Symmetries
• A brief note on the G${}_2$ Affleck-Kennedy-Lieb-Tasaki chain
• Topological to magnetically ordered quantum phase transition in antiferromagnetic spin ladders with long-range interactions
• Multipartite entanglement in quantum phase transitions
• Retrieval of free-Majorana wavefunctions for the finite Kitaev chain using an appropriate momentum representation
• Generalized symmetry enriched criticality in (3+1)d
• A Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph
• Detecting underlying symmetry-protected topological phases via strange correlators and edge engineering
• Spontaneously Broken Non-Invertible Symmetries in Transverse-Field Ising Qudit Chains
• Bosonic realization of SU(3) chiral Haldane phases
• Matrix product state classification of 1D multipole symmetry protected topological phases
• Phases and phase transitions of an $S=3/2$ chain on metallic and semi-metallic surfaces
• Resource complexity of Symmetry Protected Topological phases
• Geometric Effects on Tunneling in Driven Quantum Systems
• Measurement-Based Quantum Computation Using the Spin-1 XXZ Model with Uniaxial Anisotropy
• Correlation functions at the topological quantum phase transition in the S=1 XXZ chain with single-ion anisotropy
• Topological phases of bosons with local parity coupling on a dimerized lattice
• Stability of the symmetry-protected topological phase and Ising transitions in a disordered U(1) quantum link model on a ladder
• Strong zero modes in integrable spin-S chains
• Persistent Haldane Phase in Carbon Tetris Chains
• Quantized edge magnetizations and their symmetry protection in one-dimensional quantum spin systems
• Vortical Quantum Memory
• Ground-State Phase Diagram of an Anisotropic S=1 Ferromagnetic-Antiferromagnetic Bond-Alternating Chain
• Anisotropy-induced spin parity effects