Classification of Gapped Symmetric Phases in 1D Spin Systems

Abstract

Quantum many-body systems divide into a variety of phases with very different physical properties. The question of what kind of phases exist and how to identify them seems hard especially for strongly interacting systems. Here we make an attempt to answer this question for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range correlated states in the same phase, we classify possible quantum phases for 1D matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if certain symmetry is required, many phases exist with different symmetry protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.

21 pages, 7 figures. Version 2, classification for parity and translation symmetry updated

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• Bosonic symmetry protected topological phases with reflection symmetry
• Heisenberg antiferromagnet on the Husimi lattice
• Brane parity orders in the insulating state of Hubbard ladders
• Non-Topological Majorana Zero Modes in Inhomogeneous Spin Ladders
• Matrix Product Operator Algebras II: Phases of Matter for 1D Mixed States
• Topological Field Theory and Matrix Product States
• Scaling of geometric phase versus band structure in cluster-Ising models
• Typical Correlation Length of Sequentially Generated Tensor Network States
• Quantum Entanglement and Topological Order in Hole-Doped Valence Bond Solid States
• Complete phase diagram and topological properties of interacting bosons in one-dimensional superlattices
• Haldane insulator protected by reflection symmetry in the doped Hubbard model on the three-legged ladder
• Emergence of spontaneous symmetry breaking in dissipative lattice systems
• (SPT-)LSM theorems from projective non-invertible symmetries
• Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
• Topological incommensurate magnetization plateaus in quasi-periodic quantum spin chains
• Emergent particles and gauge fields in quantum matter
• Topological Crystalline Magnets: Symmetry-Protected Topological Phases of Fermions
• Product vacua with boundary states and the classification of gapped phases
• Intrinsically Interacting Higher-Order Topological Superconductors
• Quasi-exact quantum computation
• Phase structure of the 1+1 dimensional massive Thirring model from matrix product states
• Haldane phase in one-dimensional topological Kondo insulators
• Holonomic quantum computing in symmetry-protected ground states of spin chains
• $C^1$-classification of gapped parent Hamiltonians of quantum spin chains
• Rational indices for quantum ground state sectors
• An index for two-dimensional SPT states
• Quantum phase transition between symmetry enriched topological phases in tensor-network states
• Matrix product states for topological phases with parafermions
• Relativistic continuous matrix product states for quantum fields without cutoff
• Exotic quantum liquids in Bose-Hubbard models with spatially-modulated symmetries
• Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip
• Quantum walks: Schur functions meet symmetry protected topological phases
• Lieb-Schultz-Mattis theorem with a local twist for general one-dimensional quantum systems
• Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements
• Symmetry protected topological edge modes and emergent partial time reversal symmetry breaking in open quantum many-body systems
• Anomalies and unnatural stability of multi-component Luttinger liquids in ${\mathbb{Z}}_n\times {\mathbb{Z}}_n$ spin chains
• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• Entanglement Properties of Gauge Theories from Higher-Form Symmetries
• Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Duality viewpoint of criticality
• Symmetry reduction induced by anyon condensation: a tensor network approach
• Optimization schemes for unitary tensor-network circuit
• Surface field theories of point group symmetry protected topological phases
• Locality at the boundary implies gap in the bulk for 2D PEPS
• Topological lattice field theories from intertwiner dynamics
• The minimal canonical form of a tensor network
• Edge modes and symmetry-protected topological states in open quantum systems
• Symmetries and anomalies of Kitaev spin-$S$ models: Identifying symmetry-enforced exotic quantum matter
• Piercing the rainbow: entanglement on an inhomogeneous spin chain with a defect
• Symmetry protected weak topological phases in a superlattice
• Phase diagram of the isotropic spin-3/2 model on the z=3 Bethe lattice
• Implementation of rigorous renormalization group method for ground space and low-energy states of local Hamiltonians
• Tensor renormalization group in bosonic field theory
• Effective field theories of topological crystalline insulators and topological crystals
• Altermagnetic Instabilities from Quantum Geometry
• Resilience of the topological phases to frustration
• Quench Dynamics of Topological Maximally-Entangled States
• Irreducible Projective Representations and Their Physical Applications
• Symmetry-protected topological phases in two-leg SU(N) spin ladder with unequal spins
• Characterizing MPS and PEPS Preparable via Measurement and Feedback
• Inaccessible entanglement in symmetry protected topological phases
• Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes
• Classification of symmetry-protected topological phases in two-dimensional many body-localized systems
• Ground-state phase diagram of a spin-1/2 frustrated XXZ ladder
• Symmetry fractionalization: Symmetry-protected topological phases of the bond-alternating spin-$1/2$ Heisenberg chain
• Low energy effective field theories of fermion liquids and mixed $U(1)\times {\mathbb{R}}^d$ anomaly
• Topological phonon analysis of the 2D buckled honeycomb lattice: an application to real materials
• Multipartite entanglement of the topologically ordered state in a perturbed toric code
• Topological qubits from valence bond solids
• Symmetry protected gapless $Z_2$ spin liquids
• Geometry of reduced density matrices for symmetry-protected topological phases
• Transmission through a potential barrier in Luttinger liquids with a topological spin gap
• Symmetry-protected topological phases and competing orders in a spin-1/2 XXZ ladder with a four-spin interaction
• Rotationally-Invariant Circuits: Universality with the exchange interaction and two ancilla qubits
• Symmetry protected phases in inhomogeneous spin chains
• Classification of symmetry-protected topological many-body localized phases in one dimension
• Tensor networks demonstrate the robustness of localization and symmetry protected topological phases
• Quantum Current and Holographic Categorical Symmetry
• Stable Symmetry-Protected Topological Phases in Systems with Heralded Noise
• Topological Crystalline Bose Insulator in Two Dimensions via Entanglement Spectrum
• Highly entangled stationary states from strong symmetries
• Long-range nonstabilizerness and phases of matter
• Field theory of symmetry-protected valence bond solid states in (2+1) dimensions
• "Haldane" phases with ultracold fermionic atoms in double-well optical lattices
• Gaussian Continuous Tensor Network States for Simple Bosonic Field Theories
• Continuum limits of Matrix Product States
• Topological versus Landau-like phase transitions
• Integrable spin chains and the Clifford group
• Symmetry-protected Topological Phases in Spinful Bosons with a Flat Band
• Entanglement sum rules in exactly solvable models
• Exact dimer phase with anisotropic interaction for one dimensional magnets
• Proposal of a spin-one chain model with competing dimer and trimer interactions
• Fractionalizing Global Symmetry on Looplike Topological Excitations
• SU(3) quantum critical model emerging from a spin-1 topological phase
• String order via Floquet interactions in atomic systems
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases
• A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III
• Symmetry-protected topological phase transition in one-dimensional Kondo lattice and its realization with ultracold atoms
• Spin-1 Haldane phase in a chain of Rydberg atoms
• Disentangling transitions in topological order induced by boundary decoherence
• Connected components of irreducible maps and 1D quantum phases
• Geometric approach to Lieb-Schultz-Mattis theorem without translation symmetry under inversion or rotation symmetry
• Gauging anomalous unitary operators
• Nonequilibrium symmetry-protected topological order: emergence of semilocal Gibbs ensembles
• Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data
• Theory of Quantum Circuits with Abelian Symmetries
• Edge theories of 2D fermionic symmetry protected topological phases protected by unitary Abelian symmetries
• Gapless Symmetry-Protected Topological States in Measurement-Only Circuits
• Symmetry-protected adiabatic quantum transistors
• Characterization of Topological Phases of Spin-1/2 Frustrated Ferromagnetic-Antiferromagnetic Alternating Heisenberg Chains by Entanglement Spectrum
• Measurement-based quantum simulation of Abelian lattice gauge theories
• Z_N Berry phase and symmetry protected topological phases of SU(N) antiferromagnetic Heisenberg chain
• Symmetry protected topological phases in two-orbital SU(4) fermionic atoms
• Statistically induced topological phase transitions in a one-dimensional superlattice anyon-Hubbard model
• Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: a Majorana-fermion approach
• Majorana end states in an interacting quantum wire
• Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi-Point Strange Correlators
• Symmetry-Protected Topological Phase for Spin-Tensor-Momentum-Coupled Ultracold Atoms
• Numerical detection of symmetry enriched topological phases with space group symmetry
• Universal resources for quantum computing
• Symmetry-resolved entanglement of 2D symmetry-protected topological states
• Error-tolerant quantum convolutional neural networks for symmetry-protected topological phases
• Filling constraints on translation invariant dipole conserving systems
• Identifying quantum phases from injectivity of symmetric matrix product states
• Boundary conditions and anomalies of conformal field theories in 1+1 dimensions
• Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry protected topological phases
• Complexity-constrained quantum thermodynamics
• Islands of Chiral Solitons in Integer Spin Kitaev Chains
• Control of superexchange interactions with DC electric fields
• Quantum Error-Detection at Low Energies
• Local order parameters for symmetry fractionalization
• A quantum phase transition from $Z_2\times Z_2$ to $Z_2$ topological order
• Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking
• Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension
• Twice Hidden String Order and Competing Phases in the Spin-1/2 Kitaev-Gamma Ladder
• Charting the space of ground states with tensor networks
• Category of SET orders
• The Asymmetric Valence-Bond-Solid States in Quantum Spin Chains: The Difference Between Odd and Even Spins
• A many-body Fredholm index for ground state spaces and Abelian anyons
• The Quantum Torus Chain
• Lieb-Schultz-Mattis theorem in higher dimensions from approximate magnetic translation symmetry
• Space of non-Fermi liquids
• Non-local finite-depth circuits for constructing SPT states and quantum cellular automata
• Classification of trivial spin-1 tensor network states on a square lattice
• Geometric Approach to 3D Interfaces at Strong Coupling
• Theory of quantum kagome ice and vison zero modes
• 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
• Quantum Berezinskii-Kosterltz-Thouless Transition for Topological Insulator
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Pivoting through the chiral-clock family
• Spin-1 Haldane chains of superconductor-semiconductor hybrids
• Robust symmetry-protected metrology with the Haldane phase
• Quantum phase transition as an interplay of Kitaev and Ising interactions
• 4D beyond-cohomology topological phase protected by $C_2$ symmetry and its boundary theories
• Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements
• Anomaly inflow, dualities, and quantum simulation of abelian lattice gauge theories induced by measurements
• Converting topological insulators into topological metals within the tetradymite family
• Supersymmetric boundaries of one-dimensional phases of fermions beyond symmetry-protected topological states
• Gappability Index for Quantum Many-Body Systems
• Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries
• Entanglement in non-critical inhomogeneous quantum chains
• Topology protection-unprotection transition: an example from multi-terminal superconducting nanostructures
• Topological quantum criticality from multiplicative topological phases
• Performance of the rigorous renormalization group for first order phase transitions and topological phases
• Efficient ground state preparation in variational quantum eigensolver with symmetry-breaking layers
• Negativity in the Generalized Valence Bond Solid State
• Higher Berry curvature from matrix product states
• Anomalies and symmetry fractionalization in reflection-symmetric topological order
• Topological phase transition and the effect of Hubbard interaction on the one-dimensional topological Kondo insulator
• Combination of Tensor Network States and Green's function Monte Carlo
• Composite symmetry protected topological order and effective models
• A Hamiltonian Approach for Obtaining Irreducible Projective Representations and the $k\cdot p$ Perturbation for Anti-unitary Symmetry Groups
• Dual-isometric Projected Entangled Pair States
• Quantum computation by teleportation and symmetry
• Anomalous domain wall condensation in a modified Ising chain
• An Interplay of Topology and Quantized Geometric Phase for two Different Symmetry-Class Hamiltonians
• Universal Characterization of Quantum Many-Body States through Local Information
• LSM and CPT
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Fermionic Symmetry Protected Topological Phase Induced by Interactions
• Gauge theory and mixed state criticality
• Exact correlations in topological quantum chains
• Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains
• Matrix-product unitaries: Beyond quantum cellular automata
• Unveiling Correlated Two-dimensional Topological Insulators through Fermionic Tensor Network States -- Classification, Edge Theories and Variational Wavefunctions
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Bulk and boundary entanglement transitions in the projective gauge-Higgs model
• Detection of gapped phases of a 1D spin chain with onsite and spatial symmetries
• Interactions and the Theta Term in One-Dimensional Gapped Systems
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Playing nonlocal games with phases of quantum matter
• Characterization of photoexcited states in the half-filled one-dimensional extended Hubbard model assisted by machine learning
• Infinite-size density matrix renormalization group with parallel Hida's algorithm
• Symmetry protected topological phases in spin-1 ladders and their phase transitions
• The resource theory of tensor networks
• Self-$G$-ality in 1+1 dimensions
• Quantum antiferromagnetic Heisenberg half-odd integer spin model as the entanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
• Fusion of one-dimensional gapped phases and their domain walls
• Triggering Boundary Phase Transitions through Bulk Measurements in 2D Cluster States
• Variational Tensor Network Operator
• Field Tensor Network States
• Entanglement of a alternating bipartition in spin chains: Relation with classical integrable models
• Observation of average topological phase in disordered Rydberg atom array
• Entanglement renormalization of thermofield double states
• Fractionalized time reversal, parity and charge conjugation symmetry in topological superconductor: a possible origin of three generations of neutrinos and mass mixing
• Quantum Simulation of Spin-1 XXZ-Heisenberg Models and the Haldane Phase with Dysprosium
• Topological end states in a one-dimensional spatially modulated interaction spinless fermion model
• Realization of the topological Hopf term in two-dimensional lattice models
• Renormalization view on resonance proliferation between many-body localized phases
• Classifying parafermionic gapped phases using matrix product states
• Composite Quantum Phases in Non-Hermitian Systems
• Topological self-organization of strongly interacting particles
• Unwinding of a one-dimensional topological superconductor
• Gauging quantum states with non-anomalous matrix product operator symmetries
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Quantum teleportation implies symmetry-protected topological order
• Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction
• A Bosonic Analog of a Topological Dirac Semi-Metal: Effective Theory, Neighboring Phases, and Wire Construction
• Floquet Amorphous Topological Orders in a One-dimensional Rydberg Glass
• The Weyl-Mott point: topological and non-Fermi liquid behavior from an isolated Green's function zero
• Topology and Spectrum in Measurement-Induced Phase Transitions
• Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions
• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• Exactly solvable chain of interacting electrons with correlated hopping and pairing
• Solvable lattice model for (2+1)D bosonic topological insulator
• Gauging Defects in Quantum Spin Systems
• Fermionic symmetry-protected topological state in strained graphene
• Symmetry-protected trivial phases and quantum phase transitions in an anisotropic antiferromagnetic spin-1 biquadratic model
• Inferring Hidden Symmetries of Exotic Magnets from Detecting Explicit Order Parameters
• Phases of Matrix Product States with Symmetric Quantum Circuits and Symmetric Measurements with Feedforward
• Topological quantum phase transition in strongly correlated Kondo insulators in 1D
• Refined Phase Diagram for a Spin-1 System Exhibiting a Haldane Phase
• Topological bosonic states on ribbons of honeycomb lattice
• Bell-type correlation at quantum phase transitions in spin-1 chain
• Complete characterization of spin chains with two Ising symmetries
• Projectification of point group symmetries with a background flux and Lieb-Schultz-Mattis theorem
• Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Quantum Computing with sine-Gordon Qubits
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Soft symmetries of topological orders
• Many-body quantum catalysts for transforming between phases of matter
• Topological Phases of Many-Body Localized Systems: Beyond Eigenstate Order
• Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model
• Tensor Renormalization Group for interacting quantum fields
• Classification of the phases of 1D spin chains with commuting Hamiltonians
• Logarithmic light cone, slow entanglement growth, and quantum memory
• Entanglement and edge effects in superpositions of many-body Fock states with spatial constraints
• Enriched classification of parafermionic gapped phases with time reversal symmetry
• Hidden $SU(2)$ Symmetries, the Symmetry Hierarchy and the Emergent Eight-Fold Way in Spin-1 Quantum Magnets
• Eleven Competing Phases in the Heisenberg-Gamma (J$\Gamma$) Ladder
• Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• One-dimensional symmetric phases protected by frieze symmetries
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• A Simple and General Equation for Matrix Product Unitary Generation
• Disentangling anomaly-free symmetries of quantum spin chains
• A Classifying Space for Phases of Matrix Product States
• Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
• Superradiance, charge density waves and lattice gauge theory in a generalized Rabi-Hubbard chain
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Quantitative Determination of the Critical Points of Mott Metal-Insulator Transition in Strongly Correlated Systems
• Topological Dipole Insulator
• Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
• Local dynamics and detection of topology in spin-1 chains
• Rainbow chains and numerical renormalisation group for accurate chiral conformal spectra
• Ground-state phase diagram of S = 1/2 Heisenberg model on 2D square-hexagon-octagon lattice
• Topological term in the non-linear $\sigma$ model of the SO(5) spin chains
• Entanglement-enabled symmetry-breaking orders
• Parent Lindbladians for Matrix Product Density Operators
• Gauging Quantum Phases: A Matrix Product State Approach
• Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems
• 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
• Dissipative Dynamics in Open Fermionic Chains
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Variational methods for characterizing matrix product operator symmetries
• Reconfigurable dissipative entanglement between many spin ensembles: from robust quantum sensing to many-body state engineering
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Simulating Topological Order on Quantum Processors
• Symmetries and field tensor network states
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Topological phases of the compass ladder model
• Spin chains, defects, and quantum wires for the quantum-double edge
• Entanglement renormalization and symmetry fractionalization
• Approximate recovery with locality and symmetry constraints
• Bulk-edge correspondence in the Haldane phase of the bilinear-biquadratic spin-1 Hamiltonian
• Characterizing the parent Hamiltonian for a complete set of orthogonal wave functions: An inverse quantum problem
• Topological Phases of Spin-1/2 Ferromagnetic--Antiferromagnetic Alternating Heisenberg Chains with Alternating Next-Nearest-Neighbour Interaction
• Interrelations among frustration-free models via Witten's conjugation
• Condensation Completion and Defects in 2+1D Topological Orders
• SPT indices emerging from translation invariance in two dimensional quantum spin systems
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Angular-time evolution for the Affleck-Kennedy-Lieb-Tasaki chain and its edge-state dynamics
• Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories
• On a bulk gap strategy for quantum lattice models
• Impurity screening by defects in (1+1)$d$ quantum critical systems
• Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks
• Non-onsite symmetry breaking: topological phase coexistence and criticality
• Layer-by-layer disentangling two-dimensional topological quantum codes
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Robust Mixed-State Cluster States and Spurious Topological Entanglement Negativity
• Resource complexity of Symmetry Protected Topological phases
• Undecidability of the spectral gap in rotationally symmetric Hamiltonians
• One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems
• Parafermionic representation of Potts-based cluster chain
• Goldstone modes and the golden spiral in the ferromagnetic spin-1 biquadratic model
• Triangular lattice models of the Kalmeyer-Laughlin spin liquid from coupled wires
• Towards a Topological Quantum Chemistry description of correlated systems: the case of the Hubbard diamond chain
• Anomalies of kagome antiferromagnets on magnetization plateaus
• Geometric Approaches to Quantum Fields and Strings at Strong Couplings
• A Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• Phases and phase transitions of an $S=3/2$ chain on metallic and semi-metallic surfaces
• Matrix product state classification of 1D multipole symmetry protected topological phases
• Bosonic realization of SU(3) chiral Haldane phases
• Symmetry-enforced minimal entanglement and correlation in quantum spin chains
• Holographic duality between bulk topological order and boundary mixed-state order
• Measurement-Based Quantum Computation Using the Spin-1 XXZ Model with Uniaxial Anisotropy
• Tensorization of neural networks for improved privacy and interpretability
• Quantitative determination of the orbital-selective Mott transition and quantum entanglement in the orbital-selective Mott phase