Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order

Abstract

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization (TEFR) approach that removes local entanglement and coarse grain the lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors $T_{inv}$ plus the symmetry group $G_{sym}$ of the tensors (i.e. the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by 2D statistical Ising model, 2D statistical loop gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a $(G_{sym},T_{inv})$ characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry protected topological phase. The $(G_{sym},T_{inv})$ characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.

25 pages, 32 figures

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• Topological crystalline antiferromagnetic state in tetragonal FeS
• Edge modes and symmetry-protected topological states in open quantum systems
• Phase transitions of the five-state clock model on the square lattice
• Boundary deconfined quantum criticality at transitions between symmetry-protected topological chains
• Classification and construction of interacting fractonic higher-order topological phases
• Spin-1/2 XXZ Heisenberg chain in a longitudinal magnetic field
• Multipartite entanglement of the topologically ordered state in a perturbed toric code
• Variational corner transfer matrix renormalization group method for classical statistical models
• Robust topological feature against non-Hermiticity in Jaynes-Cummings Model
• Construction of Non-Hermitian Parent Hamiltonian from Matrix Product States
• Non-perturbative linked-cluster expansions for the trimerized ground state of the spin-one Kagome Heisenberg model
• Nodes and Spin Windings for Topological Transitions in Light-Matter Interactions: \\ Anisotropic Quantum Rabi Model as a Born Abstract Artist
• Global optimization of tensor renormalization group using the corner transfer matrix
• Topological phonon analysis of the 2D buckled honeycomb lattice: an application to real materials
• Edge stability and edge quantum criticality in 2D interacting topological insulators
• Translation invariant topological superconductors on lattice
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Symmetry-protected topological phases and competing orders in a spin-1/2 XXZ ladder with a four-spin interaction
• Long-range entanglement and topological excitations
• Low-energy physics of isotropic spin-1 chains in the critical and Haldane phases
• Symmetry-protected topological phase in a one-dimensional correlated bosonic model with a synthetic spin-orbit coupling
• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Finite Size Scaling of Topological Entanglement Entropy
• Geometry of reduced density matrices for symmetry-protected topological phases
• Tangential finite-size scaling of the Gaussian topological transition in the quantum spin-1 anisotropic chain
• Hidden Magnetism in Periodically Modulated One Dimensional Dipolar Fermions
• Field theory of symmetry-protected valence bond solid states in (2+1) dimensions
• Collective Monte Carlo updates through tensor network renormalization
• Symmetry protected phases in inhomogeneous spin chains
• Functional Tensor Network Solving Many-body Schrödinger Equation
• Gapless edge states and their stability in two-dimensional quantum magnets
• Demonstrating the wormhole mechanism of the entanglement spectrum via a perturbed boundary
• Hidden Order and Dynamics in Supersymmetric Valence Bond Solid States -- Super-Matrix Product State Formalism
• "Haldane" phases with ultracold fermionic atoms in double-well optical lattices
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
• Some aspects of Affleck-Kennedy-Lieb-Tasaki models: tensor network, physical properties, spectral gap, deformation, and quantum computation
• Homogeneous and domain wall topological Haldane conductors with dressed Rydberg atoms
• Signatures of broken parity and time-reversal symmetry in generalized string-net models
• Universal quantum Fisher information and simultaneous occurrence of Landau-class and topological-class transitions in non-Hermitian Jaynes-Cummings models
• Quantum phase transitions and string orders in the spin-1/2 Heisenberg-Ising alternating chain with Dzyaloshinskii-Moriya interaction
• Time-reversal symmetry breaking and gapped surface states due to spontaneous emergence of new order in $d$-wave nanoislands
• Symmetry-protected topological phase transition in one-dimensional Kondo lattice and its realization with ultracold atoms
• Computing classification of interacting fermionic symmetry-protected topological phases using topological invariants
• Link between the hierarchy of fractional quantum Hall states and Haldane's conjecture for quantum spin chains
• Renormalization group approach to symmetry protected topological phases
• Lattice realization of a bosonic integer quantum Hall state - trivial insulator transition and relation to the self-dual line in the easy-plane NCCP1 model
• Symmetry-protected Topological Phases in Spinful Bosons with a Flat Band
• Quantum entanglement of fermionic symmetry-enriched quantum critical points in one dimension
• $\pi$-fluxes, semi-metals and flat bands in artificial materials
• Measurement-based quantum simulation of Abelian lattice gauge theories
• Spin dynamics of the anisotropic spin-1 antiferromagnetic chain at finite magnetic fields
• Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems
• Spin Winding and Topological Nature of Transitions in Jaynes-Cummings Model with Stark Non-linear Coupling
• Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension
• Automorphic equivalence preserves the split property
• Gapless Symmetry-Protected Topological States in Measurement-Only Circuits
• Identification of topological phases using classically-optimized variational quantum eigensolver
• Cross derivative: a universal and efficient method for phase transitions in classical spin models
• Average-exact mixed anomalies and compatible phases
• Dynamical spin excitations of topological Haldane gapped phase in the $S=1$ Heisenberg antiferromagnetic chain with single-ion anisotropy
• Edge theories of 2D fermionic symmetry protected topological phases protected by unitary Abelian symmetries
• Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data
• Hidden Order in Quantum Many-body Dynamics of Driven-Dissipative Nonlinear Photonic Lattices
• Reduced basis surrogates for quantum spin systems based on tensor networks
• Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems
• Z_N Berry phase and symmetry protected topological phases of SU(N) antiferromagnetic Heisenberg chain
• Matrix product state fixed points of non-Hermitian transfer matrices
• High-pressure inelastic neutron scattering study of the anisotropic $S=1$ spin chain [Ni(HF${}_2$)(3-Clpyradine)${}_4$]BF${}_4$
• Symmetry-protection of multiphoton states of light
• Comparative study of the density matrix embedding theory for the Hubbard models
• Boundary conditions and anomalies of conformal field theories in 1+1 dimensions
• Entanglement Signature of Hinge Arcs, Fermi Arcs, and Crystalline Symmetry Protection in Higher-Order Weyl Semimetals
• Tensor network calculation of the logarithmic correction exponent in the XY model
• Infinite boundary conditions for response functions and limit cycles in iDMRG, demonstrated for bilinear-biquadratic spin-1 chains
• Hierarchy of Entanglement Renormalization and Long-Range Entangled States
• Universal resources for quantum computing
• Quantum Fisher information and polaron picture for identification of transition coupling in quantum Rabi model
• Unitary Designs of Symmetric Local Random Circuits
• Multi-impurity method for the bond-weighted tensor renormalization group
• Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models
• Higgs Condensates are Symmetry-Protected Topological Phases: II. $U(1)$ Gauge Theory and Superconductors
• Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry protected topological phases
• Identifying quantum phases from injectivity of symmetric matrix product states
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid
• Sign structure and ground state properties for a spin-$S$ $t$-$J$ chain
• Twice Hidden String Order and Competing Phases in the Spin-1/2 Kitaev-Gamma Ladder
• Improved coarse-graining methods on two dimensional tensor networks including fermions
• Tensor renormalization group study of (1+1)-dimensional U(1) gauge-Higgs model at $\theta =\pi$ with Lüscher's admissibility condition
• Efficient variational quantum circuit structure for correlated topological phases
• Tensor Renormalization Group Centered About a Core Tensor
• Interaction driven topological phase transitions of hardcore bosons on a two-leg ladder
• Characterization of randomness in quantum circuits of continuous gate sets
• Enforced symmetry breaking by invertible topological order
• Magic entanglement renormalization for quantum fields
• Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point
• The Asymmetric Valence-Bond-Solid States in Quantum Spin Chains: The Difference Between Odd and Even Spins
• Bulk and Surface Critical Behaviors of quantum Heisenberg antiferromagnet on a two-dimensional coupled diagonal ladders
• Superuniversality from disorder at two-dimensional topological phase transitions
• Phase transition of four-dimensional Ising model with tensor network scheme
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Critical behavior of the extended Hubbard model with bond dimerization
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model
• Anomaly inflow, dualities, and quantum simulation of abelian lattice gauge theories induced by measurements
• 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
• Tensor Network Renormalization Study on the Crossover in Classical Heisenberg and $\mathrm{R}{\mathrm{P}}^2$ Models in Two Dimensions
• Generalized Kramers-Wannier Duality from Bilinear Phase Map
• Magnetization plateau of the $S=2$ antiferromagnetic Heisenberg chain with anisotropies
• Properties of the Hamiltonian Renormalisation and its application to quantum mechanics on the circle
• Theory of quantum kagome ice and vison zero modes
• Reentrance of Topological Phase in Spin-1 Frustrated Heisenberg Chain
• Perfect discretizations as a gateway to one-loop partition functions for 4D gravity
• Symmetry-protected intermediate trivial phases in quantum spin chains
• Stability of invertible, frustration-free ground states against large perturbations
• Converting topological insulators into topological metals within the tetradymite family
• The Gauge Theory of Measurement-Based Quantum Computation
• 4D beyond-cohomology topological phase protected by $C_2$ symmetry and its boundary theories
• Non-Abelian Three-Loop Braiding Statistics for 3D Fermionic Topological Phases
• Higher-order tensor renormalization group approach to lattice glass model
• Quantum Berezinskii-Kosterltz-Thouless Transition for Topological Insulator
• Tensor network method for reversible classical computation
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Exotic surface behaviors induced by geometrical settings of the two-dimensional dimerized quantum XXZ model
• A Parallel Computing Method for the Higher Order Tensor Renormalization Group
• Loop-TNR analysis of CP(1) model with theta term
• Continuous phase transitions between fractional quantum Hall states and symmetry-protected topological states
• Reflection Structures and Spin Statistics in Low Dimensions
• Fermionic Symmetry Protected Topological Phase Induced by Interactions
• Evaluating the Jones polynomial with tensor networks
• Geometric Entanglement and Quantum Phase Transition in Generalized Cluster-XY models
• Measuring the Boundary Gapless State and Criticality via Disorder Operator
• Deriving the non-perturbative gravitational dual of quantum Liouville theory from BCFT operator algebra
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Anomalies and symmetry fractionalization in reflection-symmetric topological order
• An Interplay of Topology and Quantized Geometric Phase for two Different Symmetry-Class Hamiltonians
• Topological phases in two-legged Heisenberg ladders with alternating interactions
• Error-correction properties of an interacting topological insulator
• Combination of Tensor Network States and Green's function Monte Carlo
• Phase boundary location with information-theoretic entropy in tensor renormalization group flows
• Exotic Haldane Superfluid Phase of Soft-Core Bosons in Optical Lattices
• Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains
• Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Existence of two distinct valence bond solid states in the dimerized frustrated ferromagnetic $J_1$-$J_1^{\prime }$-$J_2$ chain
• Interaction-induced phase transitions at topological quantum criticality of an extended Su-Schrieffer-Heeger model
• Electrically tunable two-channel Kondo fixed points in helical liquids
• Detection of gapped phases of a 1D spin chain with onsite and spatial symmetries
• Spectral properties for a family of two-dimensional quantum antiferromagnets
• Bulk and boundary entanglement transitions in the projective gauge-Higgs model
• Symmetry protected topological phases in spin-1 ladders and their phase transitions
• Fractionalized time reversal, parity and charge conjugation symmetry in topological superconductor: a possible origin of three generations of neutrinos and mass mixing
• Weakly interacting one-dimensional topological insulators: a bosonization approach
• Electric and Magnetic Properties of Higher-Spin Kondo-Heisenberg Models at Strong Coupling
• A Monte Carlo algorithm for spin foam intertwiners
• Single-layer tensor network approach for three-dimensional quantum systems
• Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group
• Self-$G$-ality in 1+1 dimensions
• Gauge theory and mixed state criticality
• Topological pump of $SU(Q)$ quantum chain and Diophantine equation
• Topological edge states in spin 1 bilinear-biquadratic model
• Extraordinary surface critical behavior induced by symmetry-protected topological state of a two-dimensional quantum magnet
• Quantum antiferromagnetic Heisenberg half-odd integer spin model as the entanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
• Topological Field Theories of Three-Dimensional Rotation Symmetric Insulators: Coupling Curvature and Electromagnetism
• Quantum criticality in dimerised anisotropic spin-1 chains
• Three-dimensional real space renormalization group with well-controlled approximations
• Inexorable Edge Kondo Breakdown in Topological Kondo Insulators
• Realization of the topological Hopf term in two-dimensional lattice models
• Volume and Topological Invariants of Quantum Many-body Systems
• The tensor renormalization group study of the general spin-S Blume-Capel model
• Quantum Inception Score
• Boundary and defect criticality in topological insulators and superconductors
• Luther-Emery liquid and dominant singlet superconductivity in the hole-doped Haldane spin-1 chain
• Impurity-driven transitions in frustrated quantum Ising ring
• Tensor-Ring Decomposition with Index-Splitting
• Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction
• Bosonic Integer Quantum Hall States without Landau Levels on Square Lattice
• Transfer Matrix and Lattice Dilatation Operator for High-Quality Fixed Points in Tensor Network Renormalization Group
• Building Symmetry Enriched Topological Phases from a Bipartite Lattice Construction and Anyon Condensation
• Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. II. Topological classification
• Haldane and Dimer phases in a frustrated spin chain: an exact groundstate and associated topological phase transition
• Topology and Spectrum in Measurement-Induced Phase Transitions
• Composite Quantum Phases in Non-Hermitian Systems
• Inferring Hidden Symmetries of Exotic Magnets from Detecting Explicit Order Parameters
• Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states
• Quantum phase transition in skewed ladders: an entanglement entropy and fidelity study
• Tensor network simulations for nonorientable surfaces
• Machine learning, quantum chaos, and pseudorandom evolution
• 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
• Finite-size scaling on the torus with periodic projected entangled-pair states
• Mixed-State Topological Order under Coherent Noise
• Universal time evolution of string order parameter in quantum critical systems with boundary invertible or non-invertible symmetry breaking
• Lowest-energy states in parity-transformation eigenspaces of SO(N) spin chain
• Symmetry-protected topological order identified via Gutzwiller-guided density-matrix-renormalization-group: $\mathrm{SO}(n)$ spin chains
• Phase structure analysis of 2d lattice CP(1) model with $\theta$ term using tensor renormalization group method
• Tensor Renormalization Group Calculations of Partition-Function Ratios
• Generalized Landau Paradigm for quantum phases and phase transitions
• Tensor Renormalization Group for interacting quantum fields
• Loop update for infinite projected entangled-pair states in two spatial dimensions
• Topological Superconductivity in Doped Symmetry Protected Topological State
• Topological quantum phase transition in strongly correlated Kondo insulators in 1D
• Quantum simulation of superexchange magnetism in linear ion crystals
• Bell-type correlation at quantum phase transitions in spin-1 chain
• Entanglement Renormalization of the class of Continuous Matrix Product States
• Canonical quantization of nonlinear sigma models with theta term, with applications to symmetry-protected topological phases
• Essential difference between 2D and 3D from the perspective of real-space renormalization group
• Revealing the topological nature of the bond order wave in a strongly correlated quantum system
• Tensor network renormalization approach to antiferromagnetic 6-state clock model on the Union Jack lattice
• One-dimensional symmetric phases protected by frieze symmetries
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Parity-dependent phase diagrams in spin-cluster two-leg ladders
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Ground-state phase diagram of S = 1/2 Heisenberg model on 2D square-hexagon-octagon lattice
• The Tangent Space to the Manifold of Critical Classical Hamiltonians Representable by Tensor Networks
• Haldane phases and phase diagrams of the S = 3/2, 1 bilinear-biquadratic Heisenberg model on the orthogonal dimer chain
• Quantitative Determination of the Critical Points of Mott Metal-Insulator Transition in Strongly Correlated Systems
• Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions
• Counter-intuitive yet efficient regimes for measurement based quantum computation on symmetry protected spin chains
• Generalized Li-Haldane Correspondence in Critical Dirac-Fermion Systems
• The AKLT models on the singly decorated diamond lattice and two degree-4 planar lattices are gapped
• Eleven Competing Phases in the Heisenberg-Gamma (J$\Gamma$) Ladder
• A Practical Introduction to Tensor Network Renormalization with TNRKit.jl
• Spectroscopic Feature of Quantum Many-Body Scar States
• Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases
• Triad representation for the anisotropic tensor renormalization group in four dimensions
• Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions
• Simulating Topological Order on Quantum Processors
• Impurity screening by defects in (1+1)$d$ quantum critical systems
• Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems
• Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities
• Fundamental role of nonlocal orders in 1D Extended Bose-Hubbard Model
• Learning topological defects formation with neural networks in a quantum phase transition
• Entropic order parameters and topological holography
• Kondo spin liquid in Kondo necklace model: Classical disordered phase versus symmetry-protected topological state
• Dynamics for the Haldane phase in the Bilinear-Biquadratic Model
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method
• S=2 Quantum Spin Chain with the Biquadratic Exchange Interaction
• Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients
• Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories
• Dynamic critical exponents as an emergent property at interacting topological quantum critical points
• An experimental algorithm for identifying the topological nature of Kondo and mixed valence insulators
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Emergent gapless spiral phases and conformal Lifshitz criticality in the cluster Ising model with off-diagonal interactions
• Applying the Triad network representation to four-dimensional ATRG method
• A chain of strongly correlated SU(2)_4 anyons: Hamiltonian and Hilbert space of states
• SPT extension of $Z_2$ quantum Ising model's ferromagnetic phase
• Post-measurement Quantum Monte Carlo
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Variational boundary based tensor network renormalization group
• Entanglement renormalization of fractonic anisotropic ${\mathbb{Z}}_N$ Laplacian models
• Efficient 2D Tensor Network Simulation of Quantum Systems
• Unsupervised Learning of Symmetry Protected Topological Phase Transitions
• Luttinger liquid parameters from tensor network data
• Grassmann tensor renormalization group for the massive Schwinger model with a $\theta$ term using staggered fermions
• Truncating loopy tensor networks by zero-mode gauge fixing
• Detecting underlying symmetry-protected topological phases via strange correlators and edge engineering
• Bosonic realization of SU(3) chiral Haldane phases
• Generalization of the Affleck-Kennedy-Lieb-Tasaki Model for Quantum Ferromagnetism
• Matrix product state classification of 1D multipole symmetry protected topological phases
• Surface topological quantum criticality: Conformal manifolds and Discrete Strong Coupling Fixed Points
• Hopf symmetry protected topological phases in the vicinity of spin orders
• Geometric Effects on Tunneling in Driven Quantum Systems
• Strong zero modes in integrable spin-S chains
• Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model
• Weak coupling theory of nickel-based 327 superconductors
• Tensor network states for the description of quantum many-body systems
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• Unveiling topological order through multipartite entanglement
• Bosonic Haldane insulator in the presence of local disorder: A quantum Monte Carlo study
• Multipartite entanglement in quantum phase transitions
• Construction of Entangled Many-body States via the Higgs Mechanism
• Universal spectrum structure at nonequilibrium critical points in the (1+1)-dimensional directed percolation
• Parafermionic representation of Potts-based cluster chain
• Towards a Topological Quantum Chemistry description of correlated systems: the case of the Hubbard diamond chain
• The enabling electronic motif for topological insulation in ABO3 perovskites and its structural stability
• Visual configuration segmentation of quantum states for phase identification in many-body systems
• Anisotropy-induced spin parity effects
• Noise-tolerant Detection of $Z_N$ Topological Orders in Quantum Many-body States
• Deconfined Quantum Phase Transition of a Higher-Order Symmetry-Protected Topological State
• Detection of 2D SPT Order with Partial Symmetries
• Spin order, spin excitations, and RIXS spectra of spin-1/2 tetramer chains
• Higher-order topological phases protected by noninvertible and subsystem symmetries