Symmetry protected topological orders and the group cohomology of their symmetry group

Abstract

Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_Ψ, H^{1+d}[G_Ψ, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_Ψ the symmetry group of the ground states.

55 pages, 42 figures, RevTeX4-1, included some new references

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• Three Dimensional Z2 Topological Phases enriched by Time-Reversal Symmetry
• Classifying local fractal subsystem symmetry protected topological phases
• TQFT, Symmetry Breaking, and Finite Gauge Theory in 3+1D
• Symmetry-Protected Topological Interfaces and Entanglement Sequences
• Symmetry enhanced boundary qubits at infinite temperature
• Lattice model constructions for gapless domain walls between topological phases
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Quantum spin systems for measurement-based quantum computation
• Multiflavor Mott insulators in quantum materials and ultracold atoms
• Classification of spin liquids on the square lattice with strong spin-orbit coupling
• On gapped phases with a continuous symmetry and boundary operators
• Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions
• Classifying Quantum Phases With The Torus Trick
• Edge physics at the deconfined transition between a quantum spin Hall insulator and a superconductor
• Higher-form Gauge Symmetries in Multipole Topological Phases
• Topological edge modes without symmetry in quasiperiodically driven spin chains
• Mixed-state quantum anomaly and multipartite entanglement
• Correlation effects on topological crystalline insulators
• Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations
• Transition from the ${\mathbb{Z}}_2$ spin liquid to antiferromagnetic order: spectrum on the torus
• Detection of symmetry-protected topological order in AKLT states by exact evaluation of the strange correlator
• Fermionic spinon theory of square lattice spin liquids near the Néel state
• Universal measurement-based quantum computation in two-dimensional SPT phases
• Unquantized anomalies in topological semimetals
• Systematic Construction of tight-binding Hamiltonians for Topological Insulators and Superconductors
• Interaction Enabled Topological Crystalline Phases
• Globally symmetric topological phase: from anyonic symmetry to twist defect
• Critical entanglement spectrum of one-dimensional symmetry protected topological phases
• Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions
• Duality, Criticality, Anomaly, and Topology in Quantum Spin-1 Chains
• Detecting Majorana fermions in quasi-one-dimensional topological phases using nonlocal order parameters
• Anyon Superconductivity from Topological Criticality in a Hofstadter-Hubbard Model
• Entanglement, non-Hermiticity, and duality
• Kennedy-Tasaki transformation and non-invertible symmetry in lattice models beyond one dimension
• Strange correlations in spin-1 Heisenberg antiferromagnets
• Universal measurement-based quantum computation in a one-dimensional architecture enabled by dual-unitary circuits
• No-go theorem for topological insulators and sure-fire recipe for Chern insulators
• Anyon and Loop Braiding Statistics in Field Theories with a Topological $\Theta -$term
• Intrinsic sign problem in fermionic and bosonic chiral topological matter
• Ground state stability, symmetry, and degeneracy in Mott insulators with long range interactions
• Parafermionic wires at the interface of chiral topological states
• Identifying Symmetry-Protected Topological Order by Entanglement Entropy
• Exactly Solvable Points and Symmetry-Protected Topological Phases of Quantum Spins on a Zig-Zag Lattice
• Projected Entangled Pair States: Fundamental analytical and numerical limitations
• Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions
• Non-Abelian Chiral Spin Liquid on the Kagome Lattice
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Charged Topological Entanglement Entropy
• Pin TQFT and Grassmann integral
• Classification of symmetry fractionalization in gapped ${\mathbb{Z}}_2$ spin liquids
• Detecting and identifying 2D symmetry-protected topological, symmetry-breaking and intrinsic topological phases with modular matrices via tensor-network methods
• Topological spin ordering via Chern-Simons superconductivity
• Density waves in strongly correlated quantum chains
• Entanglement convertibility by sweeping through the quantum phases of the alternating bonds $XXZ$ chain
• Topological order, symmetry, and Hall response of two-dimensional spin-singlet superconductors
• Topological Invariants and Ground-State Wave Functions of Topological Insulators on a Torus
• Observing Floquet topological order by symmetry resolution
• Disorder effects in correlated topological insulators
• A classification of magnetic frustration and metamaterials from topology
• On adiabatic cycles of quantum spin systems
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Measurement-based quantum computation in finite one-dimensional systems: string order implies computational power
• Space group symmetry fractionalization in a family of exactly solvable models with Z2 topological order
• Universal quantum computing using $({\mathbb{Z}}_d{)}^3$ symmetry-protected topologically ordered states
• String flux mechanism for fractionalization in topologically ordered phases
• Unwinding Short-Range Entanglement
• Refined Holographic Entanglement Entropy for the AdS Solitons and AdS black Holes
• Topological modular forms and the absence of all heterotic global anomalies
• Tensor network state approach to quantum topological phase transitions and their criticalities of ${\mathbb{Z}}_2$ topologically ordered states
• Quantum computational advantage with string order parameters of 1D symmetry-protected topological order
• Hidden order and symmetry protected topological states in quantum link ladders
• Spin-orbit Coupling in Optical lattices
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Detecting subsystem symmetry protected topological order via entanglement entropy
• Topology of many-body edge and extended quantum states in an open spin chain: 1/3--plateau, Kosterlitz-Thouless transition, and Luttinger liquid
• Gapped symmetric edges of symmetry protected topological phases
• Fate of exceptional points under interactions: Reduction of topological classifications
• Quantum Hall states for Rydberg atoms with laser-assisted dipole-dipole interactions
• Entangled end states with fractionalized spin projection in a time-reversal-invariant topological superconducting wire
• Two dimensional Symmetry Protected Topological Phases with PSU(N) and time reversal symmetry
• Quantum phase transition and critical behavior between the gapless topological phases
• Spin quantum Hall effects in a spin-1 topological paramagnet
• Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup
• Braiding statistics and classification of two-dimensional charge-$2m$ superconductors
• `Gauging' time reversal symmetry in tensor network states
• Hidden order and flux attachment in symmetry protected topological phases: a Laughlin-like approach
• Ultra Unification
• Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications
• Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements
• Anomaly and Cobordism Constraints Beyond the Standard Model: Topological Force
• Bosonic Crystalline Symmetry Protected Topological Phases Beyond the Group Cohomology Proposal
• Free-fermion descriptions of parafermion chains and string-net models
• Fractionalization on the Surface: Is Type-II Terminated $1T$-TaS${}_2$ Surface an Anomalously Realized Spin Liquid?
• Topological Bootstrap: Fractionalization From Kondo Coupling
• Symmetry Protected Quantum State Renormalization
• Gifts from long-range interaction: Emergent gapless topological behaviors in quantum spin chain
• Fractional corner charges in a 2D super-lattice Bose-Hubbard model
• Product Vacua and Boundary State Models in d Dimensions
• Bridging coupled wires and lattice Hamiltonian for two-component bosonic quantum Hall states
• Perfect Metal Phases of One-Dimensional and Anisotropic Higher-Dimensional Systems
• Spontaneous symmetry breaking from anyon condensation
• Restricted Boltzmann Machines and Matrix Product States of 1D Translational Invariant Stabilizer Codes
• Unwinding Fermionic SPT Phases: Supersymmetry Extension
• Global phase diagram of two-component Bose gases in antiparallel magnetic fields
• Walker-Wang models and axion electrodynamics
• Bilayer Graphene as a platform for Bosonic Symmetry Protected Topological States
• Homotopical Foundations of Parametrized Quantum Spin Systems
• Detecting Topological Order with Ribbon Operators
• Folding approach to topological orders enriched by mirror symmetry
• Organizing symmetry-protected topological phases by layering and symmetry reduction: a minimalist perspective
• A UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• Quantum phase transition between topologically distinct quantum critical points
• Interacting Symmetry-Protected Topological Phases Out of Equilibrium
• Bifurcating subsystem symmetric entanglement renormalization in two dimensions
• Global anomaly detection in two-dimensional symmetry-protected topological phases
• Scaling of geometric phase versus band structure in cluster-Ising models
• Adiabatic Quantum Transistors
• Theory of oblique topological insulators
• Continuous phase transition between bosonic integer quantum Hall liquid and trivial insulator: evidences for deconfined quantum criticality
• Lecture Notes on Topological Crystalline Insulators
• Bosonic symmetry protected topological phases with reflection symmetry
• Heisenberg antiferromagnet on the Husimi lattice
• Symmetry-Enriched Criticality in a Coupled Spin-Ladder
• Disentangling interacting symmetry protected phases of fermions in two dimensions
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• String order parameters for 1d Floquet Symmetry Protected Topological Phases
• Topological edge modes and phase transition in the critical fermionic chain with long-range interaction
• Anomalous symmetry protected topological states in interacting fermion systems
• Topological Edge and Interface states at Bulk disorder-to-order Quantum Critical Points
• Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
• Exactly Solvable Model for Two Dimensional Topological Superconductor
• Complete crystalline topological invariants from partial rotations in (2+1)D invertible fermionic states and Hofstadter's butterfly
• Model Realization and Numerical Studies of a Three-Dimensional Bosonic Topological Insulator and Symmetry-Enriched Topological Phases
• Topological phases in small quantum Hall samples
• Fractonic Higher-Order Topological Phases in Open Quantum Systems
• Universal finite-size amplitude and anomalous entanglement entropy of $z=2$ quantum Lifshitz criticalities in topological chains
• Topological Nature of High Temperature Superconductivity
• Intrinsically Interacting Higher-Order Topological Superconductors
• Correlated Fragile Topology: a Parton Approach
• Compatible braidings with Hopf links, multi-loop, and Borromean rings in $(3+1)$-dimensional spacetime
• Quasi-exact quantum computation
• (SPT-)LSM theorems from projective non-invertible symmetries
• Multipole and fracton topological order via gauging foliated SPT phases
• Spectral Gap and Edge Excitations of $d$-dimensional PVBS models on half-spaces
• SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement
• Modularity of supersymmetric partition functions
• Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip
• Gauging Lie group symmetry in (2+1)d topological phases
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point
• Anomaly indicators for topological orders with $U(1)$ and time-reversal symmetry
• Multiversality and Unnecessary Criticality in One Dimension
• Visualizing a Bosonic Symmetry Protected Topological Phase in an Interacting Fermion Model
• Relativistic continuous matrix product states for quantum fields without cutoff
• An index for two-dimensional SPT states
• Matrix product states for topological phases with parafermions
• Criticality at the Haldane-insulator charge-density-wave quantum phase transition
• Quantum phase transition between symmetry enriched topological phases in tensor-network states
• Effective response theory for Floquet topological systems
• Gapped domain walls between 2+1D topologically ordered states
• Detecting Subsystem Symmetry Protected Topological Order Through Strange Correlators
• Surface field theories of point group symmetry protected topological phases
• Symmetry reduction induced by anyon condensation: a tensor network approach
• Symmetry protected topological edge modes and emergent partial time reversal symmetry breaking in open quantum many-body systems
• An infinite family of 3d Floquet topological paramagnets
• Topological lattice field theories from intertwiner dynamics
• Quantum State Tomography with Locally Purified Density Operators and Local Measurements
• Gauging (3+1)-dimensional topological phases: an approach from surface theories
• Entanglement Properties of Gauge Theories from Higher-Form Symmetries
• Chiral and Geometric Anomalies in Finite Systems
• Unoriented 3d TFTs
• Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups
• Changing anyonic ground degeneracy with engineered gauge fields
• From orbifolding conformal field theories to gauging topological phases
• $Z_2$ fractionalized Chern/topological insulators in an exactly soluble correlated model
• Antiferromagnetic magnons from fractionalized excitations
• Line defects in Three dimensional Symmetry Protected Topological Phases
• Bosonic Short Range Entangled states Beyond Group Cohomology classification
• Strong-to-weak spontaneous symmetry breaking meets average symmetry-protected topological order
• Topological defects in Floquet systems: Anomalous chiral modes and topological invariant
• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• Localized Plasmons in Topological Insulators
• Classification and construction of interacting fractonic higher-order topological phases
• Lattice realization of compact $U(1)$ Chern-Simons theory with exact 1-symmetries
• Symmetry-protected topological phases in two-leg SU(N) spin ladder with unequal spins
• Disentangling the Generalized Double Semion Model
• Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories
• Higher order topological matter and fractional chiral states
• Renyi entropies for classical stringnet models
• Boundary deconfined quantum criticality at transitions between symmetry-protected topological chains
• Gaining insights on anyon condensation and 1-form symmetry breaking across a topological phase transition in a deformed toric code model
• Worldsheet matter for electric flux strings
• Floquet topological transitions in extended Kane-Mele models with disorder
• Symmetries and anomalies of Kitaev spin-$S$ models: Identifying symmetry-enforced exotic quantum matter
• Linking Entanglement and Discrete Anomaly
• Altermagnetic Instabilities from Quantum Geometry
• Exactly solvable models for U(1) symmetry-enriched topological phases
• Irreducible Projective Representations and Their Physical Applications
• Effective field theories of topological crystalline insulators and topological crystals
• Emergence of the XY-like phase in the deformed spin-3/2 AKLT systems
• Bosonic integer quantum Hall effect as topological pumping
• Interaction-induced topological phase transition at finite temperature
• Topological gauge theory for mixed Dirac stationary states in all dimensions
• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Symmetry protected gapless $Z_2$ spin liquids
• No-Go Theorem for Boson Condensation in Topologically Ordered Quantum Liquids
• Supersymmetric Many-Body Systems from Partial Symmetries: Integrability, Localization and Scrambling
• Symmetry-protected topological phase in a one-dimensional correlated bosonic model with a synthetic spin-orbit coupling
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Inaccessible entanglement in symmetry protected topological phases
• Classification of symmetry-protected topological phases in two-dimensional many body-localized systems
• Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
• Symmetry analysis of bond-alternating Kitaev spin chains and ladders
• Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes
• Symmetry-protected topological phases and transition in a frustrated spin-$\frac{1}{2}$ XXZ chain
• Construction of Non-Hermitian Parent Hamiltonian from Matrix Product States
• Geometry of reduced density matrices for symmetry-protected topological phases
• Searching for two-dimensional Weyl superconductors in heterostructures
• Topological Dipole Conserving Insulators and Multipolar Responses
• 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
• Generic "Unnecessary" Quantum Critical Points with Minimal Degrees of Freedom
• Extended quantum field theory, index theory and the parity anomaly
• Three-dimensional Topological Insulators and Bosonization
• Topological Crystalline Bose Insulator in Two Dimensions via Entanglement Spectrum
• Symmetry protected phases in inhomogeneous spin chains
• Strange correlators for topological quantum systems from bulk-boundary correspondence
• Topological edge states in single layers of honeycomb materials with strong spin-orbit coupling
• Stable Symmetry-Protected Topological Phases in Systems with Heralded Noise
• Criteria for protected edge modes with ${\mathbb{Z}}_2$ symmetry
• Classification and surface anomaly of glide symmetry protected topological phases in three dimensions
• Field theory of symmetry-protected valence bond solid states in (2+1) dimensions
• Quantum entanglement of fermionic symmetry-enriched quantum critical points in one dimension
• Topological versus Landau-like phase transitions
• Phase diagrams of spin-$S$ Kitaev ladders
• Minimal Fractional Topological Insulator in half-filled conjugate moiré Chern bands
• Interacting Topological Insulators with Synthetic Dimensions
• Quantum spin liquids of Rydberg excitations in a honeycomb lattice induced by density-dependent Peierls phases
• Symmetry-protected Topological Phases in Spinful Bosons with a Flat Band
• SU(3) quantum critical model emerging from a spin-1 topological phase
• Some aspects of Affleck-Kennedy-Lieb-Tasaki models: tensor network, physical properties, spectral gap, deformation, and quantum computation
• Gauging anomalous unitary operators
• Interacting Floquet topological phases in three dimensions
• Entanglement in quantum field theory via wavelet representations
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Fractionalizing Global Symmetry on Looplike Topological Excitations
• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• Non-Pauli topological stabilizer codes from twisted quantum doubles
• Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases
• Signatures of broken parity and time-reversal symmetry in generalized string-net models
• Entanglement properties of spin models in triangular lattices
• $\pi$-fluxes, semi-metals and flat bands in artificial materials
• Fracton-like phases from subsystem symmetries
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
• Vanishing Hall conductance for commuting Hamiltonians
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Symmetry-protected topological phase transition in one-dimensional Kondo lattice and its realization with ultracold atoms
• 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
• Computing classification of interacting fermionic symmetry-protected topological phases using topological invariants
• Construction and Field Theory of Bosonic Symmetry Protected Topological states beyond Group Cohomology
• Gaussian Continuous Tensor Network States for Simple Bosonic Field Theories
• Exact dimer phase with anisotropic interaction for one dimensional magnets
• Unconventional quantum phase transitions in a one-dimensional Lieb-Schultz-Mattis system
• Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems
• Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension
• Correlation-Assisted Quantized Charge Pumping
• Anomaly inflow for dipole symmetry and higher form foliated field theories
• Average-exact mixed anomalies and compatible phases
• Edge theories of 2D fermionic symmetry protected topological phases protected by unitary Abelian symmetries
• Symmetry protected topological phases in two-orbital SU(4) fermionic atoms
• Dynamical spin excitations of topological Haldane gapped phase in the $S=1$ Heisenberg antiferromagnetic chain with single-ion anisotropy
• Interacting Valley Chern Insulator and its Topological Imprint on Moiré Superconductors
• Hierarchies of resources for measurement-based quantum computation
• Symmetry-protected adiabatic quantum transistors
• Quantum Hall phase diagram of two-component Bose gases: Intercomponent entanglement and pseudopotentials
• 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
• Measurement-based quantum simulation of Abelian lattice gauge theories
• Disentangling (2+1)d Topological States of Matter with the Entanglement Negativity
• Cusps near Topological Phase Transitions: Signatures of Majorana fermions and interactions with fluctuations
• Interacting Edge States of Fermionic Symmetry-Protected Topological Phases in Two Dimensions
• Categories of quantum liquids II
• Finite Temperature Entanglement Negativity of Fermionic Symmetry Protected Topological Phases and Quantum Critical Points in One Dimension
• Boundary conditions and anomalies of conformal field theories in 1+1 dimensions
• Topological Quantum Critical Points in Strong Coupling limits: Global Symmetries and Strongly Interacting Majorana Fermions
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Gauging U(1) symmetry in (2+1)d topological phases
• Exact projected entangled pair ground states with topological Euler invariant
• Symmetry-resolved entanglement of 2D symmetry-protected topological states
• Four revolutions in physics and the second quantum revolution -- a unification of force and matter by quantum information
• Construction of Variational Matrix Product States for the Heisenberg Spin-1 Chain
• Islands of Chiral Solitons in Integer Spin Kitaev Chains
• Unsupervised Interpretable Learning of Phases From Many-Qubit Systems
• A unified diagrammatic approach to topological fixed point models
• Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry protected topological phases
• Superuniversality from disorder at two-dimensional topological phase transitions
• Exotic Invertible Phases with Higher-Group Symmetries
• Topological Quantum Criticality in Superfluids and Superconductors: Surface criticality, Thermal properties, and Lifshitz Majorana fields
• Disconnected entanglement entropy as a marker of edge modes in a periodically driven Kitaev chain
• Boundary Criticality of Topological Quantum Phase Transitions in $2d$ systems
• Interplay between short and long-range entanglement in symmetry protected phases
• Harnessing symmetry-protected topological order for quantum memories
• Two-dimensional Anomaly, Orbifolding, and Boundary States
• Signatures of quantum criticality in the complex inverse temperature plane
• Local order parameters for symmetry fractionalization
• Quantum Error-Detection at Low Energies
• Twice Hidden String Order and Competing Phases in the Spin-1/2 Kitaev-Gamma Ladder
• Category of SET orders
• Structure of the Entanglement Entropy of (3+1)D Gapped Phases of Matter
• Quantum simulation from the bottom up: the case of rebits
• Explicit cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf-Witten invariants
• Exact Fisher zeros and thermofield dynamics across a quantum critical point
• Universal finite size scaling around tricriticality between topologically ordered, SPT, and trivial phases
• Sign structure and ground state properties for a spin-$S$ $t$-$J$ chain
• Chern number matrix of the non-Abelian spin-singlet fractional quantum Hall effect
• Enforced symmetry breaking by invertible topological order
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• On Exotic Consistent Anomalies in (1+1)$d$: A Ghost Story
• The tenfold way redux: Fermionic systems with $N$-body interactions
• Generalized Kramers-Wannier Duality from Bilinear Phase Map
• Anomaly inflow, dualities, and quantum simulation of abelian lattice gauge theories induced by measurements
• Effective response theory for zero energy Majorana bound states in three spatial dimensions
• Symmetry-protected intermediate trivial phases in quantum spin chains
• Orbital Frustration and Emergent Flat Bands
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• A $H^3(G,\mathbb{T})$-valued index of symmetry protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems
• Quantum Phases and Collective Excitations in Bose-Hubbard Models with Staggered Magnetic Flux
• Non-local finite-depth circuits for constructing SPT states and quantum cellular automata
• Higher prequantum geometry
• Pivoting through the chiral-clock family
• Spectral Signatures of Non-Trivial Topology in a Superconducting Circuit
• Bulk-to-boundary anyon fusion from microscopic models
• Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model
• Geometric Approach to 3D Interfaces at Strong Coupling
• Mordell integrals and Giveon-Kutasov duality
• Defect bulk-boundary correspondence of topological skyrmion phases of matter
• Extended Functorial Field Theories and Anomalies in Quantum Field Theories
• The Gauge Theory of Measurement-Based Quantum Computation
• Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements
• 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
• $3d$ One-form Mixed Anomaly and Entanglement Entropy
• Distinct Floquet topological classifications from color-decorated frequency lattices with space-time symmetries
• Superconductivity in a Topological Lattice Model with Strong Repulsion
• A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases
• Fermionic Symmetry Protected Topological Phase Induced by Interactions
• Topological quantum criticality from multiplicative topological phases
• A Hamiltonian Approach for Obtaining Irreducible Projective Representations and the $k\cdot p$ Perturbation for Anti-unitary Symmetry Groups
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview
• Anomalies and symmetry fractionalization in reflection-symmetric topological order
• An invariant of symmetry protected topological phases with on-site finite group symmetry for two-dimensional Fermion systems
• Towards topological fixed-point models beyond gappable boundaries
• Fingerprints of bosonic symmetry protected topological state in a quantum point contact
• Solvable Lattice Hamiltonians with Fractional Hall Conductivity
• Non-Local Order Parameters as a Probe for Phase Transitions in the Extended Fermi-Hubbard Model
• The Partially-Split Hall Bar: Tunneling in the Bosonic Integer Quantum Hall Effect
• Classical topological paramagnetism
• Continuous phase transitions between fractional quantum Hall states and symmetry-protected topological states
• Non-Abelian Fusion Rules from an Abelian System
• Continuous transition between Ising magnetic order and a chiral spin liquid
• Anomalous domain wall condensation in a modified Ising chain
• Topological Orders with Global Gauge Anomalies
• LSM and CPT
• Pressure-Driven Topological Phase Transition in the Yb Chalcogenides YbO and YbS
• Gauge interactions and topological phases of matter
• Topological phase transition and the effect of Hubbard interaction on the one-dimensional topological Kondo insulator
• Edge mode locality in perturbed symmetry protected topological order
• Role of anomalous symmetry in 0-$\pi$ qubits
• Fusion of one-dimensional gapped phases and their domain walls
• Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases
• Weakly interacting one-dimensional topological insulators: a bosonization approach
• Bulk and boundary entanglement transitions in the projective gauge-Higgs model
• Quantum Simulation of Spin-1 XXZ-Heisenberg Models and the Haldane Phase with Dysprosium
• Floquet Flux Attachment in Cold Atomic Systems
• Gauge theory and mixed state criticality
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• Quantum antiferromagnetic Heisenberg half-odd integer spin model as the entanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• The resource theory of tensor networks
• Time reversal invariant gapped boundaries of the double semion state
• Observation of average topological phase in disordered Rydberg atom array
• Universality and quantized response in bosonic nonfractionalized tunneling
• Topological states in strongly correlated systems
• Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase
• Symmetry-protected topological phase transitions and robust chiral order on a tunable zigzag lattice
• Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction
• Fractionalized time reversal, parity and charge conjugation symmetry in topological superconductor: a possible origin of three generations of neutrinos and mass mixing
• Topological Field Theories of Three-Dimensional Rotation Symmetric Insulators: Coupling Curvature and Electromagnetism
• Mirror anomaly in fermionic topological orders
• Kitaev Building-block Construction for Inversion-Protected Higher-order Topological Superconductors
• Interacting topological phases in thin films of topological mirror Kondo insulators
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases II: The (2+1)-Dimensional Case
• Characterization and Classification of Fermionic Symmetry Enriched Topological Phases
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Electric and Magnetic Properties of Higher-Spin Kondo-Heisenberg Models at Strong Coupling
• Bound states of three fermions forming symmetry-protected topological phases
• Topologically ordered time crystals
• Lattice construction of exotic invertible topological phases
• Self-$G$-ality in 1+1 dimensions
• A bosonic topological phase in a paired superfluid
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Dynamical generalization of Yetter's model based on a crossed module of discrete groups
• A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions
• Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction
• Topology and Spectrum in Measurement-Induced Phase Transitions
• Composite Quantum Phases in Non-Hermitian Systems
• Quantum teleportation implies symmetry-protected topological order
• Universal contributions to charge fluctuations in spin chains at finite temperature
• The low-energy TQFT of the generalized double semion model
• Building Symmetry Enriched Topological Phases from a Bipartite Lattice Construction and Anyon Condensation
• The Ultraviolet Structure of Quantum Field Theories. Part 1: Quantum Mechanics
• Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. II. Topological classification
• Hydrodynamics of nonlinear gauge-coupled quantum fluids
• Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking
• Gauging quantum states with non-anomalous matrix product operator symmetries
• The Ultraviolet Structure of Quantum Field Theories. Part 2: What is Quantum Field Theory?
• G-Extensions of Quantum Group Categories and Functorial SPT
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Classifying parafermionic gapped phases using matrix product states
• Gauging Defects in Quantum Spin Systems
• Volume and Topological Invariants of Quantum Many-body Systems
• Inexorable Edge Kondo Breakdown in Topological Kondo Insulators
• Bosonic Integer Quantum Hall States without Landau Levels on Square Lattice
• Solvable lattice model for (2+1)D bosonic topological insulator
• Realization of the topological Hopf term in two-dimensional lattice models
• Instanton Density Operator in Lattice QCD from Higher Category Theory
• On 'categories' of quantum field theories
• Complete characterization of spin chains with two Ising symmetries
• Quantum measurement in two-dimensional conformal field theories: Application to quantum energy teleportation
• Topological Phases of Many-Body Localized Systems: Beyond Eigenstate Order
• Qualitative features of the rearrangement of molecular energy spectra from a "wall-crossing" perspective
• Quantum Circuits Reproduce Experimental Two-dimensional Many-body Localization Transition Point
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• Engineering Hierarchical Symmetries
• Ferromagnetic Haldane state and dimer multiplet state of quantum ferromagnets
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Signatures of topological quantum phase transitions in driven and dissipative qubit-arrays
• Modular extension of topological orders from congruence representations
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Universal time evolution of string order parameter in quantum critical systems with boundary invertible or non-invertible symmetry breaking
• Entanglement renormalization for chiral topological phases
• The Symmetry Enriched Center Functor is Fully Faithful
• Phases of Matrix Product States with Symmetric Quantum Circuits and Symmetric Measurements with Feedforward
• Many-body quantum catalysts for transforming between phases of matter
• Dihedral twist liquid models from emergent Majorana fermions
• Symmetry-protected topological order identified via Gutzwiller-guided density-matrix-renormalization-group: $\mathrm{SO}(n)$ spin chains
• Representation Theory for Massless Quasiparticles in Bogoliubov-de Gennes Systems
• Topological Superconductivity in Doped Symmetry Protected Topological State
• Redundant string symmetry-based error correction: Demonstrations on quantum devices
• Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate beyond the $n^{1/3}$ Distance Barrier
• One-dimensional symmetric phases protected by frieze symmetries
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• The fate of supersymmetry in quantum field theories
• Eleven Competing Phases in the Heisenberg-Gamma (J$\Gamma$) Ladder
• Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
• Parity-dependent phase diagrams in spin-cluster two-leg ladders
• Systematic construction of stabilizer codes via gauging abelian boundary symmetries
• Canonical quantization of nonlinear sigma models with theta term, with applications to symmetry-protected topological phases
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Multi-flavor string-net models
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Symmetry-deformed toric codes and the quantum dimer model
• Magic Boundaries of 3D Color Codes
• Enriched classification of parafermionic gapped phases with time reversal symmetry
• Adiabatic Continuity of the Spinful Quantum Hall States
• Geometric and computational aspects of chiral topological quantum matter
• Nonlocal order parameter of pair superfluids
• Assembling Kitaev honeycomb spin liquids from arrays of 1D symmetry protected topological phases
• Quantum Restored Symmetry Protected Topological Phases
• Topological and Error-Correcting Properties for Symmetry-Protected Topological Order
• Generalized Statistics on Lattices
• Generalized Li-Haldane Correspondence in Critical Dirac-Fermion Systems
• Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization
• Ground-state phase diagram of S = 1/2 Heisenberg model on 2D square-hexagon-octagon lattice
• Haldane phases and phase diagrams of the S = 3/2, 1 bilinear-biquadratic Heisenberg model on the orthogonal dimer chain
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• Disentangling anomaly-free symmetries of quantum spin chains
• $Z_2\times Z_2$ symmetry and $Z_4$ Berry phase of bosonic ladder
• Approximate recovery with locality and symmetry constraints
• C^1-Classification of gapped parent Hamiltonians of quantum spin chains with local symmetry
• Generating domain walls between topologically ordered phases using quantum double models
• Impurity screening by defects in (1+1)$d$ quantum critical systems
• Dynamic critical exponents as an emergent property at interacting topological quantum critical points
• SPT extension of $Z_2$ quantum Ising model's ferromagnetic phase
• Topology of Fermi Surfaces and anomaly inflows
• Emergent gapless spiral phases and conformal Lifshitz criticality in the cluster Ising model with off-diagonal interactions
• Anomalies of global symmetries on the lattice
• Bosonic SET to SPT Transition under Dyonic LSM Theorem
• Entropic order parameters and topological holography
• Artificial Gauge Field Engineered Excited-State Topology: Control of Dynamical Evolution of Localized Spinons
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking
• Abelian Topological Order on Lattice Enriched with Electromagnetic Background
• Topological gravity in 3+1D and a possible origin of dark matte
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• The emergence of 3+1D Einstein gravity from topological gravity
• Dynamical mass generation in QED${}_3$: A non-perturbative approach
• Symmetries and field tensor network states
• Entanglement renormalization and symmetry fractionalization
• Threefold Way for Typical Entanglement
• Perturbative and global anomalies in bosonic analogues of integer quantum Hall and topological insulator phases
• Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems
• Condensation Completion and Defects in 2+1D Topological Orders
• Kondo spin liquid in Kondo necklace model: Classical disordered phase versus symmetry-protected topological state
• Refined Characterization of Lattice Chern Insulators by Bulk Entanglement Spectrum
• Phase transitions in (2 + 1)D subsystem-symmetric monitored quantum circuits
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks
• SPT indices emerging from translation invariance in two dimensional quantum spin systems
• Bosonic Halperin fractional quantum Hall effect at filling factor $\nu =2/5$
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Fractionally Quantized Electric Polarization and Discrete Shift of Crystalline Fractional Chern Insulators
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Intertwined order of generalized global symmetries
• Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories
• Instanton effects in lattice models of bosonic symmetry-protected topological states
• Theory of Hofstadter Superconductors
• Higher-order topological phases protected by noninvertible and subsystem symmetries
• Geometric Approaches to Quantum Fields and Strings at Strong Couplings
• Perspectives on Anomaly Resolution
• Detection of 2D SPT Order with Partial Symmetries
• Subsystem symmetries, critical Bose surface, and immobile excitations in an extended compass model
• Hofstadter Butterflies in Topological Insulators
• Ultracold atomic lattice systems for simulating topological phases: A review
• Non-invertible translation from Lieb-Schultz-Mattis anomaly
• Topological phases of bosons with local parity coupling on a dimerized lattice
• A Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph
• Construction of Entangled Many-body States via the Higgs Mechanism
• Spurious Strange Correlators in Symmetry-Protected Topological Phases
• Topological Manipulations On $\mathbb{R}$ Symmetries Of Abelian Gauge Theory
• Measurement-Based Quantum Computation Using the Spin-1 XXZ Model with Uniaxial Anisotropy
• Tensorization of neural networks for improved privacy and interpretability
• Matrix product state classification of 1D multipole symmetry protected topological phases
• Symmetric entanglers for non-invertible SPT phases
• Tensor network states for the description of quantum many-body systems
• Twisted quantum doubles are sign problem-free
• Detecting underlying symmetry-protected topological phases via strange correlators and edge engineering
• Universal entanglement correction induced by relevant deformations at the quantum critical point
• Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge
• Phases and Quantum Phase Transitions in Anisotropic Antiferromagnetic Kitaev-Heisenberg-$\Gamma$ magnet
• Combined spontaneous symmetry-breaking and symmetry-protected topological order from cluster charge interaction
• Generalized symmetry enriched criticality in (3+1)d
• Exploiting $\vartheta -$functions for the identification of topological materials
• Symmetry-Protected Topological relationship between $SU(3)$ and $SU(2)\times U(1)$ in Two Dimension
• Parafermionic representation of Potts-based cluster chain
• Crystallographic Interacting Topological Phases and Equivariant Cohomology: To assume or not to assume
• Distinct topological excitonic insulators characterized by quantum geometry
• Anomalies of kagome antiferromagnets on magnetization plateaus
• Topological Properties and Characterizations
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• Index theory and noncommutative geometry of topological insulator