2D symmetry protected topological orders and their protected gapless edge excitations

Abstract

Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry protected topological orders exist. In this paper, we present a model in a 2D interacting spin system with nontrivial on-site $Z_2$ symmetry protected topological order. The order is nontrivial because we can prove that the 1D system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the $Z_2$ group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if $Z_2$ symmetry is not broken. We prove the latter by developing the tool of matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and revealing the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with on-site $Z_2$ symmetry protected topological order.

15 pages, 9 figures

References in corpus (8)

• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• String order and symmetries in quantum spin lattices
• Fractional topological insulators in three dimensions
• Zero modes of two-dimensional chiral p-wave superconductors

Cited by in corpus (292)

• Symmetry protected topological orders and the group cohomology of their symmetry group
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Zoo of quantum-topological phases of matter
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Symmetry Protected Topological phases of Quantum Matter
• Braiding statistics approach to symmetry-protected topological phases
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Pre-thermal phases of matter protected by time-translation symmetry
• Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions
• Classification of topological phases in periodically driven interacting systems
• Symmetry protected topological phases from decorated domain walls
• Integer quantum Hall effect for bosons: A physical realization
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• Many-body Localization and Symmetry Protected Topological Order
• Correlation effects in two-dimensional topological insulators
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Subsystem symmetry protected topological order
• Topological phases protected by point group symmetry
• Choreographed entangle dances: topological states of quantum matter
• Gauging spatial symmetries and the classification of topological crystalline phases
• Symmetry-protected phases for measurement-based quantum computation
• A Time-Reversal Invariant Topological Phase at the Surface of a 3D Topological Insulator
• Topological phases with generalized global symmetries
• Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry
• The effect of interactions on 2D fermionic symmetry-protected topological phases with Z2 symmetry
• Intrinsically Gapless Topological Phases
• Gapless Symmetry Protected Topological Order
• Matrix Product Unitaries: Structure, Symmetries, and Topological Invariants
• Hierarchy of topological order from finite-depth unitaries, measurement and feedforward
• Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems
• A theory of 2+1D fermionic topological orders and fermionic/bosonic topological orders with symmetries
• Abelian Floquet symmetry-protected topological phases in one dimension
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Symmetry Protected Topological States of Interacting Fermions and Bosons
• Symmetry Protected Topological phases and Generalized Cohomology
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids
• Long-range entanglement from measuring symmetry-protected topological phases
• Higher order symmetry-protected topological states for interacting bosons and fermions
• Topological invariants for gauge theories and symmetry-protected topological phases
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Fractal Symmetric Phases of Matter
• Anomalies and Bounds on Charged Operators
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Topological Phase Transition without Gap Closing
• Scaling of the entanglement spectrum near quantum phase transitions
• Topological Luttinger Liquids from Decorated Domain Walls
• Non-Fermi liquids as ersatz Fermi liquids: general constraints on compressible metals
• Quantum Phase Transition between Integer Quantum Hall States of Bosons
• Topological color code and symmetry-protected topological phases
• Hierarchy of universal entanglement in 2D measurement-based quantum computation
• Many-body localization, symmetry, and topology
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Lecture Notes on Generalized Symmetries and Applications
• Symmetric Fracton Matter: Twisted and Enriched
• Phases of correlated spinless fermions on the honeycomb lattice
• Unconventional Surface Critical Behaviors Induced by Quantum Phase Transition from Two-Dimensional Affleck-Kennedy-Lieb-Tasaki Phase to Néel Order
• Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions
• Crystalline topological phases as defect networks
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Layer construction of 3D topological states and string braiding statistics
• Symmetry-resolved entanglement in symmetry-protected topological phases
• Bosonic Integer Quantum Hall effect in an interacting lattice model
• Symmetric Mass Generation
• An order parameter for impurity systems at quantum criticality
• Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions
• Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Quantum phase transitions between bosonic symmetry protected topological phases in two dimensions: emergent $QE{D}_3$ and anyon superfluid
• Photonic Dirac monopoles and skyrmions: spin-1 quantization
• A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
• Symmetry protected topological order at nonzero temperature
• Realization of Fractional Chern Insulators in the Thin-Torus-Limit with Ultracold Bosons
• Cluster state as a non-invertible symmetry protected topological phase
• Bosonic Anomalies, Induced Fractional Quantum Numbers and Degenerate Zero Modes: the anomalous edge physics of Symmetry-Protected Topological States
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Efficient Representation of Topologically Ordered States with Restricted Boltzmann Machines
• Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions
• Classification of topological phases with finite internal symmetries in all dimensions
• Short-range entanglement and invertible field theories
• A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model
• Interaction effects on 1D fermionic symmetry protected topological phases
• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
• Fractonic Chern-Simons and BF theories
• Quasi-Topological Phases of Matter and Topological Protection
• Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases
• Bulk-boundary correspondence for three-dimensional symmetry-protected topological phases
• A K matrix Construction of Symmetry Enriched Phases of Matter
• Symmetry Fractionalization in Two Dimensional Topological Phases
• Duality and bosonization of (2+1)d Majorana fermions
• Non-Perturbative Regularization of 1+1D Anomaly-Free Chiral Fermions and Bosons: On the equivalence of anomaly matching conditions and boundary gapping rules
• Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry
• Universal symmetry-protected topological invariants for symmetry-protected topological states
• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Detecting edge degeneracy in interacting topological insulators through entanglement entropy
• Lattice models that realize ${\mathbb{Z}}_n$-1-symmetry protected topological states for even $n$
• Engineering Surface Critical Behavior of (2+1)-Dimensional O(3) Quantum Critical Points
• Real-space recipes for general topological crystalline states
• Symmetric tensor networks and practical simulation algorithms to sharply identify classes of quantum phases distinguishable by short-range physics
• Detecting two dimensional symmetry protected topological order in a ground state wave function
• Minimalist approach to the classification of symmetry protected topological phases
• Characterizing Topological Order with Matrix Product Operators
• Fermionic projected entangled-pair states and topological phases
• Symmetry protection of measurement-based quantum computation in ground states
• Quantum gyro-electric effect: Photon spin-1 quantization in continuum topological bosonic phases
• Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries
• Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory
• Strong planar subsystem symmetry-protected topological phases and their dual fracton orders
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Boundary conformal field theory and symmetry protected topological phases in $2+1$ dimensions
• Fragile topological phases in interacting systems
• ${\mathbb{Z}}_N$ Symmetries, Anomalies, and the Modular Bootstrap
• A systematic construction of gapped non-liquid states
• Symmetry-protected sign problem and magic in quantum phases of matter
• Rokhsar-Kivelson Models of Bosonic Symmetry-Protected Topological States
• Sequential Quantum Circuits as Maps between Gapped Phases
• A Solution to the 1+1D Gauged Chiral Fermion Problem
• Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
• Matrix Product Representation of Locality Preserving Unitaries
• A Non-Perturbative Definition of the Standard Models
• Stroboscopic Symmetry-Protected Topological Phases
• Topological and symmetry-enriched random quantum critical points
• Exact realization of Integer and Fractional Quantum Hall Phases in U(1)xU(1) models in (2+1)d
• Chiral symmetry on the edge of 2D symmetry protected topological phases
• Symmetry Protected Topological Charge in Symmetry Broken Phase: Spin-Chern, Spin-Valley-Chern and Mirror-Chern Numbers
• Microscopic Realization of 2-Dimensional Bosonic Topological Insulators
• A generalization of the injectivity condition for Projected Entangled Pair States
• Classifying phases protected by matrix product operator symmetries using matrix product states
• Topologically-Protected Long Edge Coherence Times in Symmetry-Broken Phases
• Categories of quantum liquids I
• Symmetry-protected self-correcting quantum memories
• Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices
• Non-local topological electromagnetic phases of matter
• Topological Phases with Average Symmetries: the Decohered, the Disordered, and the Intrinsic
• Evidence for deconfined $U(1)$ gauge theory at the transition between toric code and double semion
• $U(1)\times U(1)$ Symmetry Protected Topological Order in Gutzwiller Wave Functions
• Wavefunctions of Symmetry Protected Topological Phases from Conformal Field Theories
• Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks
• Topological Field Theory with Haagerup Symmetry
• Subsystem symmetry enriched topological order in three dimensions
• Anomalies in bosonic SPT edge theories: connection to F-symbols and a method for calculation
• Quantum Phase Transitions Between a Class of Symmetry Protected Topological States
• Realizing topological surface states in a lower-dimensional flat band
• Classification of Matrix-Product Unitaries with Symmetries
• Quantum Phase Transitions Between Bosonic Symmetry Protected Topological States Without Sign Problem: Nonlinear Sigma Model with a Topological Term
• Fractional Quantum Hall Effect in Weyl Semimetals
• Field Theories for Gauged Symmetry Protected Topological Phases: Abelian Gauge Theories with non-Abelian Quasiparticles
• Smooth gauge and Wannier functions for topological band structures in arbitrary dimensions
• Robustness of a topological phase: Topological color code in parallel magnetic field
• Symmetry-protected topologically ordered states for universal quantum computation
• Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1D
• Symmetry-protected many-body Aharonov-Bohm effect
• Robustness in Projected Entangled Pair States
• Anomalies and entanglement renormalization
• From trivial to topological paramagnets: The case of ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_2^3$ symmetries in two dimensions
• One dimensional gapped quantum phases and enriched fusion categories
• Kagome chiral spin liquid as a gauged U(1) symmetry protected topological phase
• Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions
• Higher-Order Entanglement and Many-Body Invariants for Higher-Order Topological Phases
• Quantum spin systems for measurement-based quantum computation
• Classification of Mixed State Topology in One Dimension
• Classifying Quantum Phases With The Torus Trick
• Mixed-state quantum anomaly and multipartite entanglement
• Universal measurement-based quantum computation in two-dimensional SPT phases
• Building models of topological quantum criticality from pivot Hamiltonians
• Short-Range Entangled Bosonic States with Chiral Edge Modes and $T$-duality of Heterotic Strings
• Gapless edge states of BF field theory and translation-symmetric Z2 spin liquids
• Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions
• Gutzwiller Projected wavefunctions in the fermonic theory of S=1 spin chains
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• Interaction Enabled Fractonic Higher-Order Topological Phases
• Constraints on order and disorder parameters in quantum spin chains
• Projected Entangled Pair States: Fundamental analytical and numerical limitations
• Topological spin ordering via Chern-Simons superconductivity
• Detecting and identifying 2D symmetry-protected topological, symmetry-breaking and intrinsic topological phases with modular matrices via tensor-network methods
• On adiabatic cycles of quantum spin systems
• Topological phase transitions at finite temperature
• Tensor network state approach to quantum topological phase transitions and their criticalities of ${\mathbb{Z}}_2$ topologically ordered states
• Lattice T-duality from non-invertible symmetries in quantum spin chains
• Detecting subsystem symmetry protected topological order via entanglement entropy
• Supersolid-Superfluid phase separation in the extended Bose-Hubbard model
• Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems
• Hidden order and flux attachment in symmetry protected topological phases: a Laughlin-like approach
• `Gauging' time reversal symmetry in tensor network states
• Symmetry Protected Quantum State Renormalization
• Perfect Metal Phases of One-Dimensional and Anisotropic Higher-Dimensional Systems
• Fractional corner charges in a 2D super-lattice Bose-Hubbard model
• Fractionalization on the Surface: Is Type-II Terminated $1T$-TaS${}_2$ Surface an Anomalously Realized Spin Liquid?
• Bosonic Crystalline Symmetry Protected Topological Phases Beyond the Group Cohomology Proposal
• Bridging coupled wires and lattice Hamiltonian for two-component bosonic quantum Hall states
• Interacting Symmetry-Protected Topological Phases Out of Equilibrium
• Hexagon-singlet solid ansatz for the spin-1 kagome antiferromagnet
• Global anomaly detection in two-dimensional symmetry-protected topological phases
• Partition Functions and Stability Criteria of Topological Insulators
• Visualizing Quasiparticles from Quantum Entanglement for general 1D phases
• (SPT-)LSM theorems from projective non-invertible symmetries
• Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory
• Holonomic quantum computing in symmetry-protected ground states of spin chains
• Phase structure of the 1+1 dimensional massive Thirring model from matrix product states
• Tensor network representations from the geometry of entangled states
• Topological transport in a spin-orbit coupled bosonic Mott insulator
• $Z_2$ fractionalized Chern/topological insulators in an exactly soluble correlated model
• Antiferromagnetic magnons from fractionalized excitations
• Classifying One-Dimensional Quantum States Prepared by a Single Round of Measurements
• Entanglement entropy and entanglement spectrum of triplet topological superconductors
• Intrinsic Contribution to Non-linear Thermoelectric Effect
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Classification of symmetry-protected topological phases in two-dimensional many body-localized systems
• Majorana Edge States for Z2 Topological Orders of the Wen-plaquette Model and the Toric-code Model
• Criteria for protected edge modes with ${\mathbb{Z}}_2$ symmetry
• Three-dimensional Topological Insulators and Bosonization
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
• Symmetry-protected Topological Phases in Spinful Bosons with a Flat Band
• Computing classification of interacting fermionic symmetry-protected topological phases using topological invariants
• Gauging anomalous unitary operators
• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• Nonequilibrium symmetry-protected topological order: emergence of semilocal Gibbs ensembles
• Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension
• Quasiparticle Berry curvature and Chern numbers in spin-orbit coupled bosonic Mott insulators
• Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi-Point Strange Correlators
• Measurement-based quantum simulation of Abelian lattice gauge theories
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Symmetry Transmutation and Anomaly Matching
• Topological antiferromagnetic spin-density-wave phase in an extended Kondo lattice model
• Monte Carlo Study of a $U(1)\times U(1)$ Loop Model with Modular Invariance
• Quantum Error-Detection at Low Energies
• Superuniversality from disorder at two-dimensional topological phase transitions
• Efficient variational quantum circuit structure for correlated topological phases
• Interplay between short and long-range entanglement in symmetry protected phases
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Anomaly inflow, dualities, and quantum simulation of abelian lattice gauge theories induced by measurements
• Fermionic Symmetry-Protected Topological Phase in a Two-Dimensional Hubbard Model
• Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• Topological transitions to Weyl states in bulk Bi${}_2$Se${}_3$: Effect of hydrostatic pressure and doping
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Phase transitions of a 2D deformed-AKLT model
• Anomalous domain wall condensation in a modified Ising chain
• LSM and CPT
• The Partially-Split Hall Bar: Tunneling in the Bosonic Integer Quantum Hall Effect
• Stability of Topological Insulators with Non-Abelian Edge Excitations
• Finite-size scaling analysis of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states
• Symmetry protected topological phases in spin-1 ladders and their phase transitions
• The resource theory of tensor networks
• Bulk and boundary entanglement transitions in the projective gauge-Higgs model
• Fusion of one-dimensional gapped phases and their domain walls
• Detection of gapped phases of a 1D spin chain with onsite and spatial symmetries
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Mirror anomaly in fermionic topological orders
• Subsystem Symmetry Fractionalization and Foliated Field Theory
• Inexorable Edge Kondo Breakdown in Topological Kondo Insulators
• Realization of the topological Hopf term in two-dimensional lattice models
• Gauging quantum states with non-anomalous matrix product operator symmetries
• Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions
• Bosonic Integer Quantum Hall States without Landau Levels on Square Lattice
• Classical and quantum spin liquids
• Preparing matrix product states via fusion: constraints and extensions
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Dihedral twist liquid models from emergent Majorana fermions
• Fractional domain wall statistics in spin chains with anomalous symmetries
• Topological Superconductivity in Doped Symmetry Protected Topological State
• Topological Phases of Many-Body Localized Systems: Beyond Eigenstate Order
• Many-body quantum catalysts for transforming between phases of matter
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization
• Generalized Statistics on Lattices
• 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
• Topological Dipole Insulator
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• From gauging to duality in one-dimensional quantum lattice models
• Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Non-onsite symmetry breaking: topological phase coexistence and criticality
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Phase diagram of the J1-J2 Heisenberg second-order topological quantum magnet
• Anomalies of global symmetries on the lattice
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Les Houches Lecture Notes on Tensor Networks
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories
• Symmetries and field tensor network states
• SPT indices emerging from translation invariance in two dimensional quantum spin systems
• Symmetry-Protected Topological relationship between $SU(3)$ and $SU(2)\times U(1)$ in Two Dimension
• Matrix product state classification of 1D multipole symmetry protected topological phases
• Detection of 2D SPT Order with Partial Symmetries
• Quantum Monte Carlo and Stabilizer States
• Spurious Strange Correlators in Symmetry-Protected Topological Phases
• Noise-tolerant Detection of $Z_N$ Topological Orders in Quantum Many-body States
• Higher-order topological phases protected by noninvertible and subsystem symmetries