Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions

Abstract

Recently, it was realized that quantum states of matter can be classified as long-range entangled (LRE) states (i.e. with non-trivial topological order) and short-range entangled (SRE) states (\ie with trivial topological order). We can use group cohomology class ${\mathcal{H}}^d(SG,R/Z)$ to systematically describe the SRE states with a symmetry $SG$ [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in $d$-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects, and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e. to measure the elements in ${\mathcal{H}}^d(G,R/Z)$ that label different SPT states. For example, 2+1D bosonic SPT states with $Z_n$ symmetry are classified by a $Z_n$ integer $m\in {\mathcal{H}}^3(Z_n,R/Z)=Z_n$. We find that $n$ identical monodromy defects, in a $Z_n$ SPT state labeled by $m$, carry a total $Z_n$-charge $2m$ (which is not a multiple of $n$ in general).

42 pages, 12 figures, 3 tables, RevTeX4-1

References in corpus (23)

• Topological Field Theory of Time-Reversal Invariant Insulators
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Topological Phases of Noncentrosymmetric Superconductors: Edge States, Majorana Fermions, and the Non-Abelian Statistics
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• A classification of symmetry enriched topological phases with exactly solvable models
• Spin Liquid States on the Triangular and Kagome Lattices: A Projective Symmetry Group Analysis of Schwinger Boson States
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Protected edge modes without symmetry
• Topological boundary conditions in abelian Chern-Simons theory
• Integer quantum Hall effect for bosons: A physical realization
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Bosonic topological insulator in three dimensions and the statistical Witten effect
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Mutual Chern-Simons theory for Z_2 topological order
• Symmetry protected Spin Quantum Hall phases in 2-Dimensions
• Constructing symmetric topological phases of bosons in three dimensions via fermionic projective construction and dyon condensation
• Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries
• Critical theories of phase transition between symmetry protected topological states and their relation to the gapless boundary theories
• Chiral symmetry on the edge of 2D symmetry protected topological phases
• Three dimensional Symmetry Protected Topological Phase close to Antiferromagnetic Neel order
• String-Net Models with $Z_N$ Fusion Algebra
• A calculus for branched spine of 3-manifolds
• Short-Range Entangled Bosonic States with Chiral Edge Modes and $T$-duality of Heterotic Strings

Cited by in corpus (75)

• Classification of topological quantum matter with symmetries
• Zoo of quantum-topological phases of matter
• Interacting topological insulators: a review
• Symmetry protected topological phases from decorated domain walls
• Self-induced topological transitions and edge states supported by nonlinear staggered potentials
• Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
• Intrinsic and emergent anomalies at deconfined critical points
• Gapless topological phases and symmetry-enriched quantum criticality
• Diagrammatics for Bose condensation in anyon theories
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Symmetry Protected Topological States of Interacting Fermions and Bosons
• Symmetry Protected Topological phases and Generalized Cohomology
• Aperiodic Weak Topological Superconductors
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Topological invariants for gauge theories and symmetry-protected topological phases
• Fractal Symmetric Phases of Matter
• Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries
• The translational side of topological band insulators
• Long-lived interacting phases of matter protected by multiple time-translation symmetries in quasiperiodically-driven systems
• Lecture Notes on Generalized Symmetries and Applications
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Long-range topological insulators and weakened bulk-boundary correspondence
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
• Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries
• Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Bosonic topological phases of matter: bulk-boundary correspondence, SPT invariants and gauging
• Topological states in a microscopic model of interacting fermions
• Bosonic Anomalies, Induced Fractional Quantum Numbers and Degenerate Zero Modes: the anomalous edge physics of Symmetry-Protected Topological States
• Classification and construction of higher-order symmetry protected topological phases of interacting bosons
• Short-range entanglement and invertible field theories
• From gauge to higher gauge models of topological phases
• Universal symmetry-protected topological invariants for symmetry-protected topological states
• Lattice models that realize ${\mathbb{Z}}_n$-1-symmetry protected topological states for even $n$
• Detecting two dimensional symmetry protected topological order in a ground state wave function
• Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulators
• Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases
• Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory
• Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
• Exact realization of Integer and Fractional Quantum Hall Phases in U(1)xU(1) models in (2+1)d
• Microscopic Realization of 2-Dimensional Bosonic Topological Insulators
• Decorated Defect Construction of Gapless-SPT States
• The characterization of topological properties in Quantum Monte Carlo simulations of the Kane-Mele-Hubbard model
• Field Theories for Gauged Symmetry Protected Topological Phases: Abelian Gauge Theories with non-Abelian Quasiparticles
• 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
• Symmetry Enrichment in Three-Dimensional Topological Phases
• Coherent generation and detection of acoustic phonons in topological nanocavities
• Relationship between Symmetry Protected Topological Phases and Boundary Conformal Field Theories via the Entanglement Spectrum
• Classifying local fractal subsystem symmetry protected topological phases
• Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations
• Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions
• A Unified Framework of Topological Phases with Symmetry
• Quantum phase transition and critical behavior between the gapless topological phases
• Interacting second-order topological insulators in one-dimensional fermions with correlated hopping
• Interaction-enabled topological phases in topological insulator-superconductor heterostructures
• Complete phase diagram and topological properties of interacting bosons in one-dimensional superlattices
• Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory
• Thermal conductance as a probe of the non-local order parameter for a topological superconductor with gauge fluctuations
• Spin-one bilinear-biquadratic model on a star lattice
• Chiral and Geometric Anomalies in Finite Systems
• Irreducible Projective Representations and Their Physical Applications
• Topological Crystalline Bose Insulator in Two Dimensions via Entanglement Spectrum
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
• Driving Topological Phases by Spatially Inhomogeneous Pairing Centers
• Practical new platform for interaction-enabled topological phases
• Observation of Fermi Arc Surface States in a Topological Metal: A New Type of 2D Electron Gas beyond Z2 Topological Insulators
• Non-local Order Parameters and Quantum Entanglement for Fermionic Topological Field Theories
• Topological physics in quantum critical systems
• A bosonic topological phase in a paired superfluid
• Solvable lattice model for (2+1)D bosonic topological insulator
• Recent progress in quantum spin liquids, fractional magnetization plateaus, and unconventional superconductivity in kagome lattices
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Emergent Symmetry-Protected Topological Phases via Polyakov Confinement in Quantum Spin Systems