Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface

Abstract

The Lieb-Schultz-Mattis theorem and its higher dimensional generalizations by Oshikawa and Hastings require that translationally invariant 2D spin systems with a half-integer spin per unit cell must either have a continuum of low energy excitations, spontaneously break some symmetries, or exhibit topological order with anyonic excitations. We establish a connection between these constraints and a remarkably similar set of constraints at the surface of a 3D interacting topological insulator. This, combined with recent work on symmetry-enriched topological phases (SETs) with on-site unitary symmetries, enables us to develop a framework for understanding the structure of SETs with both translational and on-site unitary symmetries, including the effective theory of symmetry defects. This framework places stringent constraints on the possible types of symmetry fractionalization that can occur in 2D systems whose unit cell contains fractional spin, fractional charge, or a projective representation of the symmetry group. As a concrete application, we determine when a topological phase must possess a "spinon" excitation, even in cases when spin rotational invariance is broken down to a discrete subgroup by the crystal structure. We also describe the phenomena of "anyonic spin-orbit coupling", which may arise from the interplay of translational and on-site symmetries. These include the possibility of on-site symmetry defect branch lines carrying topological charge per unit length and lattice dislocations inducing on-site symmetry protected degeneracies.

27 pages, 9 figures; published version

References in corpus (9)

• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Fractionalized Fermi liquids
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• 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
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• Fractionalization, topological order, and quasiparticle statistics
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Constraints on topological order in Mott Insulators

Cited by in corpus (160)

• Pre-thermal phases of matter protected by time-translation symmetry
• Valence Bonds in Random Quantum Magnets: Theory and Application to YbMgGaO4
• Topological phases protected by point group symmetry
• Building crystalline topological phases from lower-dimensional states
• Gauging spatial symmetries and the classification of topological crystalline phases
• Intrinsic and emergent anomalies at deconfined critical points
• Intrinsically Gapless Topological Phases
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Average Symmetry-Protected Topological Phases
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Higher order symmetry-protected topological states for interacting bosons and fermions
• Floquet topological phases with symmetry in all dimensions
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Lieb-Schultz-Mattis Theorem and its generalizations from the Perspective of the Symmetry Protected Topological phase
• Non-Fermi liquids as ersatz Fermi liquids: general constraints on compressible metals
• Fractonic Matter in Symmetry-Enriched U(1) Gauge Theory
• Ising ferromagnet to valence bond solid transition in a one-dimensional spin chain: Analogies to deconfined quantum critical points
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Multipolar Topological Field Theories: Bridging Higher Order Topological Insulators and Fractons
• Crystalline topological phases as defect networks
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases
• Symmetry protected topological order at nonzero temperature
• Classification and construction of higher-order symmetry protected topological phases of interacting bosons
• Emergent Symmetry and Conserved Current at a One Dimensional Incarnation of Deconfined Quantum Critical Point
• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
• Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality
• Electric polarization as a nonquantized topological response and boundary Luttinger theorem
• Twisted boundary condition and Lieb-Schultz-Mattis ingappability for discrete symmetries
• Three-dimensional superconductors with hybrid higher order topology
• Dynamical Signature of Symmetry Fractionalization in Frustrated Magnets
• Bulk-boundary correspondence for intrinsically-gapless SPTs from group cohomology
• Anomaly matching and symmetry-protected critical phases in $SU(N)$ spin systems in 1+1 dimensions
• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Higher order topological superconductors as generators of quantum codes
• Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter
• Fermionic symmetry fractionalization in (2+1)D
• Surface critical behaviors of coupled Haldane chains
• Quantum Spin Liquid with Even Ising Gauge Field Structure on Kagome Lattice
• Higher-group symmetry in finite gauge theory and stabilizer codes
• Decorated Defect Construction of Gapless-SPT States
• Topological Phases with Average Symmetries: the Decohered, the Disordered, and the Intrinsic
• A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers
• Ground state degeneracy on torus in a family of ${\mathbb{Z}}_N$ toric code
• Non-zero momentum requires long-range entanglement
• Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Dyonic Lieb-Shultz-Mattis Theorem and Symmetry Protected Topological Phases in Decorated Dimer Models
• Quantum many-body topology of quasicrystals
• Deconfined criticality and a gapless ${\mathbb{Z}}_2$ spin liquid in the square lattice antiferromagnet
• Rotation Symmetry-Protected Topological Phases of Fermions
• Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries
• Higher-Order Entanglement and Many-Body Invariants for Higher-Order Topological Phases
• Chern-Simons Composite Fermion Theory of Fractional Chern Insulators
• Mixed-state quantum anomaly and multipartite entanglement
• Emergent anomalies and generalized Luttinger theorems in metals and semimetals
• Unquantized anomalies in topological semimetals
• Duality, Criticality, Anomaly, and Topology in Quantum Spin-1 Chains
• Diagnosing Fractionalization from the Spin Dynamics of $Z_2$ Spin Liquids on the Kagome Lattice by Quantum Monte Carlo Simulations
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• Lieb-Schultz-Mattis type constraints on Fractonic Matter
• Classification of symmetry fractionalization in gapped ${\mathbb{Z}}_2$ spin liquids
• Doping a fractional quantum anomalous Hall insulator
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications
• Non-diagonal anisotropic quantum Hall states
• A UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases
• Anomaly cascade in (2+1)D fermionic topological phases
• Disentangling interacting symmetry protected phases of fermions in two dimensions
• Fermi Surface Symmetric Mass Generation
• Metal-Insulator Transition with Charge Fractionalization
• Complete crystalline topological invariants from partial rotations in (2+1)D invertible fermionic states and Hofstadter's butterfly
• (SPT-)LSM theorems from projective non-invertible symmetries
• Topological Edge and Interface states at Bulk disorder-to-order Quantum Critical Points
• Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
• Intrinsically Interacting Higher-Order Topological Superconductors
• Gauging Lie group symmetry in (2+1)d topological phases
• Surface field theories of point group symmetry protected topological phases
• Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Role of boundary conditions in the full counting statistics of topological defects after crossing a continuous phase transition
• Anomalies and unnatural stability of multi-component Luttinger liquids in ${\mathbb{Z}}_n\times {\mathbb{Z}}_n$ spin chains
• Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories
• Filling constraints on fermionic topological order in zero magnetic field
• Detecting Quantum Anomalies in Open Systems
• Symmetry protected gapless $Z_2$ spin liquids
• Low energy effective field theories of fermion liquids and mixed $U(1)\times {\mathbb{R}}^d$ anomaly
• Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
• Anomalous localization at the boundary of an interacting topological insulator
• Evidence for the topological order in a kagome antiferromagnet
• Topological Dipole Conserving Insulators and Multipolar Responses
• Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
• Fractional Chern insulator edges and layer-resolved lattice contacts
• Definition and Classification of Fermi Surface Anomalies
• Interacting Topological Insulators with Synthetic Dimensions
• Geometric approach to Lieb-Schultz-Mattis theorem without translation symmetry under inversion or rotation symmetry
• Translation symmetry-enriched toric code insulator
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Symmetry-protected Topological Phases in Spinful Bosons with a Flat Band
• Minimal Fractional Topological Insulator in half-filled conjugate moiré Chern bands
• Majorana lattices from the quantized Hall limit of a proximitized spin-orbit coupled electron gas
• Average-exact mixed anomalies and compatible phases
• Unconventional quantum phase transitions in a one-dimensional Lieb-Schultz-Mattis system
• Filling constraints on translation invariant dipole conserving systems
• Gauging U(1) symmetry in (2+1)d topological phases
• Machine Learning the Operator Content of the Critical Self-Dual Ising-Higgs Gauge Model
• Boundary conditions and anomalies of conformal field theories in 1+1 dimensions
• Measuring Topological Order
• Interacting Edge States of Fermionic Symmetry-Protected Topological Phases in Two Dimensions
• Symmetry Transmutation and Anomaly Matching
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Ferromagnets, a New Anomaly, Instantons, and (noninvertible) Continuous Translations
• Rotational Symmetry Protected Edge and Corner States in Abelian topological phases
• Enforced symmetry breaking by invertible topological order
• Translation constraints on quantum phases with twisted boundary conditions
• Gappability Index for Quantum Many-Body Systems
• Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries
• Classification of symmetry-enriched topological quantum spin liquids
• Theory of quantum kagome ice and vison zero modes
• Anomalous domain wall condensation in a modified Ising chain
• Anomalous Continuous Translations
• Fractionalization as an alternate to charge ordering in electronic insulators
• Thermodynamic signature of the $\mathrm{SU}(4)$ spin-orbital liquid and symmetry fractionalization from the Lieb-Schultz-Mattis theorem
• Anomaly constraint on chiral central charge of (2+1)d topological order
• (3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases
• Unveiling UV/IR Mixing via Symmetry Defects: A View from Topological Entanglement Entropy
• Projective Representations, Bogomolov Multiplier, and Their Applications in Physics
• Building Symmetry Enriched Topological Phases from a Bipartite Lattice Construction and Anyon Condensation
• Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking
• Two-dimensional Paired Topological Superfluids of Rydberg Fermi Gases
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Universal contributions to charge fluctuations in spin chains at finite temperature
• Lieb-Schultz-Mattis constraints for the insulating phases of the one-dimensional SU($N$) Kondo lattice model
• Spin dynamics of the planar kagome lattice ferromagnet with four-site ring exchange processes
• Topological Superconductivity in Doped Symmetry Protected Topological State
• Crystalline invariants of fractional Chern insulators
• Soft symmetries of topological orders
• Spinless and spinful charge excitations in moiré Fractional Chern Insulators
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Disentangling anomaly-free symmetries of quantum spin chains
• Absence of localization in Weyl semimetals
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Numerical identification and gapped boundaries of abelian fermionic topological order
• Ordering the topological order in the fractional quantum Hall effect
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Entanglement-enabled symmetry-breaking orders
• Quantum Degenerate Majorana Surface Zero Modes in Two-Dimensional Space
• Anomalies of global symmetries on the lattice
• Bosonic SET to SPT Transition under Dyonic LSM Theorem
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• When Wannier centers jump: Critical points between atomic insulating phases
• Theory of Hofstadter Superconductors
• Parity anomaly from LSM: exact valley symmetries on the lattice
• Non-invertible translation from Lieb-Schultz-Mattis anomaly