Building crystalline topological phases from lower-dimensional states

Abstract

We study the classification of symmetry protected topological (SPT) phases with crystalline symmetry (cSPT phases). Focusing on bosonic cSPT phases in two and three dimensions, we introduce a simple family of cSPT states, where the system is comprised of decoupled lower-dimensional building blocks that are themselves SPT states. We introduce a procedure to classify these block states, which surprisingly reproduces a classification of cSPT phases recently obtained by Thorngren and Else using very different methods, for all wallpaper and space groups. The explicit constructions underlying our results clarify the physical properties of the phases classified by Thorngren and Else, and expose additional structure in the classification. Moreover, the states we classify can be completely characterized by point group SPT (pgSPT) invariants and related weak pgSPT invariants that we introduce. In many cases, the weak invariants can be visualized in terms of translation-symmetric stacking of lower-dimensional pgSPT states. We apply our classification to propose a Lieb-Shultz-Mattis type constraint for two-dimensional spin systems with only crystalline symmetry, and establish this constraint by a dimensional reduction argument. Finally, the surprising matching with the Thorngren-Else classification leads us to conjecture that all SPT phases protected only by crystalline symmetry can be built from lower-dimensional blocks of invertible topological states. We argue that this conjecture holds if we make a certain physically reasonable but unproven assumption.

arXiv comment18+14 pages. S.-J.H. and H.S. are co-first-authors. v2: published version

References in corpus (46)

• Topological Insulators
• Topological insulators and superconductors
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Periodic table for topological insulators and superconductors
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Classifying quantum phases using Matrix Product States and PEPS
• Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials
• Lieb-Schultz-Mattis in Higher Dimensions
• Three-Dimensional Topological Insulators
• Symmetry Protected Topological phases of Quantum Matter
• Braiding statistics approach to symmetry-protected topological phases
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Commensurability, excitation gap and topology in quantum many-particle systems on a periodic lattice
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface
• Filling constraints for spin-orbit coupled insulators in symmorphic and non-symmorphic crystals
• Topological phases protected by point group symmetry
• Topological Order and Absence of Band Insulators at Integer Filling in Non-Symmorphic Crystals
• Symmetry Protected Topological States of Interacting Fermions and Bosons
• Theory of interacting topological crystalline insulators
• Breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• All Majorana Models with Translation Symmetry are Supersymmetric
• Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions
• Symmetry-protected Topological Phases, Generalized Laughlin Argument and Orientifolds
• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Distinct trivial phases protected by a point-group symmetry in quantum spin chains
• Topological Phases Protected By Reflection Symmetry and Cross-cap States
• CPT theorem and classification of topological insulators and superconductors
• Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators
• Coupled-layer description of topological crystalline insulators
• Lieb-Schultz-Mattis theorems for symmetry protected topological phases
• Degeneracy of the ground-state of antiferromagnetic spin-1/2 Hamiltonians
• Some remarks on the Lieb-Schultz-Mattis theorem and its extension to higher dimensions
• Correlation effects on topological crystalline insulators
• Interaction Enabled Topological Crystalline Phases
• Bosonic symmetry protected topological phases with reflection symmetry
• Space group symmetries and low lying excitations of many-body systems at integer fillings
• Classification and surface anomaly of glide symmetry protected topological phases in three dimensions
• Topological Crystalline Bose Insulator in Two Dimensions via Entanglement Spectrum
• Ground state degeneracy in quantum spin systems protected by crystal symmetries
• Anomalies and symmetry fractionalization in reflection-symmetric topological order

Cited by in corpus (133)

• Higher-order topological insulators and superconductors protected by inversion symmetry
• Quantitative mappings between symmetry and topology in solids
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Symmetry indicators and anomalous surface states of topological crystalline insulators
• Fracton-Elasticity Duality
• All Topological Bands of All Nonmagnetic Stoichiometric Materials
• Topological Materials Discovery from Crystal Symmetry
• Rotation Anomaly and Topological Crystalline Insulators
• Intrinsic and emergent anomalies at deconfined critical points
• Symmetry Enforced Chiral Hinge States and Surface Quantum Anomalous Hall Effect in Magnetic Axion Insulator ${\text{Bi}}_{2-x}{\text{Sm}}_x{\text{Se}}_3$
• Boundary-obstructed topological phases
• Non-Hermitian second-order skin and topological modes
• Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems
• Average Symmetry-Protected Topological Phases
• Higher order symmetry-protected topological states for interacting bosons and fermions
• Refined symmetry indicators for topological superconductors in all space groups
• Connecting higher-order topological insulators to lower-dimensional topological insulators
• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
• Topological states from topological crystals
• Ising ferromagnet to valence bond solid transition in a one-dimensional spin chain: Analogies to deconfined quantum critical points
• Topological states on fractal lattices
• Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Multipolar Topological Field Theories: Bridging Higher Order Topological Insulators and Fractons
• Crystalline topological phases as defect networks
• Topological Defect Networks for Fractons of all Types
• Crystal-to-Fracton Tensor Gauge Theory Dualities
• Shift insulators: rotation-protected two-dimensional topological crystalline insulators
• Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms
• Symmetry-based indicators for topological Bogoliubov-de Gennes Hamiltonians
• Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases
• Symmetry indicators of band topology
• Classification and construction of higher-order symmetry protected topological phases of interacting bosons
• Higher-order topological band structures
• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
• Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models
• Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Three-dimensional superconductors with hybrid higher order topology
• Topological classification and diagnosis in magnetically ordered electronic materials
• Real-space recipes for general topological crystalline states
• Higher order topological superconductors as generators of quantum codes
• Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology
• Minimalist approach to the classification of symmetry protected topological phases
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• 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
• Fragile topological phases in interacting systems
• A systematic construction of gapped non-liquid states
• `Unhinging' the surfaces of higher-order topological insulators and superconductors
• Generalized Wen-Zee Terms
• A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers
• Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems
• qeirreps: an open-source program for Quantum ESPRESSO to compute irreducible representations of Bloch wavefunctions
• Non-zero momentum requires long-range entanglement
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Bulk characterization of topological crystalline insulators: stability under interactions and relations to symmetry enriched U(1) quantum spin liquids
• Quantum many-body topology of quasicrystals
• Delocalization Transition of Disordered Axion Insulator
• The classification of surface states of topological insulators and superconductors with magnetic point group symmetry
• Searching for Fracton Orders via Symmetry Defect Condensation
• Faithful derivation of symmetry indicators: A case study for topological superconductors with time-reversal and inversion symmetries
• Construction and classification of point group symmetry protected topological phases in 2D interacting fermionic systems
• Rotation Symmetry-Protected Topological Phases of Fermions
• Direct dynamical characterization of higher-order topological insulators with nested band inversion surfaces
• $N$-band Hopf insulator
• Higher-Order Entanglement and Many-Body Invariants for Higher-Order Topological Phases
• Topological Crystalline Phases in a Disordered Inversion-Symmetric Chain
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• Interacting topological quantum chemistry of Mott atomic limits
• Symmetry-protected quantization of complex Berry phases in non-Hermitian many-body systems
• Lieb-Schultz-Mattis type constraints on Fractonic Matter
• Topological polarization, dual invariants, and surface flat band in crystalline insulators
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Bosonic Crystalline Symmetry Protected Topological Phases Beyond the Group Cohomology Proposal
• Organizing symmetry-protected topological phases by layering and symmetry reduction: a minimalist perspective
• Higher-order topological superconductors from Weyl semimetals
• Spin chain network construction of chiral spin liquids
• Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
• Intrinsically Interacting Higher-Order Topological Superconductors
• Fractonic Higher-Order Topological Phases in Open Quantum Systems
• (SPT-)LSM theorems from projective non-invertible symmetries
• Complete crystalline topological invariants from partial rotations in (2+1)D invertible fermionic states and Hofstadter's butterfly
• Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory
• Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups
• Surface field theories of point group symmetry protected topological phases
• Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Effective field theories of topological crystalline insulators and topological crystals
• Classification and construction of interacting fractonic higher-order topological phases
• Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories
• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Topological classification of intrinsic 3D superconductors using anomalous surface construction
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Geometric approach to Lieb-Schultz-Mattis theorem without translation symmetry under inversion or rotation symmetry
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors
• Symmetry-protected Topological Phases in Spinful Bosons with a Flat Band
• Interacting Edge States of Fermionic Symmetry-Protected Topological Phases in Two Dimensions
• Filling constraints on translation invariant dipole conserving systems
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Higher-order topological superconductivity in a topological metal 1T${}^{\prime }$-MoTe${}_2$
• Ferromagnets, a New Anomaly, Instantons, and (noninvertible) Continuous Translations
• Classification of time-reversal symmetric topological superconducting phases for conventional pairing symmetries
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Reentrance of Topological Phase in Spin-1 Frustrated Heisenberg Chain
• Classification of symmetry-enriched topological quantum spin liquids
• Many-body multipole index and bulk-boundary correspondence
• Defect bulk-boundary correspondence of topological skyrmion phases of matter
• 4D beyond-cohomology topological phase protected by $C_2$ symmetry and its boundary theories
• Anomalous Crystal Shapes of Topological Crystalline Insulators
• Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries
• Computing the ${\mathbb{Z}}_2$ Invariant in Two-Dimensional Strongly-Correlated Systems
• Anomalous Continuous Translations
• Quantized polarization in a generalized Rice-Mele model at arbitrary filling
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Exceptional second-order topological insulators
• Crystalline invariants of fractional Chern insulators
• Projectification of point group symmetries with a background flux and Lieb-Schultz-Mattis theorem
• Intrinsic (Axion) Statistical Topological Insulator
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• (2+1)D topological phases with RT symmetry: many-body invariant, classification, and higher order edge modes
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Disclinations, dislocations, and emanant flux at Dirac criticality
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Frustration-free free fermions
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Frustration-free free fermions and beyond
• Hofstadter Butterflies in Topological Insulators
• Topological Invariants of a Filling-Enforced Quantum Band Insulator
• Construction of Entangled Many-body States via the Higgs Mechanism
• When Wannier centers jump: Critical points between atomic insulating phases