Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena

Abstract

We propose that symmetry protected topological (SPT) phases with crystalline symmetry are formulated by equivariant generalized homologies $h_n^G(X)$ over a real space manifold $X$ with $G$ a crystalline symmetry group. The Atiyah-Hirzebruch spectral sequence unifies various notions in crystalline SPT phases such as the layer construction, higher-order SPT phases and Lieb-Schultz-Mattis type theorems. Our formulation is applicable to interacting systems with onsite and crystalline symmetries as well as free fermions.

arXiv comment66 pages, many figures and tables

References in corpus (79)

• Unpaired Majorana fermions in quantum wires
• Classification of topological insulators and superconductors in three spatial dimensions
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Topological Crystalline Insulators
• Quantized Electric Multipole Insulators
• Periodic table for topological insulators and superconductors
• Generalized Global Symmetries
• Higher-Order Topological Insulators
• Topological quantum chemistry
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Reflection symmetric second-order topological insulators and superconductors
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Topological phases of fermions in one dimension
• Classifying quantum phases using Matrix Product States and PEPS
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Braiding statistics approach to symmetry-protected topological phases
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Topology of crystalline insulators and superconductors
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• Theta, Time Reversal, and Temperature
• Fermion Path Integrals And Topological Phases
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• Commensurability, excitation gap and topology in quantum many-particle systems on a periodic lattice
• Classification of topological insulators and superconductors in the presence of reflection symmetry
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Topological classification with additional symmetries from Clifford algebras
• Twisted equivariant matter
• Classification of interacting electronic topological insulators in three dimensions
• Reflection positivity and invertible topological phases
• Filling constraints for spin-orbit coupled insulators in symmorphic and non-symmorphic crystals
• Exactly Soluble Model of a 3D Symmetry Protected Topological Phase of Bosons with Surface Topological Order
• Building crystalline topological phases from lower-dimensional states
• Theta-terms in nonlinear sigma-models
• Topology of nonsymmorphic crystalline insulators and superconductors
• Rotation Anomaly and Topological Crystalline Insulators
• Intrinsic and emergent anomalies at deconfined critical points
• Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases
• Spin-charge Separated Solitons in a Topological Band Insulator
• Symmetry Protected Topological States of Interacting Fermions and Bosons
• Spin Charge Separation in the Quantum Spin Hall State
• Higher order symmetry-protected topological states for interacting bosons and fermions
• Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Symmetry protection of critical phases and global anomaly in $1+1$ dimensions
• All Majorana Models with Translation Symmetry are Supersymmetric
• On the cobordism classification of symmetry protected topological phases
• On the K-theoretic classification of topological phases of matter
• Sufficient Conditions for Topological Order in Insulators
• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
• Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D
• Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases
• Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions
• Classification and construction of higher-order symmetry protected topological phases of interacting bosons
• Symmetry-protected Topological Phases, Generalized Laughlin Argument and Orientifolds
• Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models
• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Minimalist approach to the classification of symmetry protected topological phases
• Distinct trivial phases protected by a point-group symmetry in quantum spin chains
• Filling-enforced constraint on the quantized Hall conductivity on a periodic lattice
• Higher Order Bosonic Topological Phases in Spin Models
• Topological Phases Protected By Reflection Symmetry and Cross-cap States
• CPT theorem and classification of topological insulators and superconductors
• Filling-Enforced Quantum Band Insulators in Spin-Orbit Coupled Crystals
• Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
• Dyonic Lieb-Shultz-Mattis Theorem and Symmetry Protected Topological Phases in Decorated Dimer Models
• Universal Relation among Many-Body Chern Number, Rotation Symmetry, and Filling
• Organizing symmetry-protected topological phases by layering and symmetry reduction: a minimalist perspective
• Bosonic symmetry protected topological phases with reflection symmetry
• Crystallographic T-duality

Cited by in corpus (33)

• Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Crystalline topological phases as defect networks
• Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups
• Generalized Wen-Zee Terms
• Non-zero momentum requires long-range entanglement
• The classification of surface states of topological insulators and superconductors with magnetic point group symmetry
• Searching for Fracton Orders via Symmetry Defect Condensation
• Rotation Symmetry-Protected Topological Phases of Fermions
• Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• AKLT models on decorated square lattices are gapped
• Bosonic Crystalline Symmetry Protected Topological Phases Beyond the Group Cohomology Proposal
• Higher-order topological superconductors from Weyl semimetals
• Effective field theories of topological crystalline insulators and topological crystals
• Interaction-induced crystalline topology of excitons
• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Global anomaly matching in higher-dimensional ${\mathbb{CP}}^{N-1}$ model
• Higher-order topological superconductivity in a topological metal 1T${}^{\prime }$-MoTe${}_2$
• Filling constraints on translation invariant dipole conserving systems
• Classification of time-reversal symmetric topological superconducting phases for conventional pairing symmetries
• A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases
• Higher Berry curvature from matrix product states
• Exciton Berryology
• Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: $E_2$ pages for 1651 magnetic space groups
• General corner charge formulas in various tetrahedral and cubic space groups
• Higher-order topological phases for time-reversal-symmetry breaking superconductivity in UTe${}_2$
• Topological Spectral Bands with Frieze Groups
• Berry Curvature of Low-Energy Excitons in Rhombohedral Graphene
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
• Frustration-free free fermions
• Frustration-free free fermions and beyond
• Classification of topological insulators and superconductors with multiple order-two point group symmetries