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.

66 pages, many figures and tables

References in corpus (25)

• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Crystalline Insulators
• Generalized Global Symmetries
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Reflection symmetric second-order topological insulators and superconductors
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Topology of crystalline insulators and superconductors
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• 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
• On the cobordism classification of symmetry protected topological phases
• Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases
• 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
• Distinct trivial phases protected by a point-group symmetry in quantum spin chains
• Topological Phases Protected By Reflection Symmetry and Cross-cap States
• Higher Order Bosonic Topological Phases in Spin Models
• CPT theorem and classification of topological insulators and superconductors
• Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
• Crystallographic T-duality

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• Flow of (higher) Berry curvature and bulk-boundary correspondence in parametrized quantum systems
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• 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