Symmetry-enforced topological parity: revisiting 2 Z classifications beyond cellwise symmetry actions

arXiv:2609.37106 · cond-mat.mes-hall, math-ph, math.MP · Submitted 2026-09-29 · Read on arXiv

cond-mat.mes-hall, math-ph, math.MP

Submitted: 2026-09-29

Updated: 2026-09-29

Terminology

• Chern Number — 28× in this paper · explained in 4 episodes
• No-Go Theorem — 11× in this paper · explained in 2 episodes
• Redundancy — 9× in this paper · explained in 2 episodes
• Core — 7× in this paper · explained in 2 episodes
• Additivity — 6× in this paper · explained in 1 episode
• Trace — 5× in this paper · explained in 2 episodes
• Hilbert Space — 3× in this paper · explained in 2 episodes
• Holonomy — 2× in this paper · explained in 2 episodes
• Topological Order — 2× in this paper · explained in 2 episodes
• Berry Curvature — 1× in this paper · explained in 3 episodes
• Bosonization — 1× in this paper · explained in 2 episodes
• Consistency — 1× in this paper · explained in 3 episodes
• Degeneracy — 1× in this paper · explained in 3 episodes
• Topological Defects — 1× in this paper · explained in 2 episodes

Sources

• Classification of topological insulators and superconductors in three spatial dimensions
• Periodic table for topological insulators and superconductors
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Classification of topological quantum matter with symmetries
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Gauging spatial symmetries and the classification of topological crystalline phases
• The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models
• Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension
• Compactly-supported Wannier functions and algebraic $K$-theory
• Dyonic Lieb-Shultz-Mattis Theorem and Symmetry Protected Topological Phases in Decorated Dimer Models
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
• Lieb-Schultz-Mattis theorems for symmetry protected topological phases
• Rotation Symmetry-Protected Topological Phases of Fermions
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Topological Insulators with Inversion Symmetry
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Topological phases: isomorphism, homotopy and $K$-theory

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