Parity Criterion and FKMM Invariant Classify Topological Insulators via TRIMs — E8 Intelligence Research

Abstract

FINDING: The Fu-Kane parity criterion and its cohomological generalization (FKMM invariant) classify time-reversal-invariant topological insulators via parity eigenvalues at time-reversal invariant momenta (TRIMs), constrained by the cube's Oh symmetry group. | MATH: Fu-Kane invariant: ν₀ = ∏_{Γᵢ∈TRIMs} δ(Γᵢ), where δ(Γᵢ) = ±1 is the parity eigenvalue product over occupied Kramers pairs; Z₂ classification for class AII (odd TRS, T² = −1). TRIMs are the 8 points {Γ, X, M, R} in the 3D Brillouin zone (BZ) — specifically Γ=(0,0,0), X=(π,0,0) and permutations, M=(π,π,0) and permutations, R=(π,π,π) — forming a cube in reciprocal space. The Oh group (order 48) acts on these 8 points; the invariant is invariant under this symmetry. Cohomological generalization: FKMM-invariant ∈ H¹(B, Z₂) for quaternionic vector bundles over involutive base spaces, extending the Z₂ to a full cohomology class. | CONNECTION: The 8 TRIMs form a cube — the vertices of a simple cubic lattice in reciprocal space. Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com