FKM Invariant as Spin Structure Obstruction in Time-Reversal Insulators — E8 Intelligence Research

Abstract

FINDING: The Fu-Kane-Mele (FKM) invariant is a cohomological obstruction — specifically a second Stiefel-Whitney class — that detects whether a time-reversal-invariant (class AII) insulator admits a globally consistent spin structure, with parity constraints at time-reversal invariant momenta (TRIMs). | MATH: FKM invariant \( \nu \in \mathbb{Z}_2 \) equals \( \int_{\text{TRIM}} w_2(TM) \mod 2 \), where \( w_2 \) is the second Stiefel-Whitney class; equivalently \( \nu = \prod_{\text{TRIM}} \delta_{a_i} \) with \( \delta_{a_i} = \pm 1 \) from parity eigenvalues. The FKMM-invariant generalizes this to quaternionic vector bundles over involutive base spaces, living in \( H^2(B,\mathbb{Z}_2) \). | CONNECTION: The \( \mathbb{Z}_2 \) parity structure at TRIMs mirrors the \( \mathbb{Z}_2 \) grading of Clifford algebras \( Cl_{0,2} \cong \mathbb{R}(2) \) — the same algebra underlying quaternion units \( i,j,k \) with \( i^2=j^2=k^2=-1 \). The TRIM lattice is a hypercubic sublattice of the Bril Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com