Z₂ Parity: Mod-2 Topological Invariant via Kramers Pairs and Surface Dirac Cones — E8 Intelligence Research

Abstract

FINDING: Z₂ parity in 3D time-reversal-invariant topological insulators is a mod-2 index, not a continuous invariant — it classifies band structures via Kramers pairs and requires odd number of surface Dirac cones. | MATH: Z₂ invariant ν ∈ {0,1} defined via Pfaffian of the time-reversal operator Θ: ν = ∏_{k∈TRIM} δ_k, where δ_k = Pf[⟨u_i(k)|Θ|u_j(k)⟩]/√Det[...] ∈ {±1}; TRIM points = 8 in 3D (2³ from base-2 lattice). Surface states obey E(k) = ±v_F|k| (Dirac cone), with parity constraint ν = N_surface_mod2. | CONNECTION: The 8 TRIM points mirror the 8-fold way of crystallographic point groups (cubic symmetry); the Z₂ group {0,1} is the simplest non-trivial cyclic group, isomorphic to the parity of the 2×2 Pauli matrix σ_z eigenvalues. The ratio 0.5 (mod-2) is the only non-trivial fraction here — no golden ratio appears. | DEPTH: 7 — profound for condensed matter (topological order), but not a new universal constant; it's a discrete symmetry classification, not a harmonic ratio. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com