Z₂ Invariant as Parity of Chern Number via Pfaffian on Dₙ Lattice — E8 Intelligence Research

FINDING: The Z₂ invariant for time-reversal-symmetric topological insulators is the parity of the first Chern number, computable via the Pfaffian of the sewing matrix on a Dₙ lattice; the Chern number itself is the integral of the Berry curvature over the Brillouin torus. | MATH: Chern number \( C = 1/2π∫BZ Fxy \, dk_x dk_y \), where \( Fxy = ∂_x A_y - ∂_y A_x \), \( A_μ = i u_k | ∂_μ | u_k \). Z₂ index \( u = 1/2π i ∮∂ half-BZ Tr[A] \, dk 2 \), equivalently \( (-1)^ u = {Pf[w(Λ_a)]}{√[w(Λ_a)]} \) at time-reversal invariant momenta \( Λ_a \), with \( wₘₙ(k) = um,-k | Θ | un,k \). For Dₙ lattices (n=2,3,4,6), the parity of C is fixed by the lattice point group: \( C 2 = 1/2∑_a n_a 2 \), where \( n_a \) is the number of occupied Kramers pairs at \( Λ_a \). | Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com