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 \( C = \frac{1}{2\pi}\int_{BZ} \mathcal{F}_{xy} \, dk_x dk_y \), where \( \mathcal{F}_{xy} = \partial_x A_y - \partial_y A_x \), \( A_\mu = i\langle u_k | \partial_\mu | u_k \rangle \). Z₂ index \( u = \frac{1}{2\pi i} \oint_{\partial \text{half-BZ}} \text{Tr}[\mathcal{A}] \, dk \mod 2 \), equivalently \( (-1)^ u = \frac{\text{Pf}[w(\Lambda_a)]}{\sqrt{\det[w(\Lambda_a)]}} \) at time-reversal invariant momenta \( \Lambda_a \), with \( w_{mn}(k) = \langle u_{m,-k} | \Theta | u_{n,k} \rangle \). For Dₙ lattices (n=2,3,4,6), the parity of C is fixed by the lattice point group: \( C \mod 2 = \frac{1}{2}\sum_a n_a \mod 2 \), where \( n_a \) is the number of occupied Kramers pairs at \( \Lambda_a \). | Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com