Z₂ Topological Invariants in Time-Reversal Invariant Superconductors via Parity Eigenvalues — E8 Intelligence Research

FINDING: Z₂ topological invariant in time-reversal invariant (TRI) superconductors is defined via a Bloch Hamiltonian mapping, yielding one invariant in 2D and four in 3D, computed from parity eigenvalues at time-reversal invariant momenta (TRIM). | MATH: For a TRI superconductor, the BdG Hamiltonian maps to a Bloch Hamiltonian \(H(k)\) with \( Θ^2 = -1 \). The Z₂ invariant \( u\) is given by \((-1)^ u = ∏i ∈ TRIM δ_i\), where \(δ_i = ∏ₘ ξ₂ₘ(Γ_i)\) (product of parity eigenvalues of occupied Kramers pairs at TRIM \(Γ_i\)). In 2D: one Z₂; in 3D: four Z₂ invariants \((\ u_0; \ u_1, \ u_2, \ u_3)\), with \(\ u_0 = ∏all 8 TRIM δ_i\) and \(\ u_j = ∏TRIM in plane k_j=0 δ_i\). | CONNECTION: TRIM points form a hypercubic lattice in momentum space — a direct crystallographic Bravais lattice (e.g., 8 points in 3D = vertices of a cube, symmetry of the cubic root system \(B_3\)). The parity product Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com