Z₂ Topological Invariants of Time-Reversal-Invariant Superconductors via Bloch Mapping — E8 Intelligence Research

FINDING: The Z₂ topological classification of time-reversal-invariant (TRI) superconductors maps to a Bloch Hamiltonian, yielding one Z₂ invariant in 2D and four in 3D, tied to the eight time-reversal-invariant momenta (TRIM) points on the 3D Brillouin zone torus. | MATH: The mapping is via the superconducting mean-field Hamiltonian \( HBdG(k) \) to a Bloch Hamiltonian \( H(k) \) with particle-hole symmetry \( C \) and time-reversal \( T \), where \( T^2 = -1 \). The Z₂ invariant in 3D is \( ν = (ν_0; ν_1, ν_2, ν_3) \), with \( ν_0 = ∑k_i ∈ TRIM νk_i 2 \) and \( ν_j = νk_j \) for the three pairs of TRIM planes. Each \( νk_i = ± 1 \) is the parity eigenvalue product at that TRIM point (Fu-Kane parity criterion). The TRIM points form a cube in momentum space: \( k_i = (n_1 π, n_2 π, n_3 π) \), \( n_j ∈ \{0,1\} \) — eight points, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com