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

Abstract

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 \( H_{\text{BdG}}(\mathbf{k}) \) to a Bloch Hamiltonian \( H(\mathbf{k}) \) with particle-hole symmetry \( \mathcal{C} \) and time-reversal \( \mathcal{T} \), where \( \mathcal{T}^2 = -1 \). The Z₂ invariant in 3D is \( \nu = (\nu_0; \nu_1, \nu_2, \nu_3) \), with \( \nu_0 = \sum_{\mathbf{k}_i \in \text{TRIM}} \nu_{\mathbf{k}_i} \mod 2 \) and \( \nu_j = \nu_{\mathbf{k}_j} \) for the three pairs of TRIM planes. Each \( \nu_{\mathbf{k}_i} = \pm 1 \) is the parity eigenvalue product at that TRIM point (Fu-Kane parity criterion). The TRIM points form a cube in momentum space: \( \mathbf{k}_i = (n_1 \pi, n_2 \pi, n_3 \pi) \), \( n_j \in \{0,1\} \) — eight points, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com