Z₂ Topological Invariants in Time-Reversal Invariant Superconductors via Parity Eigenvalues — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: 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 \(H(\mathbf{k})\) with \( \Theta^2 = -1 \). The Z₂ invariant \( u\) is given by \((-1)^ u = \prod_{i \in \text{TRIM}} \delta_i\), where \(\delta_i = \prod_{m} \xi_{2m}(\Gamma_i)\) (product of parity eigenvalues of occupied Kramers pairs at TRIM \(\Gamma_i\)). In 2D: one Z₂; in 3D: four Z₂ invariants \((\ u_0; \ u_1, \ u_2, \ u_3)\), with \(\ u_0 = \prod_{\text{all 8 TRIM}} \delta_i\) and \(\ u_j = \prod_{\text{TRIM in plane } k_j=0} \delta_i\). | 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
Author: Caldin, Andrew Stewart
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221316909
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