Topological \(\mathbb{Z}_2\) Invariants in 2D and 3D Time-Reversal-Invariant Superconductors via C*-Algebraic Classification — E8 Intelligence Research

Abstract

FINDING: Topological classification of time-reversal-invariant superconductors yields a single \(\mathbb{Z}_2\) invariant in 2D and four \(\mathbb{Z}_2\) invariants in 3D, with a parallel C*-algebraic scheme classifying topological phases via homotopy classes of sections over pure-state fiber bundles, and a complete enumeration of Weyl group sections satisfying braid relations. | MATH: 2D: \(\nu \in \mathbb{Z}_2\); 3D: \((\nu_1,\nu_2,\nu_3,\nu_4) \in \mathbb{Z}_2^4\). Bloch Hamiltonian mapping: \(H_{\text{MF}}(\mathbf{k}) \mapsto H_{\text{Bloch}}(\mathbf{k})\), with particle-hole symmetry \(C H(\mathbf{k}) C^{-1} = -H(-\mathbf{k})\) and time-reversal \(T H(\mathbf{k}) T^{-1} = H(-\mathbf{k})\), \(T^2=-1\). Weyl group \(B_3\): order \(2^3 \cdot 3! = 48\), root system \(\{\pm e_i \pm e_j, \pm e_i\}\), Cartan matrix \(\begin{pmatrix}2&-1&0\\-1&2&-2\\0&-1&2\end{pmatrix}\). Sections of Weyl group: count and partial order from arXiv:1912.07665 — braid relations \(s_i s_{i+1} s_i = s_{i+1} s_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com