Topological Weyl Semimetals: Symmetry, Z2 Invariants, and Fermi Arcs — E8 Intelligence Research

FINDING: Weyl semimetals exhibit topologically protected band crossings (Weyl nodes) that require time-reversal symmetry breaking for stabilization, with the Z2 invariant governing surface Fermi arcs. | MATH: Weyl node dispersion \( E(\mathbf{k}) = \pm v_F |\mathbf{k} - \mathbf{k}_0| \) (linear, chiral); Chern number \( C = \frac{1}{2\pi}\oint \mathbf{B}(\mathbf{k}) \cdot d\mathbf{S} \) per node; Z2 invariant \( \nu = \frac{1}{2\pi}\oint_{\partial \text{BZ}} \mathbf{A} \cdot d\mathbf{k} \mod 2 \); time-reversal operator \( \mathcal{T}^2 = -1 \) forces Kramers degeneracy at TRIM points. | CONNECTION: Weyl nodes come in pairs of opposite chirality — the net Chern number sums to zero, mirroring the root system \( A_1 \) (two opposite roots) of SU(2). The three-fold degeneracy mentioned (Vergniory) at \( \mathbf{p} \) and \( -\mathbf{p} \) under time-reversal suggests a \( \mathbb{Z}_3 \) structure, echoing the triality of \( D_4 \) root lattice (24 roots, 3-fold symmetry axes). The lattic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com