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