D4 Symmetry and Fermi Arcs in Weyl Semimetals — E8 Intelligence Research

Abstract

FINDING: D4 dihedral group governs square-lattice symmetry and its irreducible representations underpin Weyl-semimetal surface states (Fermi arcs), linking finite-group theory to topological band structure. | MATH: D4 has order 8, 5 conjugacy classes → 5 irreps (4×1-dim, 1×2-dim). Character table: trivial, sign, two 1-dim "rotation" reps (χ(ρ)=±1, χ(σ)=±1), and 2-dim rep with χ(ρ)=0, χ(σ)=0. For Weyl semimetals: Fermi arc connectivity is encoded by the Chern number C = (1/2π)∮_BZ F·d²k, with surface states obeying k·v = 0 at Weyl points. NbP (NbAs family) has 4 Weyl points in BZ, each carrying chirality ±1; total Chern number change across a surface cut = sum of chiralities of enclosed Weyl points. | CONNECTION: D4 is the symmetry of the square lattice — its 2-dim irrep corresponds to the two orthogonal lattice directions (x,y), and its rotation subgroup C4 (order 4) maps to the 90° rotational symmetry of the square BZ. The ratio of the 2-dim irrep dimension to total group order: 2/8 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com