Crystalline Symmetry-Protected Edge Conductance in Hexagonal Topological Insulators — E8 Intelligence Research

FINDING: Topological insulators on hexagonal/C3v lattices exhibit quantized edge conductance protected by crystalline symmetry, not just time-reversal symmetry. | MATH: Chern number \( C = \frac{1}{2\pi}\int_{BZ} F_{xy} d^2k \in \mathbb{Z} \); for C3v-symmetric lattices, mirror Chern number \( C_m = \frac{1}{2}(C_+ - C_-) \) (where \( C_\pm \) are Chern numbers in mirror-even/odd sectors); edge conductance \( G = \frac{e^2}{h} \cdot (C_m \mod 2) \); hexagonal lattice basis vectors \( \mathbf{a}_1 = a(1,0), \mathbf{a}_2 = a(\tfrac{1}{2}, \tfrac{\sqrt{3}}{2}) \) with \( a \) the lattice constant. | CONNECTION: **Strong geometric harmony** — C3v symmetry is the point group of the hexagonal lattice, whose reciprocal lattice is also hexagonal (self-dual). The ratio of the two inequivalent high-symmetry points \( K \) and \( K' \) in the Brillouin zone: \( |K|/|K'| = 1 \) (degenerate by time-reversal). The nearest-neighbor hopping phase \( \phi = 2\pi/3 \) (120° = \( 2\pi/3 \) rad) appears i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com