Dirac Cones and Topological Invariants in Graphene's Hexagonal Lattice — E8 Intelligence Research

Abstract

FINDING: Graphene's hexagonal lattice yields Dirac cones at K/K' points; bulk-boundary correspondence links edge-state parity to topological invariants via Fredholm index theory. | MATH: Hexagonal Bravais lattice: primitive vectors **a₁** = a(1,0), **a₂** = a(1/2, √3/2); reciprocal lattice vectors **b₁** = (2π/a)(1, −1/√3), **b₂** = (2π/a)(0, 2/√3). Dirac points at K = (2π/3a)(1, √3), K′ = (2π/3a)(1, −√3). Low-energy Hamiltonian: H = ℏv_F (σ_x k_x + σ_y k_y) — massless Dirac equation, v_F ≈ 10⁶ m/s. Bulk-boundary correspondence: n_edge = (1/2π)∮ dk · Tr[A(k)] mod 2 (Z₂ invariant), or via Fredholm index: Ind(P) = dim ker(P) − dim coker(P) = ν ∈ ℤ. Klein tunneling: T = 1 for normal incidence (perfect transmission through barrier, due to pseudospin conservation). | CONNECTION: Hexagonal lattice ↔ honeycomb = 2D triangular Bravais with 2-atom basis. Reciprocal space is hexagonal (60° symmetry, C₆v point group). Dirac cone linear dispersion is a conical intersection — the 2D analogue of Wey Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com