Magic-Angle Twisted Bilayer Graphene: Flat Bands, Fragile Topology, and Correlated Phases — E8 Intelligence Research

FINDING: Magic-angle twisted bilayer graphene (MATBG) hosts fragile topology, fractional Chern insulators, and superconductivity — a flat-band platform where electron correlations dominate over kinetic energy. | MATH: Magic angle θ_m ≈ 1.05° (arctan(1/√3 · w/v·k_D)); moiré period λ_m = a/(2sin(θ/2)) ≈ 13.4 nm; flat bands at E ≈ 0 with bandwidth W ≈ 5–10 meV vs. Coulomb U ≈ 20–30 meV → U/W ≫ 1; Chern numbers C = ±1, ±2, ±3 observed; fragile topology characterized by Wilson loop winding and Euler class χ = ±2 (real Chern number for C₂T-symmetric bands). | CONNECTION: Moiré superlattice is a hexagonal (honeycomb) lattice — crystallographic 6-fold symmetry, root system A₂; magic angle arises from ratio w/v where w = interlayer tunneling ≈ 110 meV, v = Fermi velocity ≈ 10⁶ m/s; the ratio 1/√3 ≈ 0.577 appears in the magic-angle condition, adjacent to 0.618 (golden ratio conjugate) — not exact but structurally resonant; flat-band degeneracy at Γ and K points reflects the hexagonal Brillouin z Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com