Fractal energy spectrum under magnetic flux — Chern numbers color-coded
Hofstadter (1976): electrons on a 2D lattice in magnetic flux Φ = p/q (in units of Φ₀) produce a fractal spectrum — the "butterfly." Each gap is labeled by an integer Chern number C (TKNN formula), which equals the number of filled Landau-like bands and gives σ_xy = Ce²/h. Fractional Chern insulators at fractional filling mimic the fractional quantum Hall effect without Landau levels.