Quasiperiodic Spectra

Hofstadter butterfly (1976): the spectrum of a 2D electron in a magnetic field B is a fractal — for rational flux p/q (in units of flux quantum), the tight-binding band splits into q sub-bands. The plot shows allowed energies E vs. flux α=p/q ∈ [0,1]; the fractal structure has Hausdorff dimension ≈ 0.5.

Fibonacci chain: a 1D quasicrystal with on-site potentials following the Fibonacci sequence (substitution rule A→AB, B→A). The spectrum is a Cantor set of measure zero — purely singular continuous, with no extended eigenstates (critical wavefunctions). The gap structure encodes the golden ratio φ = (1+√5)/2 via gap labeling.