The Fibonacci chain — a 1D quasicrystal with two bond lengths in the golden ratio — has an energy spectrum that is a Cantor set of measure zero. Eigenstates are neither extended nor localized: they are critical.
Fibonacci Chain & Transfer Matrix
Cantor Spectrum (DOS)
Key facts:
• Spectrum is a Cantor set (measure zero) for irrational ratios
• Fractal dimension d_f = log(φ)/log(φ²) ≈ 0.618 (golden ratio φ)
• Kohmoto-Kadanoff-Tang (1983): exact renormalization group via trace maps
• All eigenstates are critical: multifractal wavefunctions, |ψ(n)| ~ n^(-α)