Fibonacci Quasicrystal — Phonon Spectrum
Tight-binding on Fibonacci chain · Fractal Cantor-set spectrum · Self-similar gaps
Fibonacci Chain Parameters
Generation n (Fib. length):
8
Strong hopping tA:
1.00
Weak hopping tB:
0.50
On-site potential V:
0.00
Recompute
Fibonacci substitution:
σ: A→AB, B→A
Produces quasiperiodic sequence with
ratio of A:B bonds → φ = (1+√5)/2 (golden ratio).
Spectrum:
Cantor-set (measure zero, uncountable). Self-similar gaps labeled by Fibonacci numbers.
Kohmoto-Kadanoff-Tang (1983):
spectrum is singular continuous — neither absolutely continuous nor pure point.
Wavefunctions: critical (power-law localized), neither extended nor Anderson-localized.