A 1D quasicrystal with two tile lengths τ:1 (golden ratio). Generated by substitution rule A→AB, B→A. Cantor-set energy spectrum and self-similar wavefunctions reveal exotic quantum properties.
Chain Structure
Energy Spectrum
Diffraction
Parameters
τ = (1+√5)/2 ≈ 1.618 A → AB, B → A F_n = F_{n-1} + F_{n-2} H = Σ t·|n⟩⟨n+1| + h.c.
Chain Properties
Sites N---
A tiles : B tiles---
Ratio → τ---
Spectral typeSingular cts.
τ = (1+√5)/21.61803
Nobel Prize 2011:
Dan Shechtman — quasicrystals
Diffraction: 5-fold symmetry
+ sharp peaks (no periodicity)
Spectrum is Cantor set:
measure zero, uncountable