Exact decimation approach to the Green’s functions of the Fibonacci-chain quasicrystal

Using real-space renormalization-group techniques, an exact decimation transformation has been derived for the parameters characterizing the phonon or tight-binding properties of a one-dimensional quasicrystal, the Fibonacci chain, for the discrete rescaling factor {τ} (the golden mean). A Green's function formalism is presented which, when combined with the rescaling transformation, provides an a priori exact determination of the Green's functions Gᵢⱼ. Iteration of the transformation provides numerical results for the density of states {ρ}(E) [or {ρ}(ω²)], and its scaling properties near the upper and lower edges of any gap have been related to the cyclic behavior of our renormalization-group mapping.