Analytical results for Scaling Properties of the Spectrum of the Fibonacci Chain

Abstract

We solve the approximate renormalisation group found by Qiu Niu and Franco Nori(Phys. Rev. Lett. 57 2057(1986)) for a tight-binding hamiltonian on the Fibonacci chain. This enables us to characterize analytically as completely as possible the spectral properties of this model, such as the spectral measure, the bandwidth distribution, the Lebesgue measure exponent, the Hausdorff dimension, the multifractal scaling, the gaps distribution as well as the long time return probability. Our results, qualitatively and quantitatively complete and unify previous works on similar models.