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

TL;DR

This paper analytically characterizes the spectral properties of the Fibonacci chain's tight-binding model using an approximate renormalization group, providing comprehensive insights into its multifractal and spectral measures.

Contribution

It offers the first complete analytical characterization of the Fibonacci chain's spectral properties, unifying previous partial results.

Findings

Analytical expressions for spectral measure and Hausdorff dimension.

Complete description of bandwidth and gaps distribution.

Insights into long-time return probability behavior.

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.