Fractal dimensions of the wavefunctions and local spectral measures on the Fibonacci chain

TL;DR

This paper develops a theoretical framework to analyze the fractal properties of wavefunctions and spectral measures in the Fibonacci chain, revealing new symmetries and providing analytical expressions that match numerical data.

Contribution

It introduces a novel analytical approach for understanding wavefunction fractality and spectral measures in quasiperiodic systems, especially in the strong modulation limit.

Findings

Analytical expressions for fractal exponents of wavefunctions

Discovery of a new symmetry under permutation of site and energy indices

Good agreement between theory and numerical data even outside perturbative regime

Abstract

We present a theoretical framework for understanding the wavefunctions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong modulation of the hopping amplitudes, are in good agreement with published numerical data. In the perturbative limit, we show a new symmetry of wavefunctions under permutation of site and energy indices. We compute the wavefunction renormalization factors and from them deduce analytical expressions for the fractal exponents corresponding to individual wavefunctions, as well as their global averages. The multifractality of wavefunctions is seen to appear at next-to-leading order in the ratio of the hopping amplitudes, $\rho$. Exponents for the local spectral density are given, in extremely good accord with numerical calculations. Interestingly, our analytical results for exponents are observed to describe the system rather well even for values of $\rho$ well outside the domain of applicability of perturbation theory.