From the Hofstadter to the Fibonacci butterfly

TL;DR

This paper demonstrates that the electronic spectrum of a Fibonacci quasiperiodic chain can be constructed as a superposition of Harper potentials, resulting in a Fibonacci butterfly spectrum with long-range interactions in reciprocal space.

Contribution

It introduces a method to generate Fibonacci spectra by adding harmonics to Harper potentials, linking quasiperiodic and fractal spectra through reciprocal space analysis.

Findings

Fibonacci spectrum can be modeled as a superposition of Harper potentials.

Energy gaps are explained via reciprocal space components.

Localization properties are analyzed through potential correlators.

Abstract

We show that the electronic spectrum of a tight-binding Hamiltonian defined in a quasiperiodic chain with an on-site potential given by a Fibonacci sequence, can be obtained as a superposition of Harper potentials. The electronic spectrum of the Harper equation is a fractal set, known as Hofstadter butterfly. Here we show that is possible to construct a similar butterfly for the Fibonacci potential just by adding harmonics to the Harper potential. As a result, the equations in reciprocal space for the Fibonacci case have the form of a chain with a long range interaction between Fourier components. Then we explore the transformation between both spectra, and specifically the origin of energy gaps due to the analytical calculation of the components in reciprocal space of the potentials. We also calculate some localization properties by finding the correlator of each potential.