Critical Level Statistics of the Fibonacci Model

TL;DR

This paper investigates the spectral properties of the Fibonacci quasiperiodic model, revealing specific power-law and exponential distributions of energy levels and gaps, and compares these with multi-scale Cantor sets to highlight qualitative differences.

Contribution

It provides a detailed numerical analysis of the spectral statistics of the Fibonacci model, identifying characteristic distribution laws and contrasting them with multi-scale Cantor sets.

Findings

Energy level distributions follow power-law and exponential laws.

Gap distributions exhibit power-law behavior at small s.

Spectral differences are observed between Fibonacci model and Cantor sets.

Abstract

We numerically analyze spectral properties of the Fibonacci model which is a one-dimensional quasiperiodic system. We find that the energy levels of this model have the distribution of the band widths $w$ obeys $P_B(w)\sim w^{\alpha}$ $(w\to 0)$ and $P_B(w) \sim e^{-\beta w}$ $(w\to\infty)$, the gap distribution $P_G(s)\sim s^{-\delta}$ $(s\to 0)$ ($\alpha,\beta,\delta >0$) . We also compare the results with those of multi-scale Cantor sets. We find qualitative differences between the spectra of the Fibonacci model and the multi-scale Cantor sets.