Critical quantum chaos and the one dimensional Harper model

TL;DR

This paper investigates the critical spectral statistics of the one-dimensional Harper model at the mobility edge, revealing a universal, scale-invariant distribution of spectral bands described by semi-Poisson statistics.

Contribution

It provides numerical evidence for a universal critical spectral distribution in the Harper model, connecting multifractal states to semi-Poisson statistics at the mobility edge.

Findings

Spectral band distribution is scale-invariant and semi-Poisson.

The tail of the distribution approaches exp(-2S) near the metal-insulator transition.

The density of states exhibits multifractal criticality with a linear number variance.

Abstract

We study the quasiperiodic Harper's model in order to give further support for a possible universality of the critical spectral statistics. At the mobility edge we numerically obtain a scale-invariant distribution of the bands $S$, which is closely described by a semi-Poisson $P(S)=4S \exp(-2S)$ curve. The $\exp (-2S)$ tail appears when the mobility edge is approached from the metal while $P(S)$ is asymptotically log-normal for the insulator. The multifractal critical density of states also leads to a sub-Poisson linear number variance $\Sigma_{2}(E)\propto 0.041E$.