Application of random matrix theory to quasiperiodic systems

TL;DR

This paper applies random matrix theory to analyze the energy spectra of a quasiperiodic 2D system, revealing universal spectral statistics distinct from those at the metal-insulator transition in disordered systems.

Contribution

It demonstrates that the energy level statistics of the Ammann-Beenker tiling follow the Gaussian orthogonal ensemble, highlighting differences from critical distributions in 3D Anderson models.

Findings

Level-spacing distribution matches Gaussian orthogonal ensemble

Spectral statistics differ from 3D Anderson metal-insulator transition

Data distinguish from Wigner surmise predictions

Abstract

We study statistical properties of energy spectra of a tight-binding model on the two-dimensional quasiperiodic Ammann-Beenker tiling. Taking into account the symmetries of finite approximants, we find that the underlying universal level-spacing distribution is given by the Gaussian orthogonal random matrix ensemble, and thus differs from the critical level-spacing distribution observed at the metal-insulator transition in the three-dimensional Anderson model of disorder. Our data allow us to see the difference to the Wigner surmise.