Energy level statistics of electrons in a 2D quasicrystal

TL;DR

This paper investigates the energy level statistics of electrons in 2D quasicrystals, revealing new spectral behaviors in perfect tilings and confirming random matrix theory predictions in disordered cases.

Contribution

It introduces a novel level statistics pattern for perfect quasicrystal tilings and analyzes the multifractal nature of their spectral measures.

Findings

Perfect tilings exhibit lognormal level spacing distributions.

Disordered tilings follow random matrix theory predictions.

Spectral measures show multifractal characteristics.

Abstract

A numerical study is made of the spectra of a tight-binding hamiltonian on square approximants of the quasiperiodic octagonal tiling. Tilings may be pure or random, with different degrees of phason disorder considered. The level statistics for the randomized tilings follow the predictions of random matrix theory, while for the perfect tilings a new type of level statistics is found. In this case, the first-, second- level spacing distributions are well described by lognormal laws with power law tails for large spacing. In addition, level spacing properties being related to properties of the density of states, the latter quantity is studied and the multifractal character of the spectral measure is exhibited.