Energy Levels of Quasiperiodic Hamiltonians, Spectral Unfolding, and Random Matrix Theory

TL;DR

This paper studies the spectral properties of a quasiperiodic Hamiltonian on the Ammann-Beenker tiling, demonstrating how spectral unfolding reveals GOE statistics in level-spacing distributions despite complex density of states.

Contribution

It introduces a spectral unfolding method for quasiperiodic Hamiltonians and analyzes its effect on level-spacing distributions, connecting spectral statistics with random matrix theory.

Findings

Unfolding recovers GOE distribution across spectrum

DOS is spiky but integrated DOS is smooth

Spectral statistics vary with energy intervals

Abstract

We consider a tight-binding Hamiltonian defined on the quasiperiodic Ammann-Beenker tiling. Although the density of states (DOS) is rather spiky, the integrated DOS is quite smooth and can be used to perform spectral unfolding. The effect of unfolding on the integrated level-spacing distribution is investigated for various parts of the spectrum which show different behaviour of the DOS. For energy intervals with approximately constant DOS, we find good agreement with the distribution of the Gaussian orthogonal random matrix ensemble (GOE) even without unfolding. For energy ranges with fluctuating DOS, we observe deviations from the GOE result. After unfolding, we always recover the GOE distribution.