Hofstadter butterfly of a quasicrystal

TL;DR

This paper explores the Hofstadter butterfly spectrum in a two-dimensional quasicrystal, revealing complex gap structures and the effects of disorder on quantum Hall states.

Contribution

It provides the first detailed analysis of the Hofstadter butterfly in a quasiperiodic Rauzy tiling, highlighting unique gap labeling and disorder effects.

Findings

Rich pattern of bulk gaps labeled by four integers

Main quantum Hall gaps are robust against phason-flip disorder

Disorder preserves only the primary quantum Hall gaps

Abstract

The energy spectrum of a tight-binding Hamiltonian is studied for the two-dimensional quasiperiodic Rauzy tiling in a perpendicular magnetic field. This spectrum known as a Hofstadter butterfly displays a very rich pattern of bulk gaps that are labeled by four integers, instead of two for periodic systems. The role of phason-flip disorder is also investigated in order to extract genuinely quasiperiodic properties. This geometric disorder is found to only preserve main quantum Hall gaps.