Finding spectral gaps in quasicrystals

TL;DR

This paper introduces a reliable algorithm for proving spectral gaps in quasicrystalline Hamiltonians, successfully applied to the Hofstadter model on Ammann-Beenker tilings, advancing understanding of spectral properties in complex systems.

Contribution

The paper presents a novel algorithm that systematically proves spectral gaps in quasicrystals, overcoming previous limitations and applicable to a broad class of systems.

Findings

Proved spectral gaps in the Hofstadter model on Ammann-Beenker tilings.

Demonstrated the algorithm's ability to find previously inconclusive gaps.

Circumvented earlier no-go theorems on spectral gap computability.

Abstract

We present an algorithm for reliably and systematically proving the existence of spectral gaps in Hamiltonians with quasicrystalline order, based on numerical calculations on finite domains. We apply this algorithm to prove that the Hofstadter model on the Ammann-Beenker tiling of the plane has spectral gaps at certain energies, and we are able to prove the existence of a spectral gap where previous numerical results were inconclusive. Our algorithm is applicable to more general systems with finite local complexity and eventually finds all gaps, circumventing an earlier no-go theorem regarding the computability of spectral gaps for general Hamiltonians.