Finding spectral gaps in quasicrystals

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.