Strictly localized states in the octagonal Ammann-Beenker quasicrystal

Abstract

Ammann-Beenker lattice is a two-dimensional quasicrystal with eightfold symmetry, which can be described as a projection of a cut from a four-dimensional simple cubic lattice. We consider the vertex tight-binding model on this lattice and investigate the strictly localized states at the center of the spectrum. We use a numerical method based on the generation of finite lattices around a given perpendicular space point and QR decomposition of the Hamiltonian to count the strictly localized states. We apply this method to count the frequency of localized states in lattices of up to 100 000 sites. We obtain an orthogonal set of compact localized states by diagonalizing the position operator projected onto the manifold spanned by the zero-energy states. We identify 20 localized state types and calculate their exact frequencies through their perpendicular space images. Unlike the Penrose lattice, all the localized state types are eightfold symmetric around an eight edge vertex, and all vertex types can support localized states. The total frequency of these 20 types gives a lower bound of ${f}_{\mathrm{LS}}=30\phantom{\rule{0.16em}{0ex}}796\ensuremath{-}21\phantom{\rule{0.16em}{0ex}}776\sqrt{2}\ensuremath{\simeq}0.085\phantom{\rule{0.16em}{0ex}}47$ for the fraction of strictly localized states in the spectrum. This value is in agreement with the numerical calculation and very close to the recently conjectured exact fraction of localized states ${f}_{\mathrm{Ex}}=3/2\ensuremath{-}\sqrt{2}\ensuremath{\simeq}0.085\phantom{\rule{0.16em}{0ex}}79$.