Perpendicular space accounting of localized states in quasicrystals

Quasicrystals can be described as projections of sections of higher dimensional periodic lattices into real space. The image of the lattice points in the projected-out dimensions, called the perpendicular space, carries valuable information about the local structure of the real space lattice. We use perpendicular space projections to analyze the strictly localized states in four quasicrystal tight-binding models. These zero energy states form a massively degenerate manifold akin to flat bands in periodic systems. For the Penrose lattice, we reproduce the six types of localized states and investigate their overlaps and linear independence. For the Ammann-Beenker and Socolar dodecagonal lattices, an infinite number of localized state types are needed to explain the numerically observed degeneracy. We also consider all pentagonal quasicrystals which have the same projection window as the Penrose lattice and show that the behavior of the zero energy manifold on the two sublattices of the quasicrystal show marked differences.