Strictly localized states on a two-dimensional Penrose lattice

Abstract

Infinitely degenerate states at an energy E=0 on a two-dimensional Penrose lattice are investigated in a tight-binding model where atomic orbitals are located at vertices of rhombuses. The states with E=0 are all strictly localized and have amplitudes only on some specific vertices, which are three-edge vertices and some non-three-edge vertices. A lower bound on the fraction of them is calculated analytically as -50\ensuremath{\tau}+81\ensuremath{\approxeq}9.83\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}2}$ [\ensuremath{\tau}=(\ensuremath{\surd}5 +1)/2], which is conjectured to be the exact fraction.