Electronic States on a Penrose Lattice

Abstract

We define a hopping Hamiltonian for independent electrons on a two-dimensional quasiperiodic Penrose lattice. This problem is then investigated numerically, up to systems of 3126 sites, and for various boundary conditions. We find the following results for the density of states: (1) There is a central peak of zero width at zero energy, consisting of about 10% of the total number of states. (2) These states are strictly localized; we calculate the wave functions explicitly. (3) The remainder of the states lie in two bands, symmetric about zero energy, separated from the localized states by a finite gap $E_0$.