Density of states for a two-dimensional Penrose lattice: Evidence of a strong Van-Hove singularity

Abstract

The density of states for a tight-binding Hamiltonian on a two-dimensional Penrose lattice is computed numerically by the continued fraction recursion method. The result shows evidence of a strong Van Hove–type singularity which is remarkable for a system possessing no long-range periodic translational order. By a method of finite-size scaling extrapolation the exponent α and amplitude C, where ρ(E)∼${\mathrm{CE}}^{\mathrm{-}\mathrm{\alpha }}$, are estimated to be α=0.09±0.05 and C=exp(3.8±0.2).