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 \ensuremath{\alpha} and amplitude C, where \ensuremath{\rho}(E)\ensuremath{\sim}${\mathrm{CE}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$, are estimated to be \ensuremath{\alpha}=0.09\ifmmode\pm\else\textpm\fi{}0.05 and C=exp(3.8\ifmmode\pm\else\textpm\fi{}0.2).