Green's functions, density of states, and dynamic structure factor for a general one-dimensional quasicrystal.

Abstract

Using an analytic method involving continued fractions, we calculate the site-site, two-time Green's functions for a general one-dimensional crystal. These Green's functions give the electron correlations for a tight-binding model in the noninteracting-electron approximation, or the displacement correlations of a vibrating system with harmonic interatomic forces. The Green's functions can be summed in closed form to give the density of states for either model. The resulting expression depends only on the trace of a ``transfer matrix,'' and on its derivative with respect to energy. We use this formula to determine the densities of states of a large class of quasiperiodic models, and we check our results against perturbation-theory calculations of the band structure of our models. From the Green's functions, we also derive an expression for the dynamic structure factor of our vibrating-lattice model, and we derive a class of iteration relations which allow us to evaluate it very rapidly for quasiperiodic chains. We give results for two different quasicrystals, and compare them against known general restrictions on the form of the dynamic structure factor.