Spectral structure of two-dimensional Fibonacci quasilattices

Abstract

By means of a decomposition-decimation method based on the renormalization-group technique, we have studied the spectral properties of two-dimensional Fibonacci quasilattices. It is found that the spectrum of two-dimensional Fibonacci quasilattices has a variety of multifurcating structures. The analytic results show that, up to the third hierarchy of the spectrum, there are three kinds of branching types: Type I corresponds to a 1:5 (one-split-into-five) subband structures; type II to a 1:3 (one-to-three) subband structure; and type III to a 1:9 (one-to-nine) subband structures. We have also predicted the branching rules of even higher hierarchies of the spectrum. These analytic results are confirmed by numerical simulations.