Perfect self-similarity of energy spectra and gap-labeling properties in one-dimensional Fibonacci-class quasilattices

Abstract

One-dimensional Fibonacci-class quasilattices are proposed and studied, which are constructed by the substitution rules B→BA , A→BAB . We have proved that this class of binary lattices is self-similar and also quasiperiodic. By the use of the renormalization-group technique, it has been proved that for all Fibonacciclass lattices the electronic energy spectra are perfect self-similar, and the branching rules of spectra are obtained. We analytically prove that each energy gap can be simply labeled by a characteristic integer, i.e., for the Fibonacci-class lattices there is a universal gap-labeling theorem @Phys. Rev. B 46, 9216 ~1992!#. @S0163-1829~97!10705-6#