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\ensuremath{\rightarrow}${\mathrm{B}}^{\mathrm{n}\mathrm{\ensuremath{-}}1}$ A, A\ensuremath{\rightarrow}${\mathrm{B}}^{\mathrm{n}\mathrm{\ensuremath{-}}1}$ AB. 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 Fibonacci-class 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)].