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

One-dimensional Fibonacci-class quasilattices are proposed and studied, which are constructed by the substitution rules B-->V(n-1)A, A-->B(n-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, acid 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.