Branching rules of the energy spectrum of one-dimensional quasicrystals

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Abstract
By means of the renormalization-group approach, we give an analytic explanation for the branching rules of the electronic energy spectrum of the on-site model for one-dimensional Fibonacci quasicrystals, which has been found by numerical simulation previously by Liu and Riklund [Phys. Rev. B 35, 6034 (1987)] and by Karmakar et al. [J. Phys. Condens. Matter 1, 1423 (1989)]. We prove that the energy spectrum has a four-subband global structure and that each subband further trifurcates following a hierarchy of splitting from one to three subsubbands. The occupation probabilities of subbands and the absolute heights of steps that distinguish the subbands in the integrated density of states are analytically obtained.