Dynamic structure factor for the Fibonacci-chain quasicrystal

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Abstract
We have performed an exact real-space renormalization-group calculation of the complete wave-vector- and frequency-dependent-response function, S(q,ω), for the one-dimensional Fibonacci-chain quasicrystal with ‘‘Goldstone’’ dynamics as in the case of phonons or magnons. We present surface plots, which are highly structured, of the full response S(q,ω), for different values of the coupling ratio. In addition, we have obtained a hierarchy of dispersion curves of ω versus q which contain features related to the gap structure of the excitation spectrum. The limiting case of equal couplings, while still nontrivial because of geometric effects, is analytically tractable and gives a useful test of the renormalization-group treatment, of which it is a special case, and when extended using degenerate perturbation theory it provides an interpretation of the general situation.