Dynamic structure factor for the Fibonacci-chain quasicrystal

Abstract

We have performed an exact real-space renormalization-group calculation of the complete wave-vector- and frequency-dependent-response function, S(q,\ensuremath{\omega}), 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,\ensuremath{\omega}), for different values of the coupling ratio. In addition, we have obtained a hierarchy of dispersion curves of \ensuremath{\omega} 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.