Propagating modes in quasicrystals

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Abstract
Quasicrystals, although lacking translational periodicity, have δ-function spectra in their diffraction patterns. An appropriate question to ask is whether there are propagating modes of well-defined wavelengths in the neighborhood of these δ functions. To test this, we have investigated the dynamic response S(k,ω) for a model system consisting of ferromagnetically coupled spins situated on the vertices of a Penrose tiling. It is shown that the dispersion relation at low energies is isotropic around the δ functions, with the spin-wave stiffness equal to that around the origin and with the weighting of the low-energy peaks of S(k,ω) correlating with the weighting of the associated Bragg peaks. Analytic relations have been obtained for the first and second moments of the S(k,ω) at low k and used to conjecture an approximate form for S(k,ω). Our results are of relevance to real quasicrystals and have implications for amorphous systems.