Hyperuniformity of Quasicrystals

Abstract: Hyperuniform systems, which include crystals, quasicrystals and special disordered systems, have attracted considerable recent attention, but rigorous analyses of the hyperuniformity of quasicrystals have been lacking because the support of the spectral intensity is dense and discontinuous. We employ the integrated spectral intensity, $Z(k)$, to quantitatively characterize the hyperuniformity of quasicrystalline point sets generated by projection methods. The scaling of $Z(k)$ as $k$ tends to zero is computed for one-dimensional quasicrystals and shown to be consistent with independent calculations of the variance, $\sigma^2(R)$, in the number of points contained in an interval of length $2R$. We find that one-dimensional quasicrystals produced by projection from a two-dimensional lattice onto a line of slope $1/\tau$ fall into distinct classes determined by the width of the projection window. For a countable dense set of widths, $Z(k) \sim k^4$; for all others, $Z(k)\sim k^2$. This distinction suggests that measures of hyperuniformity define new classes of quasicrystals in higher dimensions as well.