Hyperuniformity of Quasicrystals

Erdal C. Oğuz, Joshua E. S. Socolar, Paul J. Steinhardt, Salvatore Torquato

Published 2016-12-06Version 1

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. We employ a new criterion for hyperuniformity to quantitatively characterize quasicrystalline point sets generated by projection methods. Reciprocal space scaling exponents characterizing the hyperuniformity of one-dimensional quasicrystals are computed and shown to be consistent with independent calculations of the scaling exponent characterizing the variance $\sigma^2(R)$ in the number of points contained in an interval of length $2R$. One-dimensional quasicrystals produced by projection from a two-dimensional lattice onto a line of slope $1/\tau$ are shown to fall into distinct classes determined by the width of the projection window. For a countable dense set of widths, $\sigma^2(R)$ is uniformly bounded for large $R$; for all others, $\sigma^2(R)$ scales like $\ln R$. This distinction provides a new classification of one-dimensional quasicrystalline systems and suggests that measures of hyperuniformity may define new classes of quasicrystals in higher dimensions as well.