Hyperuniformity Classes of Quasiperiodic Tilings via Diffusion Spreadability

Abstract: Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent $\alpha > 0$, that characterizes the power-law scaling behavior of the structure factor $S(\mathbf{k})$ as a function of wavenumber $k\equiv|\mathbf{k}|$ in the vicinity of the origin, e.g., $S(\mathbf{k})\sim|\mathbf{k}|^{\alpha}$ in cases where $S(\mathbf{k})$ varies continuously with $k$ as $k\rightarrow0$. In this paper, we show that the spreadability is an effective method for determining $\alpha$ for quasiperiodic systems where $S(\mathbf{k})$ is discontinuous and consists of a dense set of Bragg peaks. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract $\alpha$ for the 1D limit-periodic period doubling chain and the 1D quasicrystalline Fibonacci chain to within $0.02\%$ of the analytically known exact results. Moreover, we obtain a value of $\alpha = 5.97\pm0.06$ for the 2D Penrose tiling, which had not been computed previously. We also show that one can truncate the small-$k$ region of the scattering information used to compute the spreadability and still obtain an accurate value of $\alpha$. The methods described here offer a simple way to characterize the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in the spectral density can be used to estimate their physical properties. (abridged)