Effect of Window Shape on the Detection of Hyperuniformity via the Local Number Variance

Abstract: Hyperuniform many particle systems in d-dimensional space, which includes crystals, quasicrystals, and some exotic disordered systems, are characterized by an anomalous suppression of density fluctuations at large length scales such that the local number variance within a "spherical" observation window grows slower than the window volume. In usual circumstances, this direct space condition is equivalent to the Fourier space hyperuniformity condition that the structure factor vanishes as the wavenumber goes to zero. In this paper, we comprehensively study the effect of aspherical window shapes with characteristic size $L$ on the direct space condition for hyperuniform systems. For lattices, we demonstrate that the variance growth rate can depend on the shape as well as the orientation of the windows, and in some cases, the growth rate can be faster than the window volume ($L^d$), which may lead one to falsely conclude that the system is non-hyperuniform solely according to the direct space condition. We analyze the effect of window orientation on the growth rate, especially for two-dimensional lattices using square windows. Based on this analysis, we explain the reason why the variance can grow faster than the window area ($L^2$) or even slower than the window perimeter ($L$). This analysis demonstrates an example of physical systems exhibiting commensurate-incommensurate transitions and is closely related to problems in number theory (e.g., Diophantine approximation and Gauss' circle problem) and discrepancy theory. We prove that for a hyperuniform system its orientationally-averaged variance, instead of the conventional one using windows with a fixed orientation, has a common large-$L$ asymptotic behavior regardless of the window shape (for convex windows), thus it can be used to resolve the window-shape dependence of the growth rate of the variance on the spherical-window condition.