Effect of window shape on the detection of hyperuniformity via the local number variance

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 (i.e. L ^d), which may lead one to falsely conclude that the system is non-hyperuniform solely according to the direct-space condition. We begin by numerically investigating the variance of two-dimensional lattices using 'superdisk' windows, whose convex shapes continuously interpolate between circles (p = 1) and squares (p → ∞), as prescribed by a deformation parameter p, when the superdisk symmetry axis is aligned with the lattice. Subsequently, we analyze the variance for lattices as a function of the window orientation, especially for two-dimensional lattices using square windows (superdisk when p → ∞). Based on this analysis, we explain the reason why the variance for d = 2 can grow faster than the window area or even slower than the window perimeter (e.g. like (L)). We then study the generalized condition of the window orientation, under which the variance can grow as fast as or faster than L^d (window volume), to the case of Bravais lattices and parallelepiped windows in ℝ^d. In the case of isotropic disordered hyperuniform systems, we prove that the large-L asymptotic behavior of the variance is independent of the window shape for convex windows. We conclude that the orientationally-averaged variance, instead of the conventional one using windows with a fixed orientation, can be used to resolve the window-shape dependence of the direct-space hyperuniformity condition. We suggest a new direct-space hyperuniformity condition that is valid for any convex window. The analysis on the window orientations 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.