Local Density Fluctuations, Hyperuniformity, and Order Metrics

TL;DR

This paper investigates the variance in point distributions within windows to understand hyperuniform systems, revealing their critical nature, optimal patterns, and proposing local variance as an order metric across dimensions.

Contribution

It introduces a new perspective on hyperuniformity as a critical point with long-range direct correlation functions and compares various lattice and disordered patterns for variance minimization.

Findings

Periodic linear arrays minimize variance in 1D hyperuniform patterns.

The body-centered cubic lattice has lower variance than face-centered cubic in 3D.

Certain disordered hyperuniform patterns have exactly evaluated correlation functions and critical exponents.

Abstract

We study the variance in the number of points contained within a window $\Omega$ of arbitrary size, and to further illuminate our understanding of {\it hyperuniform} systems, i.e., point patterns that do not possess long-wavelength fluctuations. For large windows, hyperuniform systems are characterized by a local variance that grows only as the surface area (rather than the volume) of the window. We show that a homogeneous point pattern in a hyperuniform state is at a ``critical-point'' of a type with appropriate scaling laws and critical exponents, but one in which the {\it direct correlation function} (rather than the pair correlation function) is long-ranged.variance.We prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one-dimensional hyperuniform patterns. We also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one, two, and three dimensions for spherical windows.Our results suggest that the local variance may serve as a useful order metric for general point patterns. Contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension, we show that for $d=3$, the body-centered cubic lattice has a smaller variance than the face-centered cubic lattice. Finally, for certain hyperuniform disordered point patterns, we evaluate the direct correlation function, structure factor, and associated critical exponents exactly.