Quantifying when hyperuniformity of a many-particle system leads to uniformity across length scales

TL;DR

The paper addresses how hyperuniformity translates into uniformity across shorter length scales by analyzing the scaled local number variance $\Sigma^2(R)$ for representative Class I–III hyperuniform systems in 2D and 3D. It develops exact expressions and asymptotic corrections for diverse models (lattices, quasicrystals, stealthy disordered states, one-component plasmas, Fermi-sphere processes, and perturbed lattices), revealing that Class I systems exhibit integer-power corrections in $1/R$, Class II systems exhibit $1/\ln R$ corrections, and Class III systems exhibit $R^{-\alpha}$ corrections with $0<\alpha<1$. A cumulative moving average $\overline{\Lambda}(L)$ is introduced to quantify convergence to large-$R$ hyperuniform behavior in finite samples, yielding practical guidance for interpreting experiments and simulations. The results illuminate how local uniformity emerges from global hyperuniformity and suggest design principles for materials with enhanced transport and optical properties due to improved uniformity at intermediate scales.

Abstract

Hyperuniform systems are distinguished by an unusually strong suppression of large-scale density fluctuations and, consequently, display a high degree of uniformity at the largest length scales. In some cases, however, enhanced uniformity is expected to be present even at intermediate and possibly small length scales. There exist three different classes of hyperuniform systems, where class I and class III are the strongest and weakest forms, respectively. We utilize the local number variance $σ_N^2(R)$ associated with a window of radius $R$ as a diagnostic to quantify the approach to the asymptotic large-$R$ hyperuniform scaling of a variety of class I, II, and III systems. We find, for all class I systems we analyzed, including crystals, quasicrystals, disordered stealthy hyperuniform systems, and the one-component plasma, a faster approach to the asymptotic scaling of $σ_N^2(R)$, governed by corrections with integer powers of $1/R$. Thus, we conclude this represents the highest degree of effective uniformity from small to large length scales. Class II systems, such as Fermi-sphere point processes, are characterized by logarithmic $1/\ln(R)$ corrections and, consequently, a lower degree of local uniformity. Class III systems, such as perturbed lattice patterns, present an asymptotic scaling of $1/R^α$, $0 < α< 1$, implying, curiously, an intermediate degree of local uniformity. In addition, our study provides insight into when experimental and numerical finite systems are representative of large-scale behavior. Our findings may thereby facilitate the design of hyperuniform systems with enhanced physical properties arising from local uniformity.

Figures (13)

• Figure 1: Two-dimensional representative configurations for each of the models that we consider at unit number density: (a) square lattice, (b) kagome lattice, (c) Penrose tiling quasicrystal, (d) $g_2$-invariant step function, (e) stealthy hyperuniform ground state with $\chi = 0.0025$, (f) stealthy hyperuniform ground state with $\chi = 0.45$, (g) one-component plasma, (h) Fermi-sphere point process, and (i) perturbed lattice with $\alpha = 0.25$ and $\delta = 0.0001$.
• Figure 2: The scaled local variance $\sigma_N^2(R)/R^2$ versus $R/D$ for the simple cubic lattice, where $D$ is the nearest-neighbor distance, and we take $\phi=\pi/6\approx 0.5236$. The average ${\overline \Lambda}(\infty)=0.83750$ is indicated as a black horizontal line.
• Figure 3: ${\overline \Lambda}(L)$ versus $L/D$ for the simple cubic lattice, where $D$ is the nearest-neighbor distance, and we take $\phi=\pi/6\approx 0.5236$.
• Figure 4: The scaled local variance $\sigma_N^2(R)/R$ versus $R/D$ for the kagome crystal, where $D$ is the nearest-neighbor distance and we take $\phi=3\pi/(8\sqrt{3})\approx 0.6802$. The average ${\overline \Lambda}(\infty)=0.48411\ldots$ is indicated as a black horizontal line.
• Figure 5: ${\overline \Lambda}(L)$ versus $L/D$ for the kagome crystal, where $D$ is the nearest-neighbor distance and we take $\phi=3\pi/(8\sqrt{3})\approx 0.6802$.