Effect of imperfections on the hyperuniformity of many-body systems

A hyperuniform many-body system is characterized by a structure factor $S\left(\mathit{k}\right)$ that vanishes in the small-wave-number limit or equivalently by a local number variance ${\sigma }_N^2\left(R\right)$ associated with a spherical window of radius $R$ that grows more slowly than $R^d$ in the large-$R$ limit. Thus, the hyperuniformity implies anomalous suppression of long-wavelength density fluctuations relative to those in typical disordered systems, i.e., ${\sigma }_N^2\left(R\right)\sim R^d$ as $R\to \infty$. Hyperuniform systems include perfect crystals, quasicrystals, and special disordered systems. Disordered hyperuniform systems are amorphous states of matter that lie between a liquid and crystal [S. Torquato et al., Phys. Rev. X 5, 021020 (2015)], and have been the subject of many recent investigations due to their novel properties. In the same way that there is no perfect crystal in practice due to the inevitable presence of imperfections, such as vacancies and dislocations, there is no “perfect” hyperuniform system, whether it is ordered or not. Thus, it is practically and theoretically important to quantitatively understand the extent to which imperfections introduced in a perfectly hyperuniform system can degrade or destroy its hyperuniformity and corresponding physical properties. This paper begins such a program by deriving explicit formulas for $S\left(\mathit{k}\right)$ in the small-wave-number regime for three types of imperfections: (1) uncorrelated point defects, including vacancies and interstitials, (2) stochastic particle displacements, and (3) thermal excitations in the classical harmonic regime. We demonstrate that our results are in excellent agreement with numerical simulations. We find that “uncorrelated” vacancies or interstitials destroy hyperuniformity in proportion to the defect concentration $p$. We show that “uncorrelated” stochastic displacements in perfect lattices can never destroy the hyperuniformity but it can be degraded such that the perturbed lattices fall into class III hyperuniform systems, where ${\sigma }_N^2\left(R\right)\sim R^{d-\alpha }$ as $R\to \infty$ and $0<\alpha <1$. By contrast, we demonstrate that certain “correlated” displacements can make systems nonhyperuniform. For a perfect (ground-state) crystal at zero temperature, increase in temperature $T$ introduces such correlated displacements resulting from thermal excitations, and thus the thermalized crystal becomes nonhyperuniform, even at an arbitrarily low temperature. It is noteworthy that imperfections in disordered hyperuniform systems can be unambiguously detected. Our work provides the theoretical underpinnings to systematically study the effect of imperfections on the physical properties of hyperuniform materials.

Figure 1

Left panel: Configurations of (a) an initially perfectly hyperuniform integer lattice and (b)–(e) imperfect lattices. The different types of imperfections include the following: (b) a single vacancy, (c) a single interstitial defect (denoted by a solid cyan five-pointed star), (d) uncorrelated stochastic particle displacements $u$ via a uniform distribution with the variance $〈u^2〉=0.05$, and (e) thermalized excitations, i.e., elastic waves at $\overline{T}=0.05$, where $\overline{T}$ is a dimensionless temperature (54). Right panel: Corresponding deviations in the number of particles $N(x_0;L)$ inside a window centered at $x_0$, i.e., $\delta N(x_0;L)\equiv N(x_0;L)-〈N(x_0;L)〉$ at two different window sizes $L$. The local number variance ${\sigma }_N^2\left(L\right)$, or equivalently the volume average of $\delta N{(x_0;L)}^2$, measures the degree of density fluctuations at a given length scale $L$. Roughly speaking, a system is nonhyperuniform if ${\sigma }_N^2\left(L\right)$ grows as the window size increases, as in cases (b), (c), and (d). By contrast, the perturbed system (d) is hyperuniform.

Figure 2

From (a)–(c): semilog plots of simulated structure factors of two-dimensional defective point configurations with point vacancies of the fraction $p$. Three types of original configurations include (a) the square lattice, (b) stealthy disordered hyperuniform configurations with $\chi =0.1$, and (c) perturbed square lattices via a distribution $f_1\left(\mathit{u};\delta ={10}^{-4},\alpha =0.8\right)$ given by (35). In (c), theoretical values of $S_0\left(k\right)$ for perturbed lattices (magenta dashed line) are calculated from (36). Insets of each panel zoom into the small-wave-number regime in linear scale, but $y$ axes are rescaled as $\left[S\right(k)-p]/(1-p)$. Note that peaks observed in (a) and (c) correspond to the first two Bragg peaks of the square lattice. Subfigure (d) is comparison of the prediction (20) to corresponding computer simulations of $S\left(0\right)$ of defective systems as functions of the vacancy concentration $p$ for three original systems.

Figure 3

Left panel: Comparison of the predictions of Eq. (25) to the corresponding computer simulations of $S\left(0\right)$ of the defective configurations with the fraction $p$ of interstitials. Right panel: Semilog plot of numerical results for the structure factor of defective point configurations with uncorrelated interstitials. The original perfectly hyperuniform systems are one-dimensional stealthy hyperuniform ground states with $\chi =0.3$ and $K=1$.

Figure 4

Log-log plots of approximate and numerical results for structure factors of perturbed point configurations, which are generated by stochastically displacing each particle by the singlet distribution $f_1\left(\mathit{u};\delta ,\alpha \right)$ defined in Eq. (35). Initial configurations include the following: (a) the integer lattice for $d=1$, (b) the square lattice for $d=2$, and (c), and (d) stealthy disordered hyperuniform point configurations with $\chi =0.1[d=1,2$ for (c) and (d), respectively]. A black solid line in each panel shows $S_0\left(k\right)$ of the initial configurations.

Figure 5

Schematics illustrating particle displacements in Einstein (left) and Debye solids (right) at a positive temperature. For illustrative purposes, displacements are exaggerated. The particles (large blue dots) in an Einstein solid “independently” experience harmonic restoring interactions (blue line) toward their equilibrium positions. By contrast, the particles in a Debye solid interact with their nearest neighbors. In summary, displacements in Einstein and Debye solids are uncorrelated and correlated, respectively, leading to different behaviors in long-wavelength density fluctuations.

Figure 6

Semilog plots of approximate and numerical results for structure factors for thermalized hypercubic lattices calculated from the Monte Carlo technique; (a) $d=1$, (b) $d=2$, and (c) $d=3$. Note that structure factors are normalized by the dimensionless temperature $\overline{T}$ defined by the relation (54). The approximate results are calculated from the formula (53). Insets in each panel are magnifications in the small-wave-number regime.