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A hyperuniform many-body system is characterized by a structure factor $S\left(\mathbit{k}\right)$ that vanishes in the small-wave-number limit or equivalently by a local number variance ${\ensuremath{\sigma}}_{N}^{2}\left(R\right)$ 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., ${\ensuremath{\sigma}}_{N}^{2}\left(R\right)\ensuremath{\sim}{R}^{d}$ as $R\ensuremath{\rightarrow}\ensuremath{\infty}$. 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 $S\left(\mathbit{k}\right)$ 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 ${\ensuremath{\sigma}}_{N}^{2}\left(R\right)\ensuremath{\sim}{R}^{d\ensuremath{-}\ensuremath{\alpha}}$ as $R\ensuremath{\rightarrow}\ensuremath{\infty}$ and $0l\ensuremath{\alpha}l1$. 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.

Spatial fluctuations of the local electric field induced by a constant applied electric field in composite media are studied analytically and numerically. It is found that the density of states for the fields exhibit sharp peaks and abrupt changes in the slope at certain critical points which are analogous to van Hove singularities in the density of states for phonons and electrons in solids. As in solids, these singularities are generally related to saddle and inflection points in the field spectra and are useful in characterizing field fluctuations. The critical points are very prominent in dispersions with a regular, ‘‘crystal-like,’’ structure. However, they broaden and eventually disappear as the disorder increases. @S0163-1829~98!50542-5# In the study of heterogeneous materials, the preponderance of work has been devoted to finding the effective transport, electromechanical, and mechanical properties of the material, 1 which amounts to knowing only the first moment of the local field. When composites are subjected to constant applied fields, the associated local fields exhibit strong spatial fluctuations. The analysis and evaluation of the distribution of the local field has received far less attention. Nonetheless, the distribution of the local field is of great fundamental and practical importance in understanding many crucial material properties such as breakdown phenomenon 2 and the nonlinear behavior of composites. 3 Much of the work on field distributions has been carried out for lattice models using numerical 4,5 and perturbation methods. 6 Recently, continuum models have been also addressed using numerical techniques. 7 In this paper, we study the local electric field fluctuations by analyzing the density of states for the fields. To illustrate the procedure, we evaluate the density of states for three different continuum models of dielectric composites: the Hashin-Shtrikman ~HS! construction, 8 periodic, and random arrays of cylinders. It is found that the density of states for the fields exhibits sharp peaks and abrupt changes in the slope at certain critical points which are analogous to van Hove singularities in the density of states for phonons and electrons in solids. This analogy is useful in quantifying field fluctuations in composites. In the case of the HashinShtrikman construction, we obtain an exact analytical expression for the density of states. We first describe the basic equations and then determine the density of states for the aforementioned examples. Consider a composite material composed of (n21) isotropic inclusions with dielectric constants e i and volume fractions f i (i52,...,n) in a uniform reference matrix of dielectric constant e 1 with volume fraction f 1 . Clearly, the local dielectric constant at position r is e(r) 5( i51

When a composite is subjected to a constant applied electric, thermal, or stress field, the associated local fields exhibit strong spatial fluctuations. In this paper, we evaluate the distribution of the local electric field (i.e., all moments of the field) for continuum (off-lattice) models of random dielectric composites. The local electric field in the composite is calculated by solving the governing partial differential equations using efficient and accurate integral equation techniques. We consider three different two-dimensional dispersions in which the inclusions are either (i) circular disks, (ii) squares, or (iii) needles. Our results show that in general the probability density function associated with the electric field for disks and squares exhibits a double-peak character. Therefore, the variance or second moment of the field is inadequate in characterizing the field fluctuations in the composite. Moreover, our results suggest that the variances for each phase are generally not equal to each other. In the case of a dilute concentration of needles, the probability density function is a singly peaked one, but the higher-order moments are appreciably larger for needles than for either disks or squares.

In the first paper of this series (DiStasio, Jr., Marcotte, Car, Stillinger, and Torquato, Phys. Rev. B 88, 134104 (2013)), we applied inverse statistical-mechanical techniques to study the extent to which targeted spin configurations on the square lattice can be ground states of finite-ranged radial spin-spin interactions. In this sequel, we enumerate all of the spin configurations within a unit cell on the one-dimensional integer lattice and the two-dimensional square lattice up to some modest size under periodic boundary conditions. We then classify these spin configurations into those that can or cannot be unique classical ground states of the aforementioned radial pair spin interactions and found the relative occurrences of these ground-state solution classes for different system sizes. As a result, we also determined the minimal radial extent of the spin-spin interaction potentials required to stabilize those configurations whose ground states are either unique or degenerate (i.e., those sharing the same radial spin-spin correlation function). This enumeration study has established that unique ground states are not limited to simple target configurations. However, we also found that many simple target spin configurations cannot be unique ground states.

In this work, we extend recent inverse statistical-mechanical methods developed for many-particle systems to the case of spin systems. For simplicity, we focus in this initial study on the two-state Ising model with radial spin-spin interactions of finite range (i.e., extending beyond nearest-neighbor sites) on the square lattice under periodic boundary conditions. Our interest herein is to find the optimal set of shortest-range pair interactions within this family of Hamiltonians, whose corresponding ground state is a targeted spin configuration such that the difference in energies between the energetically closest competitor and the target is maximized. For an exhaustive list of competitors, this optimization problem is solved exactly using linear programming. The possible outcomes for a given target configuration can be organized into the following three solution classes: unique (nondegenerate) ground state (class I), degenerate ground states (class II), and solutions not contained in the previous two classes (class III). We have chosen to study a general family of striped-phase spin configurations comprised of alternating parallel bands of up and down spins of varying thicknesses and a general family of rectangular block checkerboard spin configurations with variable block size, which is a generalization of the classic antiferromagnetic Ising model. Our findings demonstrate that the structurally anisotropic striped phases, in which the thicknesses of up- and down-spin bands are equal, are unique ground states for isotropic short-ranged interactions. By contrast, virtually all of the block checkerboard targets are either degenerate or fall within class III solutions. The degenerate class II spin configurations are identified up to a certain block size. We also consider other target spin configurations with different degrees of global symmetries and order. Our investigation reveals that the solution class to which a target belongs depends sensitively on the nature of the target radial spin-spin correlation function. In the future, it will be interesting to explore whether such inverse statistical-mechanical techniques could be employed to design materials with desired spin properties.

We introduce the concept of nearly hyperuniform network (NHN) structures as alternatives to the conventional continuous random network (CRN) models for amorphous tetrahedrally coordinated solids, such as amorphous silicon (a-Si). A hyperuniform solid has a structure factor $S(k)$ that approaches zero as the wavenumber $k\ensuremath{\rightarrow}0$. We define a NHN as an amorphous network whose structure factor $S(k\ensuremath{\rightarrow}0)$ is smaller than the liquid value at the melting temperature. Using a novel implementation of the Stillinger-Weber potential for the interatomic interactions, we show that the energy landscape for a spectrum of NHNs includes a sequence of local minima with an increasing degree of hyperuniformity [smaller $S(k\ensuremath{\rightarrow}0)$] that is significantly below the frozen-liquid value and that correlates with other measurable features in $S(k)$ at intermediate and large $k$ and with the width of the electronic band gap.

We present a method to calculate upper bounds on the photonic band gaps of two-component photonic crystals. The method involves calculating both upper and lower bounds on the frequency bands for a given structure, and then maximizing over all possible two-component structures. We apply this method to a number of examples, including a one-dimensional photonic crystal (or ``Bragg grating'') and two-dimensional photonic crystals (in both the TM and TE polarizations) with both four and sixfold rotational symmetries. We compare the bounds to band gaps of numerically optimized structures and find that the bounds are extremely tight. We prove that the bounds are ``sharp'' in the limit of low dielectric contrast ratio between the two components. This method and the bounds derived here have important implications in the search for optimal photonic band-gap structures.

We evaluate third-order bounds on the effective conductivity ${\ensuremath{\sigma}}_{e}$ and effective bulk modulus ${K}_{e}$ of a random dispersion of equal-sized impenetrable spheres in a matrix up to sphere volume fractions near the random close-packing value. The third-order bounds, which incorporate an integral ${\ensuremath{\zeta}}_{2}$ that depends upon the three-point probability function of the two-phase medium, are shown to significantly improve upon second-order Hashin-Shtrikman bounds, which do not utilize this information, for a wide range of phase property values and volume fractions. The physical significance of the microstructural parameter ${\ensuremath{\zeta}}_{2}$ for general microstructures is briefly discussed. The third-order bounds on ${\ensuremath{\sigma}}_{e}$ and ${K}_{e}$ are found to be sharp enough to yield good estimates of the bulk properties for a wide range of sphere volume fractions, even when the phase property values differ by as much as two orders of magnitude. Moreover, when the spheres are highly conducting or highly rigid relative to the matrix, the third-order lower bound on the respective effective property provides a useful estimate of it for a wide range of sphere volume fractions.

We derive a new perturbation expansion for the effective conductivity tensor sigma/sub e/ of a macroscopically anisotropic d-dimensional two-phase composite of arbitrary microstructure. The nth-order tensor coefficients ssA/sub n//sup (//sup i//sup )/ of the expansion (termed n-point microstructural parameters) are given explicitly in terms of integrals over the set of n-point probability functions (associated with the ith phase) which statistically characterize the microstructure. Macroscopic anisotropy arises out of some asymmetry in the microstructure, i.e., due to statistical anisotropy (e.g., a distribution of oriented, nonspherical inclusions in a matrix, layered media, such as sandstones and laminates, etc.). General and useful properties of the n-point microstructural parameters are established, and contact is made with the formal results of Milton. We then derive rigorous nth-order bounds on sigma/sub e/ (from our perturbation expansion) that depend upon the n-point parameters ssA/sub n//sup (//sup i//sup )/ for n = 1, 2, 3, and 4. This is the first time that such bounds (for n>1) have been explicitly given in terms of the ssA/sub n//sup (//sup i//sup )/.