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Hyperuniform States of Matter Hyperuniform States of Matter Salvatore Torquato Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics,Princeton University,Princeton, New Jersey 08544, USA Abstract Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of long- wavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered sys- tems, such as ordinary fluids and amorphous solids. All perfect crystals, perfect quasicrystals and special disordered systems are hyperuniform. Thus, the hyperuniformity concept enables a unified framework to classify and structurally characterize crystals, quasicrystals and the exotic disordered varieties. While disordered hyperuniform systems were largely unknown in the scientific community over a decade ago, now there is a realization that such systems arise in a host of contexts across the physical, materials, chemical, mathematical, engineering, and biological sciences, in- cluding disordered ground states, glass formation, jamming, Coulomb systems, spin systems, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, number theory, stochastic point processes, integral and stochastic geometry, the immune system, and photoreceptor cells. Such unusual amor- phous states can be obtained via equilibrium or nonequilibrium routes, and come in both quantum-mechanical and classical varieties. The connections of hyperuniform states of matter to many different areas of fundamental science appear to be profound and yet our theoretical understanding of these unusual systems is only in its infancy. The purpose of this review article is to introduce the reader to the theoretical foundations of hyperuniform ordered and disordered systems. Special focus will be placed on fundamental and practical aspects of the disordered kinds, includ- ing our current state of knowledge of these exotic amorphous systems as well as their formation and novel physical properties. Contents 1 Introduction 4 1.1 Disordered Hyperuniform Systems Are Distinguishable States of Matter . .5 1.2 Hyperuniformity of Heterogeneous Materials . .7 1.3 Further Generalizations of the Hyperuniformity Concept . .7 1.4 Overview . .8 2 Basic Definitions and Preliminaries 9 2.1 Point Processes . .9 2.2 Lattices and Periodic Point Configurations . 12 2.3 Two-Phase Heterogeneous Media . 14 3 Local Number Fluctuations 16 3.1 Ensemble-Average Formulation of the Number Variance . 16 3.1.1 Hyperspherical Windows . 17 3.2 Number Variance for a Single Point Configuration . 19 arXiv:1801.06924v1 [cond-mat.stat-mech] 22 Jan 2018 3.3 Number Variance for a Single Periodic Point Configurations . 19 4 Local Volume-Fraction Fluctuations in Two-Phase Media 21 1 5 Mathematical Foundations of Hyperuniformity: Point Configurations 22 5.1 Vanishing of Normalized Infinite-Wavelength Number Fluctuations . 22 5.2 Distinctions Between Equilibrium and Nonequilibrium Infinite-Wavelength Density Fluctuations . 22 5.3 Asymptotics Via Ensemble-Average Formulation and Three Hyperuniformity Classes . 23 5.3.1 Asymptotics of the Number Variance . 24 5.3.2 Asymptotics from Power-Law Structure Factors . 26 5.4 Asymptotics for a Single Point Configuration . 28 5.5 Asymptotics for Periodic Point Configurations . 28 5.6 Hyperuniform Order Metrics . 30 5.7 Nonspherical Windows . 33 6 Mathematical Foundations of Hyperuniformity: Two-Phase Media 35 6.1 Vanishing of Infinite-Wavelength Volume-Fraction Fluctuations . 35 6.1.1 Asymptotic Behavior of the Volume-Fraction Variance for Two-Phase Systems . 36 6.1.2 Three Classes of Hyperuniform Two-Phase Systems . 36 6.2 Asymptotics From Power-Law Spectral Densities . 37 6.3 Design of Hyperuniform Two-Phase Materials with Prescribed Spectral Densities . 37 6.4 Interrelations Between Number and Volume-Fraction Variances . 37 6.5 Spectral Densities for Packings . 38 6.6 Nonspherical Windows . 40 7 Hyperuniformity as a Critical-Point Phenomenon and Scaling Laws 40 7.1 Direct Correlation Function for Hyperuniform Systems and New Critical Exponents . 41 7.2 Critical behavior of g2-Invariant Hyperuniform Point Configurations . 43 7.2.1 Step-Function g2 ......................................... 44 7.2.2 Step+Delta Function g2 ..................................... 47 8 Hyperuniform Disordered Classical Ground States 48 8.1 Stealthy Configurations via Collective-Coordinate Optimizations . 48 8.2 Can Disordered Stealthy Particle Configurations Tolerate Arbitrarily Large Holes? . 52 8.3 Configurations with Designed Hyperuniform Structure Factors via Collective Coordinates . 52 8.3.1 Underconstrained Systems . 52 8.3.2 Perfect-Glass Paradigm . 53 8.4 Ensemble Theory for Stealthy Hyperuniform Disordered Ground States . 54 8.4.1 General Results . 55 8.4.2 Pseudo Hard Spheres in Fourier Space and Pair Statistics . 56 8.4.3 Number Variance and Translational Order Metric τ ....................... 57 8.4.4 Entropically Favored (Most Probable) States and Ground-State Manifold . 59 8.4.5 Comparison of Stealthy Configurations to Some Common States of Matter . 60 8.4.6 Excited (Vibrational) States . 60 9 1D Hyperuniform Patterns Derived from Random Matrices and the Zeta Function 61 10 Hyperuniform Determinantal Point Processes 66 10.1 General Considerations . 66 10.2 Fermi-Sphere Point Processes . 67 10.3 Two-Dimensional Ginibre Ensemble: One-Component Plasma . 69 10.4 Weyl-Heisenberg Ensemble . 71 10.5 Multicomponent Hard-Sphere Plasmas . 71 2 11 Classical Nonequilibrium Systems 72 11.1 Ordered and Disordered Jammed Particle Packings . 72 11.1.1 Jamming Categories . 72 11.1.2 Geometric-Structure Approach and Order Maps . 73 11.1.3 Jamming-Hyperuniformity Conjecture . 74 11.1.4 Numerical Simulations of MRJ-Like States . 76 11.1.5 Critical Slowing Down as a Disordered Jammed State is Approached . 77 11.1.6 “Effective” Hyperuniformity Criterion . 78 11.1.7 Toward the Ideal MRJ State . 78 11.1.8 Is Hyperuniformity Also a Signature of MRJ Packings of Aspherical Particles? . 78 11.2 Driven Nonequilibrium Systems and Critical Absorbing States . 79 11.2.1 Hyperuniformity in Absorbing-State Models . 80 11.2.2 Interpretation of the Ideal MRJ Sphere Packing as a Critical-Absorbing State . 82 12 Natural Disordered Hyperuniform Systems 82 12.1 Avian Photoreceptor Cells . 82 12.2 Receptor Organization in the Immune System . 85 13 Generalizations of the Hyperuniformity Concept 85 13.1 Fluctuations in the Interfacial Area . 86 13.2 Random Scalar Fields . 87 13.3 Random Vector Fields . 89 13.4 Statistical Anisotropic Structures . 91 13.5 Spin Systems . 94 14 Novel Bulk Physical Properties of Disordered Hyperuniform Materials 94 14.1 Thermodynamic Properties . 94 14.2 Wave Characteristics . 94 14.3 Transport Properties . 96 14.4 Mechanical Properties . 96 14.5 Structurally Anisotropic Materials . 96 15 Effect of Imperfections on Hyperuniform States 97 16 Nearly Hyperuniform Systems 99 16.1 Growing Length Scale on Approach to a Hyperuniform State . 99 16.2 Amorphous Silicon and Electronic Band Gaps . 99 16.3 Polymeric Materials: Liquid State and Glass Formation . 100 16.3.1 Equilibrium Liquids . 100 16.3.2 Glassy Systems . 101 17 Discussion and Outlook 102 Appendix AGaussCircle Problem and Its Generalizations 104 3 1. Introduction The quantitative characterization of density fluctuations in many-particle systems is a fundamental and practical problem of great interest in the physical, materials, mathematical and biological sciences. It is well known that density fluctuations contain crucial thermodynamic and structural information about many-particle system both away from [1–6] and near critical points [7–12]. The one-dimensional point patterns associated with the eigenvalues of random matrices and energy levels of integrable quantum systems have been characterized by their density fluctuations [13– 15]. The quantification of density fluctuations has been used to reveal the fractal nature of structures within living cells [16]. The measurement of galaxy density fluctuations is a powerful way to quantify and study the large-scale structure of the Universe [17, 18]. Structural fluctuations play a critical role in charge transfer in DNA [19], thermodynamics of polyelectrolytes [20, 21], and dynamics in glass formation [22]. Depletion phenomena that occur in various molecular systems near interfaces are due to density fluctuations [23, 24]. Knowledge of density fluctuations in vibrated granular media has been used to probe the structure and collective motions of the grains [25]. Local density fluctuations have been recently used to distinguish the spatial distribution of cancer cells from that of normal, healthy cells [26]. The unusually large suppression of density fluctuations at long wavelengths (large length scales) is central to the hyperuniformity concept, whose broad importance for condensed matter physics and materials science was brought to the fore only about a decade ago in a study that focused on fundamental theoretical aspects, including how it provides a unified means to classify and structurally characterize crystals, quasicrystals and special disordered point config- urations [27]. Hyperuniform systems are poised at an exotic critical point in which the direct correlation function, defined via the Ornstein-Zernike relation [6], is long-ranged [27], in diametric contrast to standard thermal critical points in which the total correlation function is long-ranged and the corresponding
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