# Hyperuniformity

A hyperuniform many-particle configuration in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) is characterized by an anomalous suppression of large-scale density fluctuations relative to those in typical disordered systems, such as liquids and structural glasses. The hyperuniformity concept generalizes the traditional notion of long-range order in many-particle systems to include all perfect crystals, perfect quasicrystals, and exotic amorphous states of matter. Disordered hyperuniform materials can have advantages over crystalline ones, such as nearly optimal, direction-independent physical properties and robustness against defects. The hyperuniformy concept has been extended to treat heterogeneous materials as well as scalar, vector and tensor fields. Additional details are provided below.

## What is Hyperuniformity?

Characterizing the local density fluctuations in a many-body system represents a fundamental problem in the physical and biological sciences. Examples include the large-scale structure of the Universe, condensed phases of matter, the structure and collective motion of grains in vibrated granular media, energy levels in integrable quantum systems, and the structure of living cells. In each of these cases, one is interested in characterizing the variance in the local number of points of a general point pattern (henceforth known as the number variance), and this problem extends naturally to higher dimensions with applications to number theory.

Of particular importance in this regard is the notion of hyperuniformity in a point pattern or many-particle configuration. A hyperuniform configuration is one in which the number variance \(\sigma_N^2(R)\) associated with the number of points (particles) in some local observation window grows more slowly than the window volume, i.e., \(R^d\), as \(R\) increases [1]. In the case of a spherical window of radius \(R\) in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) (see Video 1 and Figure 1), hyperuniformity means that the number variance \(\sigma_N^2(R)\) grows more slowly than \(R^d\). This property implies that hyperuniform point patterns are characterized by vanishing infinite-wavelength density fluctuations (when appropriately scaled) and encompass all crystals, quasicrystals, and special disordered many-particle systems [1]. Thus, hyperuniformity generalizes our traditional notions of long-range translational and orientational order by also incorporating these exotic amorphous states of matter.

Video 1: An animation showing how local density fluctuations in stealthy disordered hyperuniform (left column) and disordered non-hyperuniform (right column) point particle configurations are measured.For so-called class I hyperuniform systems, the number variance obeys the following large-R asymptotic formula:

where \(A_N\) and \(B_N\) are the dimensionally dependent “volume” and “surface-area” coefficients which are given by volume integrals involving the total correlation function \(h(\mathbf{r})\) [1]. A key signature of hyperuniform systems is that they have volume coefficient \(A_N=0\) which is mathematically equivalent to having a structure factor \(S(k)\) that vanishes in the limit that wavevector \(|\mathbf{k}|→0\). Using either the large-\(R\) behavior of \(\sigma_N^2(R)\) or the small-\(k\) behavior of \(S(\mathbf{k})\), one may gauge the degree to which large-scale density fluctuations are suppressed in a system and rank the order in crystals, quasicrystals, and the aforementioned special disordered systems [1,2]. We have also demonstrated that hyperuniformity signals the onset of an “inverted” critical point in which the direct correlation function (rather than the standard pair correlation function) becomes long-ranged [1,3].

Consider the following small-\(\mathbf{k}\) scaling for the structure factor:

For hyperuniform systems, the exponent \(\alpha\) is positive (\(\alpha>0\)) and its value determines three different large-\(R\) scaling behaviors of the number variance:

R^{d-1} & \text{$\alpha>1$ (class I)}\\

R^{d-1}\ln R & \text{$\alpha=1$ (class II)}\\

R^{d-\alpha} & \text{$\alpha<1$ (class III)}. \end{cases}\)

These scalings of \(\sigma_N^2(R)\) define three classes of hyperuniformity, with classes I and III describing the strongest and weakest forms of hyperuniformity, respectively. States of matter that belong to class I include all perfect crystals, many perfect quasicrystals, and “randomly” perturbed crystal structures, classical disordered ground states of matter, as well as systems out of equilibrium. Class II hyperuniform systems include some quasicrystals, the positions of the prime numbers, and many disordered classical and quantum states of matter. Examples of class III hyperuniform systems include classical disordered ground states, random organization models, and perfect glasses [3].

Hyperuniform disordered structures can be regarded as a **new state of disordered matter** in that they behave more like crystals or quasicrystals in the manner in which they suppress density fluctuations on large length scales, and yet are also like liquids and glasses in that they are statistically isotropic structures with no Bragg peaks. Thus, hyperuniform disordered materials can be regarded to possess a **“hidden order”** that is not apparent on short length scales.

Figure 1: Schematics depicting observation window \(\Omega\) used to diagnose hyperuniformity in point patterns and two-phase media. Upper left: a periodic point pattern. Upper right: A periodic heterogeneous medium obtained by decorating the point pattern with circles. Lower left: a disordered point pattern. Lower right: The corresponding disordered heterogeneous medium.

We have also extended the notion of hyperuniformity to include two-phase random heterogeneous media as well [2-4]. Hyperuniform random media do not possess infinite-wavelength volume fraction fluctuations, implying that the variance in the local volume fraction \(\sigma_N^2(R)\) in an observation window of radius \(R\) decays faster than the reciprocal of the window volume \(R^d\) as \(R\) increases. Analogous coefficients \(A_V\) and \(B_V\) for two-phase systems may be defined in terms of volume integrals over the autocovariance function \(\chi_V(r)\). Furthermore, a hyperuniform two-phase medium is distinguished from nonhyperuniform ones by having \(A_V=0\) which is mathematically equivalent to the spectral density \(\tilde{\chi}_{_V}(\mathbf{k})\), which is defined as the Fourier transform of \(\chi_V(r)\), vanishing in the limit that wavevector \(|\mathbf{k}|\to0\).

Beyond local fluctuations in number density and volume fractions for point patterns and two-phase media, respectively, we have generalized hyperuniformity to treat local fluctuations in the interfacial area of heterogeneous media which is critical to the characterization of physical properties that depend on the geometry of the phase interface. We have also extended the hyperuniformity concept to local fluctuations in the intensities of random scalar fields, and the intensities and orientations in divergence-free random vector fields. Additionally, we have examined hyperuniformity in statistically anisotropic many-particle and two-phase systems. Most notably, we found that random vector fields and statistically anisotropic systems can exhibit directionally-dependent hyperuniformity (see Figure 4). Such “directional hyperuniform” systems could be realized as materials with exotic anisotropic physical properties. For more information, see Ref. [5].

Figure 2: (left) A statistically anisotropic point pattern. (right) The structure factor of the point pattern on the left which clearly shows that hyperuniformity depends on the direction in which the origin \(\mathbf{k}=\mathbf{0}\) is approached. Figure taken from Ref. [5].

- S. Torquato and F. H. Stillinger, Local Density Fluctuations, Hyperuniform Systems, and Order Metrics,
**Physical Review E**,**68**, 041113 (2003). - C. E. Zachary and S. Torquato, Hyperuniformity in Point Patterns and Two-Phase Random Heterogeneous Media,
**Journal of Statistical Mechanics: Theory and Experiment**, P12015 (2009). - S. Torquato, Hyperuniform States of Matter,
**Physics Reports**,**745**, 1-95 (2018). - S. Torquato, Disordered Hyperuniform Heterogeneous Materials,
**Journal of Physics: Condensed Matter**,**28**, 414012 (2016). - S. Torquato, Hyperuniformity and its Generalizations,
**Physical Review E**,**94**, 022122 (2016).

## Institute for Advanced Study High Energy Theory Seminar: Hyperuniformity in Classical and Quantum States of Matter

Here is a recording of Professor Torquato’s seminar on hyperuniformity given at the Institute for Advanced Study on February 25th, 2022.

Abstract: The study of hyperuniform states of matter is an emerging multidisciplinary field, influencing and linking developments across the physical sciences, mathematics and biology. A hyperuniform many-particle system in \(d\)-dimensional Euclidean space is characterized by an anomalous suppression of large-scale density fluctuations relative to those in typical disordered systems, such as liquids and amorphous solids. As such, the hyperuniformity concept generalizes the traditional notion of long-range order to include not only all perfect crystals and quasicrystals, but also exotic disordered states of matter, thus providing a unified framework to quantitatively categorize such phases of matter. Disordered hyperuniform states have attracted great attention across many fields over the last two decades because they have the character of crystals on large length scales but are isotropic like liquids. I will begin by reviewing the hyperuniformity concept. Then I will give an overview of how hyperuniformity is linked to the Epstein zeta function of number theory, sphere packing problems, eigenvalues of random matrices, nontrivial zeros of the Riemann zeta function, spatial distribution of the prime numbers, free fermions, Laughlin’s incompressible quantum states, and photoreceptor mosaics in avian retina.

## Hyperuniformity is a Signature of Maximally Random Jammed Packings

In an initial study, we showed that so-called maximally random jammed (MRJ) packings of identical three-dimensional spheres, which can be viewed as a prototypical glass [1], are hyperuniform such that pair correlations decay asymptotically with scaling \(r^{-4}\), which we call *quasi-long-range* correlations [2,3]. Such correlations are to be contrasted with typical disordered systems in which pair correlaitons decay exponentially fast. More recently, we have shown that quasi-long-range pair correlations that decay asymptotically with scaling \(r^{-(d+1)}\) in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\), trademarks of certain quantum systems and cosmological structures, are a universal signature of a wide class of maximally random jammed (MRJ) hard-particle packings, including nonspherical particles with a polydispersity in size [4-6].

More generally, Torquato and Stillinger [7] suggested that certain defect-free strictly jammed packings of identical spheres are hyperuniform. Specifically, they conjectured that any strictly jammed saturated infinite packing of identical spheres is hyperuniform. A saturated packing of hard spheres is one in which there is no space available to add another sphere. Importantly, the Torquato-Stillinger conjecture excludes MRJ packings that may have a mechanically rigid backbone but possess “rattlers” (particles that are not locally jammed but are free to move about a confining cage) because a strictly jammed packing, by definition, cannot contain rattlers. It has been suggested that the ideal MRJ state is rattler-free, implying that the packing is more disordered without the presence of rattlers [8]. Recently, Torquato presented a refined variant of the conjecture: “Any strictly jammed infinite packing of identical spheres that is defect-free is hyperuniform” [9]. Notably, the Torquato-Stillinger conjecture is supported by various numerical experiments; for example, see a recent letter by Rissone, Corwin, and Parisi [10].

- A. B. Hopkins, F. H. Stillinger, and S. Torquato, Nonequilibrium Static Diverging Length Scales on Approaching a Prototypical Model Glassy State,
**Physical Review E**,**86**, 021505 (2012). - A. Donev , F. H. Stillinger, and S. Torquato, Unexpected Density Fluctuations in Disordered Jammed Hard-Sphere Packings,
**Physical Review Letters**,**95**, 090604 (2005). - C. E. Zachary and S. Torquato, Anomalous Local Coordination, Density Fluctuations, and Void Statistics in Disordered Hyperuniform Many-Particle Ground States,
**Physical Review E**,**83**, 051133 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato, Hyperuniform Long-Range Correlations are a Signature of Disordered Jammed Hard-Particle Packings,
**Physical Review Letters**,**106**, 178001 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato, Hyperuniformity, Quasi-Long-Range Correlations, and Void Space Constraints in Maximally Random Jammed Particle Packings. I. Polydisperse Spheres,
**Physical Review E**,**83**, 051308 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato, Hyperuniformity, Quasi-Long-Range Correlations, and Void Space Constraints in Maximally Random Jammed Particle Packings. II. Anisotropy in Particle Shape,
**Physical Review E**,**83**, 051309 (2011). - S. Torquato and F. H. Stillinger,
*Local Density Fluctuations, Hyperuniform Systems, and Order Metrics*,**Physical Review E**,**68**, 041113 1-25 (2003). - S. Atkinson, G. Zhang, A. B. Hopkins, and S. Torquato, Critical Slowing Down and Hyperuniformity on Approach to Jamming,
**Physical Review E**,**94**, 012902 (2016). - S. Torquato, Structural characterization of many-particle systems on approach to hyperuniform states,
**Physical Review E**,**103**052126 (2021). - P. Rissone, E. I. Corwin, and G. Parisi, Long-Range Anomalous Decay of the Correlation in Jammed Packings,
**Physical Review Letters**,**127**038001 (2021).

## Growing Length Scales in Supercooled Liquids and Glasses

We have recently demonstrated that quasi-long-range pair correlations are present well before overcompressed hard-sphere systems or supercooled liquids reach their glass transition in Refs. [1] and [2], respectively. These quasi-long-range correlations translate into a long-ranged direct correaltion function, which enables us extract a length scale that is *nonequilibrium* in nature. We show that this nonequilibrium static length scale grows on approach to the glassy state. This provides an alternative view of the nature of the glass transition.

- A. Donev , F. H. Stillinger, and S. Torquato, Unexpected Density Fluctuations in Disordered Jammed Hard-Sphere Packings,
**Physical Review Letters**,**95**, 090604 (2005). - É. Marcotte, F. H. Stillinger, and S. Torquato, Nonequilibrium Static Growing Length Scales in Supercooled Liquids on Approaching the Glass Transition,
**Journal of Chemical Physics**,**138**, 12A508 (2013).

## Disordered Multihyperuniformity in Avian Photoreceptor Cell Patterns and Models

The evolution of animal eyes has been an intense subject of research since Darwin. The purpose of a visual system is to sample light in such a way as to provide an animal with actionable knowledge of its surroundings that will permit it to survive and reproduce. Cone photoreceptor cells in the retina are responsible for detecting colors and they are often spatially arranged in a regular array (e.g., insects, some fish and reptiles), which is often a superior arrangement to sample light. In the absence of any other constraints, classical sampling theory tells us that the triangular lattice (i.e., a hexagonal array) is the best arrangement.

Diurnal birds have one of the most sophisticated cone visual systems of any vertebrate, consisting of four types of single cone (violet, blue, green and red) which mediate color vision and double cones involved in luminance detection; see Figure 1. Given the utility of the perfect triangular-lattice arrangement of photoreceptors for vision, the presence of disorder in the spatial arrangement of avian cone patterns was puzzling.

Our investigation in collaboration with Joseph Corbo at Washington University presents a stunning example of how fundamental physical principles can constrain and limit optimization in a biological system [1]. By analyzing the chicken cone photoreceptor system consisting of five different cell types using a variety of sensitive microstructural descriptors, we found that the disordered photoreceptor patterns are “hyperuniform” (as defined above), a property that had heretofore been identified in a unique subset of physical systems, but had never been observed in any living organism.

Disordered hyperuniform structures can be regarded as a new exotic state of disordered matter in that they behave more like crystals or quasicrystals in the manner in which they suppress density fluctuations on large length scales, and yet are also like liquids and glasses in that they are statistically isotropic structures with no Bragg peaks. Thus, hyperuniform disordered materials can be regarded to possess a “hidden order” that is not apparent on short length scales.

Remarkably, the patterns of both the total population and the individual cell types are simultaneously hyperuniform, which has never been observed in any system before, physical or not. We term such patterns “multi-hyperuniform” because multiple distinct subsets of the overall point pattern are themselves hyperuniform. We devised a unique multiscale cell packing model in two dimensions that suggests that photoreceptor types interact with both short- and long-ranged repulsive forces and that the resultant competition between the types gives rise to the aforementioned singular spatial features characterizing the system, including multi-hyperuniformity (see Figure 1). These findings suggest that a disordered hyperuniform pattern may represent the most uniform sampling arrangement attainable in the avian system, given intrinsic packing constraints within the photoreceptor epithelium. In addition, they show how fundamental physical constraints can change the course of a biological optimization process. Our results suggest that multi-hyperuniform disordered structures have implications for the design of materials with novel physical properties and therefore may represent a fruitful area for future research. For more information, see Ref. [1].

Figure 1: (left) Experimentally obtained configurations representing the spatial arrangements of centers of the chicken cone photoreceptors (violet, blue, green, red and black cones). (right) Simulated point configurations representing the spatial arrangements of chicken cone photoreceptors. The photoreceptor types interact with both short- and long-ranged repulsive forces such that the resultant competition between the types gives rise to the aforementioned singular spatial features characterizing the system, including multi-hyperuniformity. The simulated patterns for individual photoreceptor species are virtually indistinguishable from the actual patterns obtained from experimental measurements. Figures taken from Ref. [1].

We have developed two additional theoretical models of these fascinating disordered multihyperuniform systems. In Ref. [2], we present a statistical-mechanical model that rigorously achieves disordered multihyperunifom many-body systems by tuning interactions in binary mixtures of nonadditive plasmas. This model provides a powerful method to generate disordered multihyperuniform systems via equilibrium mixtures and will facilitate the future study of their potentially unique photonic, phononic, electronic, and transport properties.

In Ref. [3], we further model the avian retina via an equimolar three-component mixture (one component to sample each primary color: red, green, and blue) of nonadditive hard disks to which a long-range logarithmic repulsion is added between like particles. Notably, this model captures the salient features of the spatial distribution of photoreceptors in avian retina; namely, the presence of disorder, multihyperuniformity, and local heterocoordination. The latter feature being critical for the efficient sampling of light.

- Y. Jiao, T. Lau, H. Hatzikirou, M. Meyer-Hermann, J. C. Corbo, and S. Torquato, Avian Photoreceptor Patterns Represent a Disordered Hyperuniform Solution to a Multiscale Packing Problem,
**Physical Review E**,**89**, 022721 (2014). - E. Lomba, J.-J. Weis, and S. Torquato, Disordered Multihyperuniformity Derived from Binary Plasmas,
**Physical Review E**,**97**, 010102(R) (2018). - E. Lomba, J.-J. Weis, L. Guisández, and S. Torquato, Minimal Statistical-Mechanical Model for Multihyperuniform Patterns in Avian Retina,
**Physical Review E**,**102**012134 (2020).

## Diagnosing Hyperuniformity in Two-Dimensional, Disordered, Jammed Packings of Soft Spheres

The task of determining whether or not an image of an experimental system is hyperuniform is experimentally challenging due to finite-resolution, noise, and sample-size effects that influence characterization measurements. We have explored these issues, employing video optical microscopy to study hyperuniformity phenomena in disordered two-dimensional jammed packings of soft spheres in Ref. [1]. Using a combination of experiment and simulation we have characterized the possible adverse effects of particle polydispersity, image noise, and finite-size effects on the assignment of hyperuniformity, and we have developed a methodology that permits improved diagnosis of hyperuniformity from real-space measurements. The key to this improvement is a simple packing reconstruction algorithm that incorporates particle polydispersity to minimize the free volume. In addition, simulations show that hyperuniformity in finite-sized samples can be ascertained more accurately in direct space than in reciprocal space. Finally, our experimental colloidal packings of soft polymeric spheres were shown to be effectively hyperuniform (see Figure 1). For a very recent discussion on methods for diagnosing hyperuniformity in finite systems, see Ref. [2].

Figure 1: A jammed packing of small and large soft poly(N-isopropyl acrylamide) (PNIPAM) micro gel particles prepared in the laboratory which we found to be effectively hyperuniform. Each PNIPAM particle has a small degree of polydispersity from its associated mean. Figure taken from Ref. [1].

- R. Dreyfus, Y. Xu, T. Still, L. A. Hough, A. G. Yodh, and S. Torquato Diagnosing Hyperuniformity in Two-Dimensional, Disordered, Jammed Packings of Soft Spheres,
**Physical Review E**,**91**, 012302 (2015). - S. Torquato, Structural characterization of many-particle systems on approach to hyperuniform states,
**Physical Review E**,**103**052126 (2021).

## Rigorous Characterization of Hyperuniform and Nonhyperuniform Point Processes

Recall that the local number variance \(\sigma_N^2(R) \) associated with a spherical sampling window of radius \(R\) enables one to diagnose hyperuniformity in many-particle systems. In Ref. [1], we conduct an extensive study of higher-order moments, including the skewness \( \gamma_1(R) \) and excess kurtosis \( \gamma_2(R) \), as well as the corresponding probability distribution function \(P[N(R)]\), of various models of hyperuniform and nonhyperuniform systems across the first three space dimensions. We derive closed-form integral expressions for \( \gamma_1(R) \) and \( \gamma_2(R) \) that encode structural information up to three- and four-body correlation functions, respectively. We also introduce a “Gaussian distance metric” \(l_2(R)\) (based on the \(L_2\) norm) which enables one to accurately determine when the full distribution function \(P[N(R)]\) for a particular system tends toward a central limit theorem (CLT). We find that the convergence to a CLT is generally the fastest for disordered hyperuniform point processes. See Ref. [1] for more information.

As hyperuniform states of matter arise in increasingly diverse fields of study, including the physical sciences, mathematics, and biology, it is also important to explore quantitative descriptors that indicate when a many particle system approaches a hyperuniform state as a function of some appropriate control parameter. In Ref. [2], we establish that the ratio of the “surface-area” coefficient \(B_N\) to the “volume” coefficient \(A_N\), \(B_N/A_N\), can be used to ascertain the extent of hyperuniform and nonhyperuniform distance-scaling regimes.

We also examined two other diagnostic measures of hyperuniformity, the hyperuniformity index \(H\) and the direct-correlation function length scale \(\xi_c\), and found that they were positively correlated with each other and the ratio \(B_N/A_N\) [2]. We also derive a Fourier representation of the surface-area coefficient in terms of the structure factor \(S(\mathbf{k})\) which is especially useful when scattering information is available experimentally or theoretically. Altogether, our findings should facilitate the diagnosis of hyperuniformity in experimentally or computationally generated samples that are necessarily of finite size.

- S. Torquato, J. Kim, and M. A. Klatt, Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions,
**Physical Review X**,**11**, 021028 (2021). - S. Torquato, Structural characterization of many-particle systems on approach to hyperuniform states,
**Physical Review E**,**103**052126 (2021).

## Stealthy Hyperuniform Systems

We have shown that so-called *stealthy* hyperuniform states can be created as disordered ground states [1,2] and that they possess novel scattering properties [1]. This has led to the discovery of the existence disordered two-phase dielectric materials with complete photonic band gaps that are comparable in size to those in photonic crystals [3]. Thus, the significance of hyperuniform materials for photonics applications has enabled us, for the first time, to broaden the class of 2D dielectric materials possessing complete and large photonic band gaps to include not only crystal and quasicrystal structures but certain hyperuniform disordered ones. The interested reader can refer to the article Advancing Photonic Functionalities that describes potential technological applications of this capability.

## Ensemble Theory of Stealthy Hyperuniform Disordered Ground States

When slowly cooling a typical liquid we expect that it will undergo a freezing transition to a solid phase at some temperature and then attain a unique perfectly ordered crystal ground-state configuration. Particles interacting with “stealthy” long-ranged pair potentials have classical ground states that are—counterintuitively—disordered, hyperuniform, and highly degenerate (see Figure 1). These exotic amorphous states of matter are endowed with novel thermodynamic and physical properties. These stealthy disordered hyperuniform systems were first realized in 2D and 3D in Refs. [4] and [5], respectively, via the collective coordinates method–a numerical computer-simulation technique [4].

The task of formulating a theory that yields analytical predictions for the structural characteristics and physical properties of stealthy degenerate ground states in multiple dimensions presents many theoretical challenges. A new type of statistical-mechanical theory, as we show in Ref. [2], must be invented to characterize these exotic states of matter. We derive general exact relations for the thermodynamic (energy, pressure, and compressibility) and structural properties that apply to any ground-state ensemble as a function of its density and dimension in the zero-temperature limit. We show how disordered, degenerate ground states arise as part of the ground-state manifold. We then specialize our results to the canonical ensemble by exploiting an ansatz that stealthy states behave remarkably like “pseudo”-equilibrium hard-sphere systems in Fourier space. This mapping enables us to obtain theoretical predictions for the pair correlation function, structure factor, and other structural characteristics that are in excellent agreement with computer simulations across one, two, and three dimensions. We also derive accurate analytical formulas for the properties of the excited states. Our results provide new insights into the nature and formation of low-temperature states of amorphous matter. Our work also offers challenges to experimentalists to synthesize stealthy ground states at the molecular level [2].

Figure 1: Schematic illustrating the inverse relationship between the direct-space number density \(\rho\) and relative fraction of constrained degrees of freedom \(\chi\) for a fixed reciprocal-space exclusion-sphere radius \(K\) (where dark blue \(\mathbf k\) points signify zero intensity with green, yellow, and red points indicating increasingly larger intensities) for a stealthy ground state. Figure taken from Ref. [2].

## Ground States of Stealthy Hyperuniform Potentials

Systems of particles interacting with so-called “stealthy” pair potentials have been shown to possess infinitely-degenerate disordered hyperuniform classical ground states with novel physical properties. Previous attempts to sample the infinitely-degenerate ground states used energy minimization techniques, introducing an algorithm dependence that is artificial in nature. Recently, an ensemble theory of stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics that was shown to be in excellent agreement with corresponding computer simulation results in the canonical ensemble (in the zero-temperature limit). In Ref. [3], we provide details and justifications of the simulation procedure, which involves performing molecular dynamics simulations at sufficiently low temperatures and minimizing the energy of the snapshots for both the high-density disordered regime, where the theory applies, as well as lower densities (see Video 1). We also use numerical simulations to extend our study to the lower-density regime and report results for the pair correlation functions, structure factors, and Voronoi cell statistics.

Video 1: Ground states of particle-systems with stealthy interactions \(\chi=0.05, 0.48, 0.70 \) from top to bottom, respectively. Starting from the random (Poisson) point configurations, the optimization by the collective-coordinates method drives these systems toward their respective ground states (the configuration with the minimum energy). Interestingly, the configurations of ground states depend on \(\chi\) of the interaction: for low \(\chi\), the ground states are counterintuitively disordered, highly degenerate, and stealthy hyperuniform, while for high \(\chi\), the ground states becomes crystalline.In the high-density regime, we verify the theoretical ansatz that stealthy disordered ground states behave like “pseudo” disordered equilibrium hard-sphere systems in Fourier space. The pair statistics obey certain exact integral conditions with very high accuracy. These results show that as the density decreases from the high-density limit, the disordered ground states in the canonical ensemble are characterized by an increasing degree of short-range order and eventually the system undergoes a phase transition to crystalline ground states. In the crystalline regime (low densities), there exist “stacked-slider configurations,” aperiodic structures that are part of the ground-state manifold, but yet are not entropically favored. We also provide numerical evidence suggesting that different forms of stealthy pair potentials produce the same ground-state ensemble in the zero-temperature limit. Our techniques may be applied to sample the zero-temperature limit of the canonical ensemble of other potentials with highly degenerate ground states.

In Ref. [7], we investigate using both numerical and theoretical techniques metastable stacked-slider phases (see Figure 2). Our numerical results enable us to devise analytical models of this phase in two, three and higher dimensions. Utilizing this model, we estimated the size of the feasible region in configuration space of the stacked-slider phase, finding it to be smaller than that of crystal structures in the infinite-system-size limit, which is consistent with our findings in Ref. [6]. In two dimensions, we also determine exact expressions for the pair correlation function and structure factor of the analytical model of stacked-slider phases and analyze the connectedness of the ground-state manifold of stealthy potentials in this density regime. We demonstrate that stacked-slider phases are distinguishable states of matter; they are nonperiodic, statistically anisotropic structures that possess long-range orientational order but have zero shear modulus. We outline some possible future avenues of research to elucidate our understanding of this unusual phase of matter.

Figure 2: A numerically obtained stacked-slider configuration (left) and the corresponding structure factor (right), where colors indicate intensity values at reciprocal lattice points. Figures taken from Ref. [7].

## Inverse Design of Disordered Stealthy Hyperuniform Spin Chains

Stealthy hyperuniform states are unique in that they are transparent to radiation for a range of wavelengths. In Ref. [8], we ask whether Ising models of magnets, called spin chains in one dimension, can possess spin interactions that enable their ground states to be disordered, stealthy, and hyperuniform. Using inverse statistical-mechanical theoretical methods, we demonstrate the existence of such states (see Figure 3), which should be experimentally realizable.

Figure 3: Different examples of hyperuniform spin chains (white = spin up, blue = spin down) and their accompanying structure factors. (a) is crystalline and thus stealthy hyperuniform while (b) and (c) are both stealthy and disordered hyperuniform. (d) is a Poisson spin pattern which is disordered hyperuniform, but not stealthy. The order metric \(\tau\) [2] is the \(L^2\)-norm of the difference between the structure factors \(S(k)\) of the spin chain and that of an uncorrelated Poisson spin pattern. Figure taken from Ref. [8].

- R. D. Batten, F. H. Stillinger and S. Torquato, Classical Disordered Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous Materials,
**Journal of Applied Physics**,**104**, 033504 (2008). - S. Torquato, G. Zhang, and F. H. Stillinger, Ensemble Theory for Stealthy Hyperuniform Disordered Ground States,
**Physical Review X**,**5**,021020 (2015). - M. Florescu, S. Torquato and P. J. Steinhardt, Designer Disordered Materials with Large, Complete Photonic Band Gaps,
**Proceedings of the National Academy of Sciences**,**106**, 20658 (2009). - O. U. Uche, F. H. Stillinger, and S. Torquato, Constraints on Collective Density Variables: Two Dimensions,
**Physical Review E**,**70**, 046122 (2004). - O. U. Uche, S. Torquato and F. H. Stillinger, Collective Coordinates Control of Density Distributions,
**Physical Review E**,**74**, 031104 (2006). - G. Zhang, F. H. Stillinger, and S. Torquato, Ground States of Stealthy Hyperuniform Potentials: I. Entropically Favored Configurations,
**Physical Review E**,**92**,022119 (2015). - G. Zhang, F. H. Stillinger, and S. Torquato, Ground States of Stealthy Hyperuniform Potentials: II. Stacked-Slider Phases,
**Physical Review E**,**92**,022120 (2015). - E. Chertkov, R. A. DiStasio, Jr., G. Zhang, R. Car, and S. Torquato, Inverse Design of Disordered Stealthy Hyperuniform Spin Chains,
**Physical Review B**,**93**, 064201 (2016).

## Fermionic (Determinantal) Point Processes

We have proven that two determinantal point processes which are intimately connected to quantum mechanical systems exhibit exact hyperuniformity. Determinantal point processes are characterized by a joint probability distribution that is given by the determinant of a finite-rank, positive, bounded, and self-adjoint operator. In Ref. [1], we introduce the “Fermi-sphere” point-processes which is a d-dimensional generalization of the one-dimensional point processes corresponding to the eigenvalues of the Gaussian Unitary Ensemble (GUE), the zeros of the Riemann zeta function, or the fermionic gas.

We further demonstrate that this disordered hyperuniform system has a structure factor which scales like \(S(\mathbf{k})\sim|\mathbf{k}|\) in the zero-wavenumber limit and corresponds exactly to the ground state of noninteracting spin-polarized fermions for which the Fermi sphere is completely filled. Notably, Feynman and Cohen as well as Reatto and Chester demonstrated that the structure factor of superfluid helium scales like \(S(\mathbf{k})\sim|\mathbf{k}|\) in the zero-wavenumber limit [2,3]. Thus, the ground state of this strongly interacting system of bosons is disordered hyperuniform as well.

In Ref. [26], we examine the Weyl-Heisenberg ensemble which is a class of determinantal point processes associated with the Schrödinger representation of the Heisenberg group. As with the Fermi-type point process, these determinantal point processes are also disordered hyperuniform with a structure factor that scales like \(S(\mathbf{k})\sim|\mathbf{k}|^2\) in the zero-wavenumber limit. Interestingly, the family of Weyl-Heisenberg ensembles includes processes where point-statistics depend on different spatial directions making them anisotropic and thus a means to study directional hyperuniformity.

## Quantum Spin Chains

As discussed previously, we have demonstrated that classical spin chains with certain long-range interactions exhibit configurations which are disordered hyperuniform [4]. In Ref. [5], we use high-precision numerical simulations to demonstrate that such classical spin chains lose their hyperuniformity when under the quantum effects of a transverse magnetic field. We also demonstrate the possibility of optimizing magnetic materials such that the quantum effects of an external transverse magnetic field increase the order of the ground state of the spin system, while still remaining more disordered than standard ferromagnetic or antiferromagnetic materials.

- S. Torquato, A Scardicchio and C. E. Zachary, Point Processes in Arbitrary Dimension from Fermionic Gases, Random Matrix Theory, and Number Theory,
**Journal of Statistical Mechanics: Theory and Experiment**, P11019 (2008). - R.P. Feynman and M. Cohen, Energy Spectrum of the Excitations in Liquid Helium,
**Physical Review**,**102**, 1189 (1956). - L. Reatto and G.V. Chester, Phonons and the Properties of a Bose System,
**Physical Review**,**155**, 88 (1968). - L. D. Abreu, J. M. Pereira, J. L. Romero and S. Torquato, The Weyl–Heisenberg Ensemble: Hyperuniformity and Higher Landau Levels,
**Journal of Statistical Mechanics: Theory and Experiment**,**2017**, 043103 (2017). - E. Chertkov, R. A. DiStasio, Jr., G. Zhang, R. Car, and S. Torquato, Inverse Design of Disordered Stealthy Hyperuniform Spin Chains,
**Physical Review B**,**93**, 064201 (2016). - A. Bose and S. Torquato, Quantum phase transitions in long-range interacting hyperuniform spin chains in a transverse field,
**Physical Review B**,**103**014118 (2021).

## Novel Physical Properties of Hyperuniform Materials

As discussed previously, disordered hyperuniform states of matter straddle crystal and liquid states in that they are like perfect crystals in their suppression of large-scale fluctuations in density while simultaneously being statistically isotropic with no Bragg peaks like liquids or glasses. The combination of these attributes can endow such materials with novel equilibrium and nonequilibrium bulk physical properties.

For example, in Ref. [1] we demonstrate that certain classical many-particle systems with soft repulsive pair interactions exhibit various unconventional physical properties; including classical disordered ground states which are hyperuniform, vanishing normal-mode frequencies, and negative thermal expansion. Importantly, these exotic states may be realized by certain systems of polymers or colloids whose interactions are effectively captured by soft pair interactions.

Interestingly, taking the point pattern formed by one of the aforementioned classical disordered hyperuniform ground states and decorating it with nonoverlapping spheres yields two-phase random media with nearly maximal effective diffusion coefficients while still being isotropic. This behavior is distinct from that of disordered nonhyperuniform systems like equilibrium hard spheres or overlapping spheres. For more information, see Ref. [2].

When certain 2D disordered hyperuniform point patterns are realized as dielectric materials, they exhibit large, complete, and isotropic photonic bandgaps. Prior to the publication of these results (see Ref. [3]), it was believed that all photonic band gap (PBG) materials must possess translational order. In the aforementioned paper, it is argued that these disordered PBG materials are possible due to their combination of hyperuniformity, uniform local topology, and short-range geometric order. Disordered hyperuniform photonic materials are discussed in detail in Ref. [4].

Figure 1: A designed disordered stealthy hyperuniform dispersion that is transparent to long-wavelength electromagnetic radiation and its associated dimensionless spectral density \(\rho\tilde{\chi}_{_V}(k)\) (scaled by the number density \(\rho\) of the particles used in the initial condition. This dispersion was made using the Fourier-space based numerical construction procedure described in Ref. [5]. Figure taken from Ref. [5].

In Ref. [5], we introduce a Fourier-space numerical construction procedure that enables one to design at will a wide variety of disordered hyperuniform two-phase materials with prescribed spectral densities. Such a procedure enables one to exert fine control over the effective physical properties of the constructed medium (see Figure 1). For example, the procedure was used to realize a stealthy disordered hyperuniform medium with nearly optimal effective thermal (or electric) conductivity while being statistically isotropic. Notably, the materials constructed via this procedure can be realized by 3D printing and lithographic techniques and have potential applications in energy-storage, batteries, and aerospace.

Recently, we have demonstrated that certain disordered hyperuniform materials exhibit desirable multifunctional behavior when realized as composite materials. Specifically, in Ref. [6], it is demonstrated that stealthy disordered hyperuniform composite materials exhibit nearly optimal, direction-independent electromagnetic and elastic wave propagation properties. These properties enable the design of low-pass filters that isotropically transmit waves up to a finite wavenumber. A potential application of these multifunctional composite materials could be in a sound-absorbing motor housing that must suppress mechanical vibrations while simultaneously transmitting thermal radiation. For a more in-depth mathematical treatment of the nonlocal field theories that describe these composites, see Refs. [7] and [8].

- R. D. Batten, F. H. Stillinger, and S. Torquato, Novel Low-Temperature Behavior in Classical Many-Particle Systems,
**Physical Review Letters**,**103**, 050602 (2009). - G. Zhang, F. H. Stillinger, and S. Torquato, Transport, Geometrical, and Topological Properties of Stealthy Disordered Hyperuniform Two-phase Systems,
**Journal of Chemical Physics**,**145**, 244109 (2016). - M. Florescu, S. Torquato and P. J. Steinhardt, Designer Disordered Materials with Large, Complete Photonic Band Gaps,
**Proceedings of the National Academy of Sciences**,**106**, 20658 (2009). - S. Yu, C. W. Qiu, Y. Chong, S. Torquato, and N. Park, Engineered disorder in photonics,
**Nature Reviews Materials**,**6**226 (2021). - D. Chen and S. Torquato, Designing Disordered Hyperuniform Two-phase Materials with Novel Physical Properties,
**Acta Materialia**,**142**, 152-161 (2018). - J. Kim and S. Torquato, Multifunctional Composites for Elastic and Electromagnetic Wave Propagation,
**Proceedings of the National Academy of Sciences of the United States of America**,**117(16)**8764-8774 (2020). - J. Kim and S. Torquato, Effective elastic wave characteristics of composite media,
**New Journal of Physics**,**22**123050 (2020). - S. Torquato and J. Kim, Nonlocal Effective Electromagnetic Wave Characteristics of Composite Media: Beyond the Quasistatic Regime,
**Physical Review X**,**11**, 021002 (2021).

## Protocols for the Creation of Large Hyperuniform Systems

Given the significant general scientific and technological interest in hyperuniform systems, the development of computational protocols for creating large hyperuniform systems is of critical importance. Two current techniques that we have developed leverage space tessellation procedures for the creation of hyperuniform dispersions and are detailed in Refs. [1] and [2] (see Figure 1).

Figure 1: A hyperuniform two-phase medium generated via the tessellation-based procedure described in Ref. [1]. Figure taken from Ref. [2].

As another synthetic route, we exploit long-ranged pair interactions in many-particle equilibrium systems to generate disordered hyperuniform states. For example, in Ref. [3], we numerically fabricate large, disordered hyperuniform point patterns via systems of binary paramagnetic colloidal particles confined in a 2D plane. In Ref. [4], we computationally demonstrate that colloids interacting via a repulsive hard-core Yukawa potential form effectively disordered hyperuniform states. Importantly, the results of both of these papers suggest a novel route to synthesize large samples of disordered hyperuniform materials at the nanoscale under standard laboratory conditions.

- J. Kim and S. Torquato, New Tessellation-Based Procedure to Design Perfectly Hyperuniform Disordered Dispersions for Materials Discovery,
**Acta Materialia**,**168**143-151 (2019). - J. Kim and S. Torquato, Methodology to Construct Large Realizations of Perfectly Hyperuniform Disordered Packings,
**Physical Review E**,**99**052141 (2019). - Z. Ma, E. Lomba, and S. Torquato, Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions,
**Physical Review Letters**,**125**068002 (2020). - D. Chen, E. Lomba and S. Torquato, Binary Mixtures of Charged Colloids: a Potential Route to Synthesize Disordered Hyperuniform Materials,
**Physical Chemistry Chemical Physics**,**20**, 17557-17562 (2018).

Questions concerning this work should be directed to Professor Torquato.