** 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 \(\sigma^2\) associated with the number of points (particles) in some local observation window grows more slowly than the window volume as the size of the window increases.

In the case of a spherical window of radius \(R\) in \(d\)-dimensional Euclidean space \(\mathcal R^d\) (see figure), hyperuniformity means that the number variance \(\sigma^2(R)\) grows more slowly than \(R^d\). This 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].

The degree to which large-scale density fluctuations are suppressed enables one to rank order crystals, quasicrystals and the aforementioned special disordered systems [1,2].

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.

We have also demonstrated that hyperuniformity also 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].

We extended the notion of hyperuniformity to include also two-phase random heterogeneous media [2].

Hyperuniform random media do not possess infinite-wavelength volume fraction fluctuations, implying that the variance in the local volume fraction in an observation window decays faster than the reciprocal of the window volume \(r^{-d}\) as \(R\) increases.

Figure 1 shows typical periodic and nonperiodic point patterns and two-phase media along with the corresponding observation windows.

[google-drive-embed url=”https://docs.google.com/uc?id=0B1EGhP8Z-3DhbmQ2R09YOTFoaU0&export=download” title=”hyperuniform.png” width=”500″ height=”497″ extra=”image” style=”embed”]

Upper: schematics indicating an observation window for a periodic point pattern (left) and a periodic heterogeneous medium (right) obtained by decorating the point pattern with circles. Lower: schematics for a disordered point pattern (left) and the corresponding disordered heterogeneous medium (right).

** 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 [6], are hyperuniform such that pair correlations decay asymptotically with scaling \(r^{-4}\), which we call *quasi-long-range* correlations [7]. 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 \(\mathcal R^d\), 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 [8-10].

** 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 [11] or supercooled liquids reach their glass transition [12]. 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.

** Disordered Multihyperuniformity in Avian Photoreceptor Cell Patterns **

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 Fig. 2a. 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 recent 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 [13]. 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 Fig. 2b. 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.

Figure 2: (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.

** 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 [14]. 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.

** Stealthy Hyperuniform Systems **

We have shown that so-called *stealthy* hyperuniform states can be created as disordered ground states [3,4] and that they possess novel scattering properties [3]. 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 [5]. 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 following article that describes potential technological applications of this capability: Advancing photonic functionalities.

** 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. These exotic amorphous states of matter are endowed with novel thermodynamic and physical properties. Previous investigations of these unusual systems relied heavily on computer-simulation techniques. 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 here, 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 [15].

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.

[google-drive-embed url=”https://docs.google.com/uc?id=0B1EGhP8Z-3DhcDV1eVpwX0MxQWc&export=download” title=”stealthy_ensemble_fig.png” width=”300″ height=”400″ extra=”image” style=”embed”]

Figure 3: 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.

** 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. 16, 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. We also use numerical simulations to extend our study to the lower-density regime. We report results for the pair correlation functions, structure factors, and Voronoi cell statistics.

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.

** Ground States of Stealthy Hyperuniform Potentials **

In Ref. 17, we investigate using both numerical and theoretical techniques metastable stacked-slider phases. 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 Ref. 16. 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.

[google-drive-embed url=”https://docs.google.com/uc?id=0B1EGhP8Z-3DhRHY0aFlfdFowNjA&export=download” title=”HU_stacked_slider_config.png” width=”500″ height=”300″ extra=”image” style=”embed”] [google-drive-embed url=”https://docs.google.com/uc?id=0B1EGhP8Z-3DhcXhkVHE5bXhOejQ&export=download” title=”HU_stacked_slider_sk.png” width=”500″ height=”300″ extra=”image” style=”embed”]

Figure 4: A numerically obtained stacked-slider configuration (left) and the corresponding structure factor (right), where colors indicate intensity values at reciprocal lattice points.

Video 1. Ground states of particle-systems with stealthy interaction of \(\chi=0.05, 0.48, 0.70 \), respectively. Starting from the random (Poisson) point configurations, the optimization by collective-coordinates method drives systems into point configurations with lower energy. Above three systems end up with the ground states (point configurations with the lowest 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.** Inverse Design of Disordered Stealthy Hyperuniform Spin Chains **

It has recently been shown that disordered hyperuniform many-particle systems represent new distinguishable states of amorphous matter that are poised between a crystal and a liquid are are endowed with novel physical and thermodynamic properties. Such systems have shown to exist as ground states, i.e., at a temperature of absolute zero. Such “stealthy” and hyperuniform states are unique in that they are transparent to radiation for a range of wavelengths. In Ref. 18, 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 do demonstrate the existence of such states, which should be experimentally realizable.

[google-drive-embed url=”https://docs.google.com/uc?id=0B1EGhP8Z-3Dhb3psNUNNX0hpVFk&export=download” title=”HU_stealthy_spins_rho_cancels.png” width=”400″ height=”400″ extra=”image” style=”embed”]

Figure 5: A visualization of how \(\rho(k)\) cancels at the smallest positive wavenumber \(k=2\pi/N\) for a stealthy spin configuration of size \(N=12\) on the 1D integer lattice. The unit vectors represent the positive spin (\(\sigma_j=+1\)) exponential \(e^{i2\pi j/N}\) terms from the collective density variables \(\rho(k=2\pi/N)=\sum_{j=1}^N\sigma_je^{i2\pi j/N}\). The negative spin terms also cancel in a similar fashion. The plot contains a gray regular polygon including all possible unit vectors. The unit vectors that are summed for the positive spins in this particular configuration are colored according to how they cancel with other vectors. In this case, the scattering cancellation can be decomposed into two doublets (green and blue) and one triplet (red).

## References:

- 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). - 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). - 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). - 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. Torquato and F. H. Stillinger,
*Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond*,**Reviews of Modern Physics**,**82**, 2633 (2010). - 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, 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). - 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). - 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). - 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). - 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, G. Zhang, and F. H. Stillinger,
*Ensemble Theory for Stealthy Hyperuniform Disordered Ground States*,**Physical Review X**,**5**,021020 (2015). - 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).

Questions concerning this work should be directed to Professor Torquato.