# Statistical Mechanics

# Self Assembly Theory via Inverse Methods

The relationship between the collective structural behavior of many-body systems and their corresponding interactions is a fundamental subject in condensed-matter theory, chemistry and materials science. The “forward” approach of statistical mechanics focuses on finding the structure and macroscopic properties of many-particle systems with specified interactions. In “inverse” statistical mechanics, the goal is to find optimized interactions that robustly and spontaneously lead to a targeted many-particle configuration of the system for a wide range of conditions. The capacity to identify realizable configurations associated with targeted functional forms for pair statistics is an inverse problem of great fundamental and practical importance. Inverse statistical mechanics has been employed to design many-body systems that self-assemble into matter with unusual structures or properties, including low-coordinated crystals, porous colloidal mesophases and complex metal alloys.

## Realizable Hyperuniform and Nonhyperuniform Particle Configurations with Targeted Spectral Functions via Effective Pair Interactions

The capacity to identify realizable many-body configurations associated with targeted functional forms for the pair correlation function \(g_2(\mathbf{r})\) or its corresponding structure factor \(S(\mathbf{k})\) is of great fundamental and practical importance. While there are obvious necessary conditions that a prescribed structure factor at number density ρ must satisfy to be configurationally realizable, sufficient conditions are generally not known due to the infinite degeneracy of configurations with different higher-order correlation functions.

Figure 1: A 3D 400-particle configuration with targeted structure factor \( S(k) = 1 – \rho(a\sqrt{\pi})^3\exp(-k^2/4\pi\rho) \) that was generated using the algorithm introduced in Ref. [1]. Figure taken from Ref. [1].

In Ref. [1], we introduce a theoretical formalism that provides a means to draw classical particle configurations from canonical ensembles with certain pairwise-additive potentials that could correspond to targeted analytical functional forms for the structure factor. This formulation enables us to devise an improved algorithm to systematically construct canonical-ensemble particle configurations with such targeted pair statistics, whenever realizable (see Figure 1). Our findings lead us to the conjecture that any realizable structure factor corresponding to either a translationally invariant equilibrium or nonequilibrium system can be attained by an equilibrium ensemble involving only effective pair interactions.

- G. Zhang and S. Torquato, Realizable Hyperuniform and Nonhyperuniform Particle Configurations with Targeted Spectral Functions via Effective Pair Interactions,
**Physical Review E**,**101**032124 (2020).

## Inversion problems for Fourier transforms of particle distributions

Collective coordinates in a many-particle system are complex Fourier components of the local particle density, and often provide useful physical insights. Given collective coordinates, it is desirable to infer the particle coordinates via inverse transformations. In principle, a sufficiently large set of collective coordinates are equivalent to particle coordinates, but the nonlinear relation between collective and particle coordinates makes the inversion procedure highly nontrivial.

Given a “target” configuration in one-dimensional (1D) Euclidean space, we investigate the minimal set of its collective coordinates that can be uniquely inverted into particle coordinates in Ref. [2]. For this purpose, we treat a finite number \(M\) of the real and/or the imaginary parts of collective coordinates of the target configuration as constraints, and then reconstruct “solution” configurations whose collective coordinates satisfy these constraints. Both theoretical and numerical investigations reveal that the number of numerically distinct solutions depends sensitively on the chosen collective-coordinate constraints and target configurations. From detailed analysis, we conclude that collective coordinates at the \( \lceil N/2 \rceil \) smallest wavevectors is the minimal set of constraints for unique inversion, where \(\lceil \cdot \rceil\) represents the ceiling function. This result provides useful groundwork to the inverse transform of collective coordinates in higher-dimensional systems.

- J. Kim, G. Zhang, F. H. Stillinger, and S. Torquato, Inversion Problems for Fourier Transforms of Particle Distributions,
**Journal of Statistical Mechanics: Theory and Experiment**,**2018**113302 (2018).

## Structural Degeneracy of Pair Statistics and Sensitivity of Pair Statistics on Pair Potentials

A traditional basic descriptor of many-particle systems has been the distribution of interparticle pair distances \(g_2(\mathbf{r})\). In the case of structureless particles at thermal equilibrium with only additive pair interactions, \(g_2(\mathbf{r})\) is sufficient to determine pressure and mean energy. However, it is usually the case that a given set of pair distances can emerge from a multiplicity of distinguishable many-particle configurations. In Ref. [1], we investigate the ways in which such a configurational detail can be overlooked. For example, we show that for many-particle systems in three dimensions, it is possible that a set of three-particle “pair-invisible” interactions can exist which modify the three-particle distribution function, but which have no effect on the pair distribution function, and thus remain undetected by conventional diffraction experiments (see Figure 1).

Figure 1: Distinct tetrahedra that can be constructed from three short pair distances (blue lines) and three larger pair distances (red lines). Configurations (a)-(d) all share the same pair statistics, but have distinct three-particle statistics. Figure taken from Ref. [1].

Using proposed metrics for sensitivity of pair statistics on pair potentials, we also show that at given number density \(\rho\) and positive temperature \(T\), very similar pair statistics may correspond to distinctly different pair potentials \(v(r)\), i.e. there is effective ambiguity of solutions to inverse problems that utilize pair information only, despite Henderson’s theorem stating that such solutions are unique (see Figure 2). [4]

Figure 2: Schematic depicting the relative sizes of the space of pair correlation functions \(g_2(\mathbf{r})\) or \(S(\mathbf{k})\) and the corresponding space of pair potentials \(v(r)\). At \( 0 \lt T \lt \inf \), Henderson’s theorem is not practically applicable if very similar pair correlation functions correspond to distinctly different pair potentials. Figure taken from Ref. [2].

- F. H. Stillinger and S. Torquato, Structural Degeneracy in Pair Distance Distributions,
**The Journal of Chemical Physics**,**150**204125 (2019). - H. Wang, F. H. Stillinger and S. Torquato, Sensitivity of Pair Statistics on Pair Potentials in Many-Body Systems,
**The Journal of Chemical Physics**,**153**124106 (2020). Data for the potentials can be found here.

## Low-Coordinated Crystal Ground States

In Refs. [1] and [2], we study self-assembly in two-dimensional systems. The most significant result therein is the discovery of an isotropic pair potential that yields the honeycomb lattice as its ground state:

The honeycomb lattice is a highly directional, three-fold coordinated structure that can be considered the two-dimensional analogue of the diamond lattice (see Figure 1). Hence, its self-assembly by a non-directional potential is indeed counterintuitive. In these papers, we also present a potential that self-assembled into the square lattice (see Figure 1). We develop two inverse statistical-mechanical schemes to optimize pair potentials for self-assembly. The first, the so-called “zero-temperature scheme” is based on maximizing the energy difference between the target lattice and a number of enumerated competitors, subject to the constraint that the target lattice is mechanically stable (phonon modes all real). The second, called the “near-melting” scheme, uses iterated molecular dynamics simulations in order to minimize the Lindemann parameter of the target lattice at temperatures near its melting point. In Ref. [3], we use the zero-temperature scheme to derive a potential that yields the simple cubic lattice in 3D.

Figure 1: (left) Particles interacting via the isotropic pair-potential \(V_{HC}(r)\) self-assembled into the honeycomb lattice. (right) Particles interacting via a different isotropic pair-potential self-assembled into the square lattice close to its melting point. Figures taken from Ref. [2].

In Ref. [4], we derive pair potentials that yield the diamond and wurtzite structures (cubic and hexagonal diamond) upon slow cooling. Due to the structural similarity between the diamond and wurtzite structures (akin to that of the face-centered-cubic and hexagonal-close-packed lattices), devising pair potentials that could differentiate between the two structures required fine tuning and more terms than were needed for the honeycomb, square and cubic systems. Although an isotropic potential which leads to self-assembly into the diamond lattice was previously found by Likos et. al., its defining feature is an extremely soft (logarithmic) core, which is not realistic for colloidal interactions. In contrast, our potential has a more realistic \( r^{-12} \) core.

- M. Rechtsman, F. H. Stillinger, S. Torquato, Optimized Interactions for Targeted Self-Assembly: Application to Honeycomb Lattice ,
**Physical Review Letters**,**95**, 228301 (2005). - M. Rechtsman, F. H. Stillinger, S. Torquato, Designed Interaction Potentials via Inverse Methods for Self-Assembly,
**Physical Review E**,**73**, 011406 (2006). Please see*Erratum*. - M. C. Rechtsman, F. H. Stillinger and S. Torquato, Self-Assembly of the Simple Cubic Lattice via an Isotropic Potential,
**Physical Review E**,**74**021404 (2006). - M. Rechtsman, F. H. Stillinger and S. Torquato, Synthetic Diamond and Wurtzite Structures Self-Assemble with Isotropic Pair Interactions,
**Physical Review E 75**, 031403 (2007).

## Negative Poisson’s Ratio Materials via Isotropic Pair Interactions

When a material is stretched along a particular direction, the material is expected to shrink in the lateral direction – like a rubber band. Conversely, the sides of a material usually bulge when compressed in a vise. Most materials change their lateral dimensions in this fashion when subjected to tension or compression. However, materials can respond in the completely opposite way – that is, they expand laterally when stretched or, equivalently, shrink laterally when compressed. Such materials are said to have negative Poisson’s ratio and are referred to as auxetic materials. Auxetic materials have technological importance. For example, they can used to improve the performance transducers, as components in microelectromechanical systems, as strain amplifiers, shock absorbers and fasteners, to mention a few examples.

In Ref. [1], we report that under tension (i.e., negative pressure), standard isotropic interactions of two- and three-dimensional many-particle systems (such as colloids) can result in elastically isotropic (nondirectional) auxetic behavior provided that the pressure of the system is negative. Matter characterized by negative pressure is unusual. An air-filled balloon will shrink in size if placed deep in an ocean consisting of ordinary water. On the other hand, such a balloon will expand in size if placed in an ocean of fluid possessing “negative pressure.” Matter under negative pressure exists; a simple example being tempered glass. A more exotic example comes from cosmology, where present thinking links the expanding Universe to a negative pressure.

The result reported in Ref. [1] is unexpected, since an inherently anisotropic behavior (auxetic behavior) arises from isotropic interactions. Indeed, most previously discovered auxetic materials exhibit complex, carefully designed anisotropic interactions. We have shown the existence of elastically isotropic auxetic behavior at zero temperature for common crystal structures in two and three dimensions, namely, the triangular lattice in two dimensions, and the face-centered cubic lattice in three dimensions. Notably, these lattices provide the densest arrangements of equal-sized spheres in these dimensions.

- M. C. Rechtsman, F. H. Stillinger and S. Torquato,
*Negative Poisson’s Ratio Materials via Isotropic Interactions*,**Physical Review Letters**,**101**, 085501 (2008).

# Unusual Ground States

## Collective Density Variables

Collective density variables have proved to be useful tools in the study of static and dynamic phenomena occurring in many-body systems. In Refs. [1] and [2], we apply the collective-coordinate approach to generate crystalline as well as disordered classical ground states for bounded or “soft” interactions using numerical optimization techniques in two and three dimensions. Such soft interactions are especially pertinent to modeling colloids, microemulsions, polymers, and other soft-matter systems.

Suppose that the particles interact pairwise with an isotropic pair potential \( v(\mathbf{r}_{jl}) \) so that the total energy is

Suppose furthermore that this pair potential has the Fourier transform:

where in the last expression the summation covers the entire set of **k**‘s. From here, it is straightforward to show that the total potential energy for the \(N\)-particle system can be exactly expressed in the following manner in terms of the real collective density variables

where the real collective density variables are

The Fourier coefficients of the density field are

and are related to \( C( \mathbf{k} ) \) by \( \rho( \mathbf{k} )\rho(-\mathbf{k}) = |\rho( \mathbf{k} )|^2 = N + 2C( \mathbf{k} ) \) and the structure factor \( S( \mathbf{k} ) \) by \( S(\mathbf{k}) = |\rho(\mathbf{k})|^2/N \). We examine features associated with collective density variables in two dimensions by using various numerical techniques to generate particle patterns in the classical ground state. Particle pair interactions are governed by a continuous, bounded potential. We place special emphasis on pair interactions whose transform \(V(\mathbf{k})\) possesses the following characteristics:

V_0 \gt 0 & 0 \le |\mathbf{k}| \le K \\

0 & K < |\mathbf{k}|. \end{cases} \)

We choose this form for simplicity recognizing at the same time that any other positive \(V(\mathbf{k})\) for \(|\mathbf{k}|\le K\) but zero otherwise would lead to the same results. Since the minimum of \(C(\mathbf{k})\) is \(-N/2\), the global minimum of the potential energy defined has the value of

if and only if there exist particle configurations that satisfy all constraints that \(C(\mathbf{k})\) be minimal. Note that \( \mathbf{Q} \) is the finite set of all wave vectors which satisfy the condition \( |\mathbf{k}| \le K \). For a system of \(N\) particles in \(d\) dimensions, there are \(dN\) degrees of freedom. We introduce the dimensionless parameter

to conveniently represent the ratio of the number of constrained degrees of freedom relative to the total number of degrees of freedom, where \(M(K)\) is the number of independently constrained wave vectors.

## The Collective Coordinates Method

In practice, one solves the minimization problem above using numerical optimization techniques. For example, in Ref. [1] we employ the conjugate gradient method to seek out particle configurations which absolutely minimize \( \Phi \) given a particular \( \chi \). In Ref. [2], we employ the “MINOP” algorithm which is better suited for collective coordinate optimizations. The animations below show the trajectories of particles, initially placed randomly inside the box, as the potential energy is minimized along a steepest descent path to a disordered ground state. The left and right systems have parameter \( \chi=0.26 \) and \( \chi=0.49 \), respectively.

Imposing the constraint that \(C(\mathbf{k})\) be minimal for \(|\mathbf{k}|\lt K \) has remarkable implications on the ground state structure. Specifically, we found that the resulting ground state configurations belonged to to one of three regions–disordered, wavy crystalline, and crystalline–depending on the number of wave vectors constrained [1]. Disordered configurations lack long-ranged order and have the lowest number of constrained wave vectors (see Figure 1). Wavy crystals have apparent short-range order but lack translational symmetry and have an intermediate number of constrained wave vectors (see Figure 2). Upon imposing a sufficient number of constraints, all the wave vectors except for those associated with Bragg scattering, are minimized. The resulting configuration and structure factor are unique to the triangular lattice (see Figure 3). In Ref. [2], we extend our analysis to three dimensions where disordered and crystalline particle patterns emerge as ground state structures (see Figure 4). Notably, no intermediate wavy-crystalline regime exists in 3D.

Figure 1: (left) A ground state of \( N=418 \) point particles in the disordered regime. The \(C(\mathbf{k})\) quantities for the wave vectors confined within the 50th shell \( (\chi=0.222488) \) have been constrained to their minimum values \( -N/2 \). (right) Associated structure factor of the disordered ground-state. Note how \( S(\mathbf{k}) \) is minimal for the concentric region around the origin while all other wave vectors are free to fluctuate. Figures and caption taken from Ref. [1].

Figure 2: (left) A ground state of \( N=418 \) point particles in the wavy crystalline regime. The \(C(\mathbf{k})\) quantities for the wave vectors confined within the 115th shell \( (\chi=0.581340) \) have been constrained to their minimum values \( -N/2 \). (right) Associated structure factor of the wave-crystalline ground-state. The explicit constraints about \( |\mathbf{k}|=0 \) impose implicit constraints on other wave vectors. Note the circular region of minimum for \(S(\mathbf{k})\), below right, and the emerging areas of minima that are implicitly constrained. Figures and caption taken from Ref. [1].

Figure 3: (left) A ground state of \( N=418 \) point particles in the crystalline regime. The \(C(\mathbf{k})\) quantities for the wave vectors confined within the 150th shell \( (\chi=0.779904) \) have been constrained to their minimum values \( -N/2 \). (right) Associated structure factor of the crystalline ground-state. Note the six-fold rotational symmetry in \( S(\mathbf{k}) \). Figures and caption taken from Ref. [1].

Figure 4: (left) A ground state of \( N=500 \) point particles in the disordered regime in 3D. The \(C(\mathbf{k})\) quantities consistent with parameter \( \chi=0.171333 \) have been constrained to their minimum values \( -N/2 \). (right) A ground state of \( N=500 \) point particles in the crystalline regime in 3D (face-centered-cubic lattice). The \(C(\mathbf{k})\) quantities consistent with parameter \( \chi=0.702666 \) have been constrained to their minimum values \( -N/2 \). Figures and caption taken from Ref. [2].

The introduction of a four-body potential energy function allows for the small-k region of the structure factor to be tailored. We specifically consider the potential function

subject to the condition \( V(\mathbf{k})=V(-\mathbf{k})\gt 0 \). In order to examine hyperuniform ground states, we select \( C_0(\mathbf{k}) = -N/2 + D|\mathbf{k}|^{\alpha} \) which corresponds to structure factors that scale like \( S(\mathbf{k})\sim|\mathbf{k}|^{\alpha} \) in the small wave vector regime. Structure factors with \( \alpha=1 \) are relevant for the Harrison-Zeldovich model of the early universe [3], superfluid Helium-4 [4,5], and disordered jammed sphere packings [6]. Selected particle configurations and their corresponding spectra for \(\alpha=1\) and \(\alpha=6\) are presented in Figures 5 and 6, respectively.

Figure 5: (left) Disordered hyperuniform ground state for \(\alpha=1\) with \(N=168\) point particles, \(K=20\pi\), \(DK^{\alpha}=75\), and \( \chi=0.470238 \). (right) The corresponding structure factor. Figures taken from Ref. [2].

Figure 6: (left) Disordered hyperuniform ground state for \(\alpha=6\) with \(N=168\) point particles, \(K=20\pi\), \(DK^{\alpha}=75\), and \( \chi=0.470238 \). (right) The corresponding structure factor.

- 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). - A. Gabrielli, M. Joyce, and F. S. Labini, Glass-like universe: Real-space correlation properties of standard cosmological models,
**Physical Review D**,**65**, 083523 (2004). - 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). - A. Donev , F. H. Stillinger, and S. Torquato, Unexpected Density Fluctuations in Disordered Jammed Hard-Sphere Packings,
**Physical Review Letters**,**95**, 090604 (2005).

## Super-Ideal Gases and Stealthy and Equi-Luminous Materials

In Ref. [1], we define and construct three classes of ground state structures: (1) stealthy hyperuniform materials which suppress scattering for a certain set of wavelengths (i.e., \(S(\mathbf{k}) = 0\) for \(0 \lt |\mathbf{k}| \lt K \) where \(K\) is positive and finite), (2) “super-ideal gases” which scatter radiation identically to that of an ensemble of ideal gases for a set of wavelengths, and (3) “equi-luminous” materials which scatter equally intensely for a prescribed set of wavelengths. We generate these configurations using the collective coordinate method described above. For these three types of ground states, we show that increasing the system size does not affect the local structure and that they are disordered in the infinite-volume limit.

## Duality Relations

In Ref. [2], we derive new duality relations that link the energy of configurations associated with a class of soft pair potentials to the corresponding energy of the dual (Fourier-transformed) potential. We apply them by showing how information about the classical ground states of short-ranged potentials can be used to draw new conclusions about the nature of the ground states of long-ranged potentials and vice versa. These duality relations also lead to bounds on the zero temperature system energies in density intervals of phase coexistence, the identification of a one-dimensional system that exhibits an infinite number of “phase transitions”, and a conjecture regarding the ground states of purely repulsive monotonic potentials. In Ref. [3], we extend these results on duality relations to include three-body and higher-order potentials.

- 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 and F. H. Stillinger, New Duality Relations for Classical Ground States,
**Physical Review Letters**,**100**, 020602 (2008). - S. Torquato, C. E. Zachary, and F. H. Stillinger, Duality Relations for the Classical Ground States of Soft-Matter Systems,
**Soft Matter**,**7**, 3780 (2011).

## Stealthy Hyperuniform Ground States

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. 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 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. [5], 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. [4]. 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. [5].

## 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. [6], 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. [6].

- 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). - G. Zhang, F. H. Stillinger, and S. Torquato, Ground States of Stealthy Hyperuniform Potentials: II. Stacked-Slider Phases,
**Physical Review E**,**92**,022120 (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). - 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).

## The Perfect Glass Paradigm

Rapid cooling of liquids below a certain temperature range can result in a transition to glassy states. The traditional understanding of glasses includes their thermodynamic metastability with respect to crystals. However, in Ref. [1] we present specific examples of interactions that eliminate the possibilities of crystalline and quasicrystalline phases, while creating mechanically stable amorphous glasses down to absolute zero temperature.

Figure 1: Snapshots of perfect glasses with \( N=2500 \) particles in 2D (left) and 3D (right). Figure taken from Ref. [1].

We show that such behavior can be observed by introducing a new ideal state of matter called a “perfect glass”, which can be regarded as the epitome of a glass since they are out of equilibrium, maximally disordered, hyperuniform, mechanically rigid with infinite bulk and shear moduli, and can never crystallize due to configuration-space trapping (see Figure 1). Our model perfect glass utilizes two-, three-, and four-body soft interactions while simultaneously retaining the salient attributes of the MRJ state. The perfect glass potentials are shown to possess disordered classical ground states with zero entropy [1]. These zero-entropy ground states are in sharp contrast with zero-entropy crystalline ground states, since the latter possess very high symmetry and long-range translational and rotational order [2].

- G. Zhang, F. Stillinger, and S. Torquato, The Perfect Glass Paradigm: Disordered Hyperuniform Glasses Down to Absolute Zero,
**Scientific Reports**,**6**, 36963 (2016). - G. Zhang, F. H. Stillinger and S. Torquato, Classical Many-Particle Systems with Unique Disordered Ground States,
**Physical Review E**,**96**, 042146 (2017).

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. In Ref. [1], we show that for a perfect (ground-state) crystal at zero temperature, increase in temperature introduces correlated displacements resulting from thermal excitations, and thus the thermalized crystal becomes nonhyperuniform, even at an arbitrarily low temperature.

- J. Kim, and S. Torquato, Effect of Imperfections on the Hyperuniformity of Many-Body Systems,
**Physical Review B**,**97**, 054105 (2018).

Questions concerning this work should be directed to Professor Torquato.