Ensemble Theory for Stealthy Hyperuniform Disordered Ground States

Ensemble Theory for Stealthy Hyperuniform Disordered Ground States

S. Torquato*

Department of Chemistry, Department of Physics,Princeton Institute for the Science and Technology of Materials,

and Program in Applied and Computational Mathematics, Princeton University,Princeton, New Jersey 08544, USA

G. Zhang and F. H. StillingerDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

(Received 9 January 2015; published 29 May 2015)

It has been shown numerically that systems of particles interacting with isotropic stealthy boundedlong-ranged pair potentials (similar to Friedel oscillations) have classical ground states that are(counterintuitively) disordered, hyperuniform, and highly degenerate. Disordered hyperuniform systemshave received attention recently because they are distinguishable exotic states of matter poised between acrystal and liquid that are endowed with novel thermodynamic and physical properties. The task offormulating an ensemble theory that yields analytical predictions for the structural characteristics and otherproperties of stealthy degenerate ground states in d-dimensional Euclidean space Rd is highly nontrivialbecause the dimensionality of the configuration space depends on the number density and there is amultitude of ways of sampling the ground-state manifold, each with its own probability measure for findinga particular ground-state configuration. The purpose of this paper is to take some initial steps in thisdirection. Specifically, we derive general exact relations for thermodynamic properties (energy, pressure,and isothermal compressibility) that apply to any ground-state ensemble as a function of in any d, and weshow how disordered degenerate ground states arise as part of the ground-state manifold. We also deriveexact integral conditions that both the pair correlation function g2r and structure factor Skmust obey forany d. We then specialize our results to the canonical ensemble (in the zero-temperature limit) by exploitingan ansatz that stealthy states behave remarkably like pseudo-equilibrium hard-sphere systems in Fourierspace. Our theoretical predictions for g2r and Sk are in excellent agreement with computer simulationsacross the first three space dimensions. These results are used to obtain order metrics, local numbervariance, and nearest-neighbor functions across dimensions. We also derive accurate analytical formulasfor the structure factor and thermal expansion coefficient for the excited states at sufficiently smalltemperatures for any d. The development of this theory provides new insights regarding our fundamentalunderstanding of the nature and formation of low-temperature states of amorphous matter. Our work alsooffers challenges to experimentalists to synthesize stealthy ground states at the molecular level.

DOI: 10.1103/PhysRevX.5.021020 Subject Areas: Soft Matter, Statistical Physics

I. INTRODUCTION

The equilibrium structure and phase behavior of softmatter systems span from the relatively simple, as found instrongly repulsive colloidal particles, to the highly com-plex, as seen in microemulsions and polymers [110]. Softmatter has been fruitfully microscopically modeled asclassical many-particle systems in which the particles (ormetaparticles) interact with effective pair potentials.Bounded (soft) effective interactions have been particularly

useful in modeling polymer systems, and they display zero-temperature ground states with a rich variety of crystallinestructures, depending on the composition of the constitu-ents and interaction parameters [1,4,710].We have previously used a collective-coordinate

approach to generate numerically exotic classical groundstates of many particles interacting with certain boundedisotropic long-ranged pair potentials in one-, two-, andthree-dimensional Euclidean space dimensions [1117] aswell as with anisotropic potentials [18]. It was shown thatthe constructed ground states across dimensions are theexpected crystal structures in a low-density regime[11,12,14,19], but above some critical density, there is aphase transition to ground states that are, counterintuitively,disordered (statistically isotropic with no long-range order),hyperuniform, and highly degenerate [20]. These unusual

*[email protected]

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published articles title, journal citation, and DOI.

PHYSICAL REVIEW X 5, 021020 (2015)

2160-3308=15=5(2)=021020(23) 021020-1 Published by the American Physical Society

amorphous states of matter have been shown to be endowedwith novel thermodynamic and physical properties[1416,2527] and belong to the more general class ofdisordered hyperuniform systems, which have beenattracting attention recently, as detailed below.The disordered ground states are highly degenerate with

a configurational dimensionality that depends on thedensity, and there are an infinite number of distinct waysto sample this complex ground-state manifold, each with itsown probability measure. For these reasons, it is theoreti-cally very challenging to devise ensemble theories thatare capable of predicting structural attributes and otherproperties of the ground-state configurations. A new typeof statistical-mechanical theory must be invented to char-acterize these exotic states of matter. The purpose of thispaper is to take some initial steps in this direction.However, to motivate the theoretical formalism, it isinstructive to first briefly review the collective-coordinatenumerical procedure that we have used to achieve disor-dered ground states.In the simplest setting, we previously examined pairwise

additive potentials vr that are bounded and integrablesuch that their Fourier transforms ~vk exist. If N identicalpoint particles reside in a fundamental region F of volumevF in d-dimensional Euclidean space Rd at positions rN r1;; rN under periodic boundary conditions, the totalpotential energy can be expressed in terms of ~vk asfollows:

rN 12vF

Xk

~vkj ~nkj2 NXk

~vk; 1

where ~nk PNj1 expik rj is the complex collec-tive density variable, which can be viewed as a nonlineartransformation from the finite set of particle coordinatesr1;; rN to the complex functions ~nk that depend on theinfinite set of wave vectors k in reciprocal space appro-priate to the fundamental cell F. The crucial idea is that if~vk is defined to be bounded and positive with support inthe radial interval 0 jkj K and if the particles arearranged so that j ~nkj2, a quantity proportional to thestructure factor Sk, is driven to its minimum value of zerofor all wave vectors where ~vk has support (exceptk 0), then it is clear from relation (1) that the systemmust be at its ground state or global energy minimum. Wehave referred to these ground-state configurations asstealthy [14] because the structure factor Sk (scatteringpattern) is zero for 0 < jkj K, meaning that they com-pletely suppress single scattering of incident radiation forthese wave vectors and, thus, are transparent at thecorresponding wavelengths [28]. Various optimizationtechniques were employed to find the globally energy-minimizing configurations within an exceedingly smallnumerical tolerance [1118]. Generally, a numericallyobtained ground-state configuration depends on the

number of particles N within the fundamental cell, initialparticle configuration, shape of the fundamental cell, andparticular optimization technique employed.As the number of k vectors for which j ~nkj is con-

strained to be zero increases, i.e., as K increases, thedimensionality of the ground-state configuration manifoldDC decreases. Because j ~nkj is inversion symmetric, thenumber of wave vectors contained in a sphere of radius Kcentered at the origin must be an odd integer, say2MK 1, and thusMK is the number of independentlyconstrained wave vectors [29]. The parameter

MKdN 1 ; 2

which is inversely proportional to density, gives a measureof the relative fraction of constrained degrees of freedomcompared to the total number of degrees of freedomdN 1 (subtracting out the system translational degreesof freedom). We show in Sec. IV B that the dimensionalityof the configuration space per particle is given by d1 2in the thermodynamic limit.It is straightforward to see why, for sufficiently small ,

ground states exist that are highly degenerate and typicallydisordered for sufficiently large N; see Fig. 1. Clearly,when the system is free of any constraints, i.e., if 0, it isa noninteracting classical ideal gas. While it is unusual tothink of classical ideal-gas configurations as ground states,at T 0, they indeed are global energy-minimizing statesthat are highly degenerate and typically disordered forlarge enough N. While the ground-state manifold containsperiodic configurations (e.g., Bravais lattices and latticeswith a basis), these are sets of zero measure in thethermodynamic limit. Clearly, if is made positive butvery small, the ground states remain disordered and highlydegenerate, even if the dimensionality of the configurationspace DC has now been suddenly reduced due to theimposed constrained degrees of freedom, the number ofwhich is determined by the radius K. From relation (2), wesee that if K is fixed, configurations with ideal-gas-like paircorrelation functions correspond to the limit 0 or,equivalently, to the limit [30]. The latter situationruns counter to traditional understanding that ideal-gasconfigurations correspond to the opposite zero-densitylimit of classical systems of particles. The reason for thisinversion of limits is due to the fact that a compression ofthe system in direct space leads to a dilation of the latticespacing in reciprocal space, as schematically shown inFig. 1. While it is not surprising that the configuration spaceis fully connected for sufficiently small , quantifying itstopology as a function of for all allowable is anoutstanding problem, which is discussed further in theConclusions.It is noteworthy that stealthy point patterns (disordered

or not) constitute a special class of so-called hyperuniform

S. TORQUATO, G. ZHANG, AND F. H. STILLINGER PHYS. REV. X 5, 021020 (2015)

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states of matter. Hyperuniform systems are characterized byvanishing (normalized) density fluctuations at large lengthscales; i.e., the structure factor Sk tends to zero in thelimit jkj 0 [31] (see Sec. II A for details). The hyper-uniformity concept provides a means of categorizingcrystals, quasicrystals, and special disordered systemsaccording to the degree to which large-scale densityfluctuations are suppressed [31,32]. Disordered hyperuni-form patterns, of which disordered stealthy systems arespecial cases, behave more like crystals in the manner theysuppress large-scale density fluctuations, and yet they alsoresemble typical statistically isotropic liquids and glasseswith no Bragg peaks. In this sense, they have a hiddenorder on large length scales that is not apparent at smalllength scales, even if short-range order is present; see Fig. 2for a vivid illustration. During the last decade, a variety ofdisordered hyperuniform states have been identified that

exist as both equilibrium and nonequilibrium phases,including maximally random jammed particle packings[3336], jammed athermal granular media [37], jammedthermal colloidal packings [38,39], cold atoms [40],transitions in nonequilibrium systems [41,42], surface-enhanced Raman spectroscopy [43], terahertz quantumcascade laser [44], wave dynamics in disordered potentialsbased on supersymmetry [45], avian photoreceptor patterns[46], and certain Coulombic systems [47]. Moreover,disordered hyperuniform materials possess novel physicalproperties potentially important for applications in photon-ics [2527,48,49] and electronics [5052].The well-known compressibility relation from statistical

mechanics [54] provides some insights about the relation-ship between temperature T and hyperuniformity forequilibrium systems at number density :

Sk 0 kBTT: 3

We see that any ground state (T 0) in which theisothermal compressibility T is bounded and positivemust be hyperuniform because the structure factor Sk 0 must be zero. This includes crystals as well as exoticdisordered ground states such as stealthy ones. However, inorder to have a hyperuniform system at positive T, theisothermal compressibility must be zero; i.e., the systemmust be incompressible [17] (see Refs. [31] and [47] forsome examples). Subsequently, we will use relation (3) todraw some conclusions about the excited states associatedwith stealthy ground states.Our general objective is the formulation of a predictive

ensemble theory for the thermodynamic and structuralproperties of stealthy degenerate disordered ground statesin arbitrary space dimension d that complements previous

Compression Dilation

Direct Space Reciprocal SpaceS(k)

FIG. 1. Schematic illustrating the inverse relationship betweenthe direct-space number density and relative fraction of con-strained degrees of freedom for a fixed reciprocal-spaceexclusion-sphere radius K (where dark blue k points signifyzero intensity with green, yellow, and red points indicatingincreasingly larger intensities) for a stealthy ground state. Acompression of a disordered ground-state configuration with afixed number of particles N in direct space leads to a dilation ofthe lattice spacing in reciprocal space. This means that during thecompression process, the k points for which j ~nkj is zeroassociated with the initial uncompressed system move out of theexclusion zone; i.e., the value of MK [cf. Eq. (2)] decreases.Since there are fewer constrained degrees of freedom (dimen-sionality of the ground-state configuration manifold increases),the disordered direct-space configuration becomes less spatiallycorrelated. For a fixed N in the limit (i.e., system volumevF 0), every k point (except the origin) is expelled from theexclusion zone, and the system tends to an ideal-gas configura-tion [30], even if it is not an ideal gas thermodynamically, asshown in Sec. IV B.

FIG. 2. A disordered nonhyperuniform configuration (leftpanel) and a disordered hyperuniform configuration (right panel).We arrive at the configuration on the right by very smallcollective displacements of the particles on the left via themethods described in Ref. [12]. (Each particle on average movesa root-mean-square distance that is about an order of magnitudesmaller than the mean-nearest-neighbor distance as measured bythe configuration proximity metric [53].) These two examplesshow that it can be very difficult to detect hyperuniformity by eye,and yet their large-scale density fluctuations are dramaticallydifferent (hidden order).

ENSEMBLE THEORY FOR STEALTHY HYPERUNIFORM PHYS. REV. X 5, 021020 (2015)

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numerical work on this topic [1116]. After providing basicdefinitions and describing a family of isotropic stealthypotentials (Secs. II and III), we derive general exactrelations for thermodynamic properties (energy, pressure,and isothermal compressibility) that apply to any well-defined ground-state ensemble as a function of the numberdensity or, equivalently, in any space dimension d(Sec. IV). We subsequently derive some exact integralconditions that both the pair correlation function g2r andstructure factor Sk must obey (Sec. V). The existence ofperiodic stealthy ground states enables us to show howdisordered degenerate ground states arise as part of theground-state manifold for sufficiently small (Sec. VI).Subsequently, we derive analytical formulas for the pairstatistics [g2r and Sk] for sufficiently small in thecanonical ensemble in the limit that the temperature T tendsto zero (Sec. VII) by exploiting an ansatz that stealthy statesbehave like pseudo-equilibrium hard-sphere systems inFourier space. Our theoretical predictions for g2r andSk are in excellent agreement with computer simulationsacross the first three space dimensions. These results arethen used to predict, with high accuracy, other structuralcharacteristics of stealthy ground states across dimensions,such as order metrics, local number variance, and nearest-neighbor functions (Secs. VIII and IX). Subsequently, wederive analytical formulas for the structure factor andthermal expansion coefficient for the associated excitedstates for sufficiently small temperatures (Sec. X). Finally,we provide concluding remarks in Sec. XI.

II. DEFINITIONS AND PRELIMINARIES

Roughly speaking, a point process in d-dimensionalEuclidean space Rd is a distribution of an infinite numberof points in Rd with the configuration r1; r2; at awell-defined number density (number of points per unitvolume). For a statistically homogeneous point process inRd at number density [55], the quantity ngnrn is theprobability density associated with simultaneouslyfinding n points at locations rn r1; r2;; rn in Rd[54]. With this convention, each n-particle correlationfunction gn approaches unity when all of the points becomewidely separated from one another. Statistical homogeneityimplies that gn is translationally invariant and henceonly depends on the relative displacements of the positionswith respect to any chosen system origin, e.g.,gn gnr12; r13;; r1n, where rij rj ri.The pair correlation function g2r is a particularly

important quantity. If the point process is also rotationallyinvariant (statistically isotropic), then g2 depends on theradial distance r jrj only, i.e., g2r g2r. Thus, itfollows that the expected number of points ZR found in asphere of radius R around a randomly chosen point of thepoint process, called the cumulative coordination function,is given by

ZR s11Z

R

0

xd1g2xdx; 4

where s1r 2d=2rd1=d=2 is the surface area ofa d-dimensional sphere of radius r. The total correlationfunction hr is trivially related to g2r as follows:

hr g2r 1: 5When there are no long-range correlations in the system,hr 0 or, equivalently, g2r 1 as jrj . Thestructure factor Sk, which plays a prominent role in thispaper, is related to the Fourier transform of hr, denotedby ~hk, via the expression

Sk 1 ~hk: 6A lattice in Rd is a subgroup consisting of integer

linear combinations of vectors that constitute a basis forRd,and thus, it represents a special subset of point processes.Here, the space can be geometrically divided into identicalregions F called fundamental cells, each of which containsjust one point specified by the lattice vector

p n1a1 n2a2 nd1ad1 ndad; 7

where ai are the basis vectors for a fundamental cell and nispans all the integers for i 1; 2;; d. We denote by vFthe volume of F. A lattice is called a Bravais lattice in thephysical sciences. Unless otherwise stated, we will use theterm lattice. Every lattice has a dual (or reciprocal) lattice in which the lattice sites are specified by the dual(reciprocal) lattice vector q p 2m for all p, wherem 0;1;2;3. The dual fundamental cell F hasvolume vF 2d=vF. This implies that the numberdensity of is related to the number density ofthe dual lattice via the expression

1=2d: 8Some common d-dimensional lattices are mathematicallydefined in Appendix A.A periodic point process (crystal) is a more general

notion than a lattice because it is obtained by placing afixed configuration of N points (where N 1) within afundamental cell F of a lattice , which is then periodicallyreplicated. Thus, the point process is still periodic undertranslations by , but the N points can occur anywhere inF; see Fig. 3.

A. Hyperuniform point processes

Consider uniformly sampling the number of points thatare contained within a spherical window of radius R of apoint process in Rd. A hyperuniform point process has theproperty that the local number variance 2R grows more


021020-4

slowly than Rd [31]. Because 2R is exactly related to ad-dimensional volume integral of the structure factor Sk(see Sec. VIII), this implies that hyperuniform states ofmatter possess infinite-wavelength density fluctuations(appropriately normalized) that vanish; i.e., Sk obeysthe condition

limjkj0

Sk 0; 9

which means they are poised at an inverted critical pointwith associated scaling exponents [31]. For a Poisson(spatially uncorrelated) point process and many disorderedpoint patterns, including typical liquids and structuralglasses, the number variance grows like the volume ofthe window, i.e., 2R Rd, implying that Sk is positiveat k 0. All perfect crystals and quasicrystals are hyper-uniform such that 2R Rd1; in other words, thevariance grows like window surface area. By contrast, itis much more unusual to find disordered systems that arealso hyperuniform. In recent years, evidence has beenemerging that disordered hyperuniform many-particle sys-tems can be regarded as new distinguishable states ofdisordered matter (see examples given in the Introduction).Whenever the structure factor goes to zero with the power-law form Sk jkj, the number variance has the follow-ing large-R asymptotic scaling that depends on the value ofthe exponent [17,32]:

2R 8 1

R : 10

Since disordered as well as ordered stealthy states can beviewed as systems in which tends to infinity, we seefrom Eq. (10) that they have the asymptotic scaling2R Rd1. We give theoretical predictions for thevariance of disordered stealthy ground states in Sec. VIII.

III. FAMILIES OF STEALTHY PAIR POTENTIALS

As we see in the next section, the specific form of astealthy potential does not affect the ground-state energymanifold, but it can affect other thermodynamic properties,such as the pressure. This has consequences in simulations

of such properties, especially with respect to convergenceissues. Hence, it is instructive to remark on some math-ematical aspects of the long-range nature of the direct-spacestealthy potentials, which are very similar to the weaklydecaying Friedel oscillations of the electron density in avariety of systems, including molten metals as well asgraphene [56,57]. As we will see, in some cases, stealthypotentials may mimic effective interactions that arise incertain polymer systems [10].Here, we will limit ourselves to pair potentials vr that

are radial functions in Rd, where r jrj (i.e., isotropic pairinteractions), and therefore, their Fourier transforms ~vkare also radial functions in Rd, where k jkj is a wavenumber. The d-dimensional Fourier transform of anyintegrable radial function fr in Rd is given by [31]

~fk 2d=2Z

0

rd1fr Jd=21krkrd=21 dr; 11

and the inverse transform of ~fk is given by

fr 12d=2Z

0

kd1 ~fk Jd=21krkrd=21 dk; 12

where Jx is the Bessel function of order .Consider the class of stealthy radial potential functions

~vk in Rd that are bounded and positive with compactsupport in the radial interval 0 k K, i.e.,

~vk VkK k; 13where, for simplicity, Vk is infinitely differentiable in theopen interval 0; K and

x

0 x < 0

1 x 014

is the Heaviside step function. The corresponding direct-space radial pair potential vr is necessarily a delocalized,long-ranged function that is integrable in Rd. Moreover,without any loss of generality, it will be assumed thatVk v0.For concreteness and purposes of illustration, we will

examine properties of two specific families of potentialsthat fall within the aforementioned wide class of stealthyinteractions: power-law and overlap potentials.

A. Power-law potentials

The power-law potentials are defined in Fourier space asfollows:

~vk v01 k=KmK k; 15where the exponent m can be any whole number. Thecorresponding direct-space potential vr will depend on dfor any given m and is exactly given by

FIG. 3. (Bravais) lattice with one particle per fundamental cell(left panel) and a periodic crystal with multiple particles perfundamental cell (right panel).


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vrv0

Kdm 1d 1=2 1F2a1; b1; b2; x

1m dd1=2 ;

16where a1 d 1=2, b1 1 m d=2, b2 1m d=2, x Kr2=4 and 1F2a1; b1; b2; x is aspecial case of the generalized hypergeometric functionpFqa1;; ap; b1;; bq; x [58]. Because the potential(16) is derived from the Fourier power-law potential (15),we will refer to Eq. (16) as the direct-space power-lawpotential. In the instance when m 0 in Eq. (15) (i.e.,simple step function), this expression for vr simplifies asfollows:

vrv0

K2r

d=2Jd=2Kr; 17

for which the large-r asymptotic behavior is given by

vrv0

K2

d1=2 cosKr d 1=4rd1=2

r :

18For any fixed d and m d, the direct-space power-lawpotential has the asymptotic form

vrv0

sr;m; drd1

r ; 19

where sr;m; d is a bounded function (a sinusoidalfunction or constant of order one). For any fixed d and1 m < d, the long-range oscillations of vr are con-trolled by an envelope that decays like 1=r, whered 1=2 < d 1.In Fig. 4, we plot the Fourier power-law potential for

selected values of m (which applies in any dimension)and the corresponding direct-space potentials for d 3.In all cases, we set v0 K 1. It is to be noted that theamplitudes of the oscillations in vr decrease as mincreases for a fixed dimension.

B. Overlap potentials

Let r;R represent the intersection volume of twoidentical d-dimensional spheres of radius R (scaled by thevolume of sphere) whose centers are separated by adistance r. This quantity is known analytically in anyspace dimension and has a variety of representations [59],including the following:

r;R cdZ

cos1r=2R

0

sindd; 20

where cd is the d-dimensional constant given by

cd 21 d=21=2d 1=2 : 21

For d 1, 2, 3, and 4, we respectively have

r;R 2R r1

r2R

; 22

r;R 2R r2

cos1

r2R

r2R

1

r2

4R2

1=2

; 23

r;R 2R r1

3

4

rR 1

16

rR

3; 24

r;R 2R r2

cos1

r2R

5r6R

1

12

rR

3

1 r2

4R2

1=2

: 25

0 0.5 1 1.5k

0

0.5

1

1.5

v(k)

m=0m=2m=4

~

10 20r

0

0.01v(r)

m=0m=2m=4

d=3

FIG. 4. Left panel: Fourier power-law potential ~vk for the special cases m 0, 2, and 4 that apply for any d. Right panel:Corresponding direct-space power-law potentials vr in the instance d 3. Here, we set v0 K 1.


021020-6

Consider the class of overlap potentials, which for anydimension is given by

~vk v0r k; R K=2: 26

Note that for d 1, the overlap potential is identical to thepower-law potential when d 1 and m 1. The thermo-dynamics of the ground-state manifold of this potential inthe special case d 2 was numerically investigated inRefs. [15] and [16]. It follows from Eq. (26) that thecorresponding direct-space overlap potential is given by

vrv0

1 d=2d=2

J2d=2Kr=2rd

; 27

which is clearly non-negative for all r. Its large-r asymp-totic behavior is given by

vrv0

41 d=2

d=21cos2Kr=2 d 1=4

rd1r;

28

revealing that the long-ranged decay of the direct-spaceoverlap potential has an envelope controlled by the inversepower law 1=rd1.Figure 5 depicts the overlap potential ~vk for the first

three space dimensions and the corresponding direct-spaceoverlap potentials vr, the latter of which vividly showsthe increasing decay rate of vr with increasing dimen-sion. The direct-space overlap potential vr is similar infunctional form to effective positive pair interactions thatarise in multilayered ionic microgels [10].

IV. ENSEMBLE THEORY FOR STEALTHYDISORDERED GROUND STATES: EXACT

RESULTS FOR THERMODYNAMIC PROPERTIES

Our general objective is the formulation of an ensembletheory for the thermodynamic and structural properties ofstealthy degenerate disordered ground states that wepreviously investigated numerically [1116]. In this sec-tion, we derive general exact relations for thermodynamic

properties that apply to any well-defined ensemble asgenerated by a particular way to sample the stealthydisordered ground-state manifold as a function of numberdensity . In the subsequent section, we derive some exactresults for the pair statistics for general ensembles.

A. Preliminaries

To begin, consider a configuration of N identicalparticles with positions rN r1;; rN in a large regionof volume V in d-dimensional Euclidean space Rd. Forparticles interacting via a pair potential vr, the totalpotential energy rN is given by

rN Xi

where ~vk and ~hk are the Fourier transforms of vr andhr, respectively, both of which are assumed to exist, k is awave vector, and Sk is the structure factor defined inrelation (6). Note that the structure factor is a non-negative,inversion-symmetric function, i.e.,

Sk 0 for all k; Sk Sk: 32

B. Ground-state energy and dimensionality of itsconfiguration space

Consider a radial (isotropic) stealthy potential function~vk with support in 0 k K of the class specified byEq. (13). In light of Eq. (31), it is clear that wheneverparticle configurations in Rd exist such that Sk is con-strained to achieve its minimum value of zero for0 k K, the system must be at its ground state or globalenergy minimum. This follows because the integrand~vkSk in the nontrivial term on the right side ofEq. (31) is identically zero because of the conflictingdemands of the step functions. When such configurationsexist, the average ground-state energy per particle in anywell-defined ensemble is given exactly by

u 2v0

1

2vr 0; 33

which is a structure-independent constant, depending onthe density , v0 ~vk 0, vr 0 and d, as explicitlyshown below. Importantly, because u is a constant inde-pendent of the structure, its value has no effect on theground-state manifold, which is generally degenerate;hence, this manifold is invariant to the specific choice ofthe stealthy function ~vk at fixed and d.We would like to express the energy (33) in terms of the

parameter , defined by relation (2), which measures therelative number of independently constrained degrees offreedom for a finite system under periodic boundaryconditions. Note that, in the thermodynamic limit, MKin Eq. (2) is simply half of the volume of a sphere of radiusK [due to the inversion symmetry of Sk] multiplied bythe density of the dual lattice [cf. Eq. (8)], i.e.,

MK v1K

2 v1K

22d ; 34

where we have used the fact that N in thisdistinguished limit. Hence, from Eq. (2), we obtain thefollowing expression for in the thermodynamic limit:

v1K2d2d ; 35

where

v1R d=2Rd

1 d=2 36

is the volume of a d-dimensional sphere (hypersphere) ofradius R. We see that for fixed K and d, which fixes thepotential, is inversely proportional to , which is thesituation that we usually consider in this paper [60].Hence, as tends to zero, tends to infinity, which

configurationally corresponds counterintuitively to theuncorrelated ideal-gas limit (Poisson distribution), as dis-cussed in the Introduction. As increases from zero, thedensity decreases and the dimensionality of the ground-state configuration manifold DC decreases. The configu-rational dimensionality per particle in the thermodynamiclimit, dC, can easily be obtained from the relation DC dN 2MK for a finite system [61]; specifically,

dC d1 2; 37

where dC limDC;NDC=N.Equations (33) and (35) yield the average ground-state

energy per particle to be

u v0

2 d

; min < ; 38

where min is the minimal density associated with the dualof the densest Bravais lattice in direct space (as elaboratedin Sec. VI), and

RRd ~vkdkv0v1K

2dvr 0

v0v1K39

is a constant whose value depends on the specific form ofthe stealthy-potential class ~vk defined by Eq. (13) andhence must lie in the interval 0; 1, where 1 corre-sponds to the step-function choice ~vk v0K k.While the system in the limit 0 ( ) correspondsconfigurationally to an ideal gas in so far as the paircorrelation function is concerned, as we will explain indetail in Sec. VII A, thermodynamically, it is nonideal; seeEq. (38) for u and Eq. (41) for the pressure.

C. Energy route to pressure and isothermalcompressibility

The pressure in the thermodynamic limit at T 0 can beobtained from the energy per particle via the relation

p 2uT: 40

Therefore, for stealthy potentials, we see from Eq. (33) thatthe ground-state pressure, for all possible values of or , isgiven by the following simple expression:


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p 2

2v0; min < : 41

Hence, the isothermal compressibility T 1=pTof such a ground state is

T v02: 42

We see that as tends to infinity, the compressibility tendsto zero.Two important remarks are in order. First, estimates of

the pressure obtained from simulations that we previouslyperformed for d 2 [15,16], as well as those carried out inthe present study across the first three space dimensions(Appendix B), are in very good agreement with the exactresult (41) across a wide range of densities, thus validatingthe accuracy of the simulations. Second, the fact that thepressure (41) is a continuous function of density impliesthat any phase transition that may take place could be acontinuous one, the implications of which are discussed inthe Conclusions.

D. Virial route to pressure and isothermalcompressibility

An alternative route to the pressure for a radial pairpotential function vr is through the virial equation,which at T 0 in the thermodynamic limit, is given by

p 2

2ds11

Z

0

rddvdr

g2rdr: 43

Although the pressure obtained via the virial route isgenerally expected to be equivalent to that obtained fromthe energy route (as described in the previous section), wewill show that, for a certain class of stealthy potentials, thepressure obtained from Eq. (43) is either ill defined ordivergent. This has practical implications for what typesof stealthy potentials can be used in constant-pressuresimulations.It is convenient to rewrite the virial relation (43) in the

following form:

p 2

2d

ZRdrdvdr

drZRdrdvdr

hrdr

2

2d

~Fk 0 12d

ZRd

~Fk ~hkdk; 44

where ~Fk is the Fourier transform of Fr rdv=dr,when it exists, and we have used Parsevals theorem anddefinition (5) for the total correlation function hr.To continue with this analysis, we make use of the

following lemma.

Consider a bounded radial function ~zk with compactsupport on the radial interval 0; K in Rd that is infinitelydifferentiable in the open interval 0; K. Therefore, itsFourier transform zr exists.Lemma 1.The Fourier transform of the radial function

wr rdz=dr in Rd is given by

~wk d ~zk d~zdk

: 45

Proof.Differentiation of ~zk [defined via Eq. (11)]with respect to k leads to the following identity:

kd~zdk

2d=2Z

0

krdzr Jd=2krkrd=21 dr: 46

The Fourier transform of wr is given by

~wk 2d=2Z

0

rddzdr

Jd=21krkrd=21 dr: 47

Integrating relation (47) by parts and using Eq. (46) provesthe lemma.Corollary.It immediately follows from Lemma 1 that

~wk has the same support as ~zk and

~wk 0 d ~zk 0; 48

meaning that the volume integral of rdz=dr over all space isproportional to the corresponding volume integral of zr.Note that by the Corollary of Lemma 1,

~F0 d ~v0, and hence we can rewrite the virialrelation (44) as

p 2

2d

d~vk 0 d2d

ZRd

~Fk ~hkdk

2

2d

d~vk 0 d

2dZRd

~Fkdk

2

2v0: 49

The second term in the second line of Eq. (49) followsbecause ~hk K k inside the exclusion sphere ofradius K [see also Eq. (52) below] and has support in thisexclusion zone by the Corollary of Lemma 1. But thissecond term must vanish in light of the trivial identity

Fr 0 rdvdr

r0

12dZRd

~Fkdk 0: 50

We see that the virial ground-state pressure (49) agrees withthat of the pressure obtained via the energy per particle[cf. Eq. (41)] provided that ~Fk exists. Since the latter is a


021020-9

stronger condition than the existence of ~vk, it is possibleto devise a stealthy function ~vk for which ~Fk does notexist and hence a virial pressure that either diverges or isnonconvergent. For example, this problem occurs for thepower-law potential (15) withm 0 (step function) for anydimension d. By contrast, the virial pressure is always welldefined for the overlap potential (26) in any dimension[62]. This example serves to illustrate the mathematicalsubtleties that can arise because of the long-ranged natureof stealthy potentials in direct space.

V. ENSEMBLE THEORY FOR STEALTHYDISORDERED GROUND STATES: EXACTINTEGRAL CONDITIONS ON THE PAIR

STATISTICS

Here, we derive some exact integral conditions that mustbe obeyed by both the pair correlation function g2r andthe structure factor Sk for stealthy ground states thatapply to general ensembles. These analytical relations canbe profitably employed to test corresponding computer-simulation results.

A. General properties

In any stealthy ground state, the structure factor attainsits minimum value Sk 0 for 0 < k K and hence hasthe form

Sk k K1 ~Qk; 51where x is the Heaviside step function defined byEq. (14) and ~Qk Sk 1 is a function that obeys theinequality ~Qk 1. Therefore, from Eq. (6), we havethat the Fourier transform of the total correlation functionhr has the form

~hk ~fk ~Pk; 52where

~fk K k 53and

~Pk k K ~Qk: 54

It is noteworthy that the function ~fk is identical to theMayer-f function for an equilibrium hard-sphere system indirect space.Taking the inverse Fourier transform of Eq. (52) yields

the direct-space total correlation function, given by

hr fr Pr; 55

where

fr K2r

d=2Jd=2Kr 56

and

Pr 12d=2Z

Kkd1 ~Qk Jd=21krkrd=21 dk

K2r

d=2Jd=2Kr ; 57

where the lower bound on Pr indicated in Eq. (57)follows from the fact that hr 1 for all r for any pointpattern. It trivially follows that since ~hk 0 RRd hrdr 1, the volume integral of Pr must be

zero, i.e.,

ZRdPrdr 0: 58

Less trivially, because the product ~vk ~Pk is zero for allk, by Parsevals theorem, we have the integral condition

ZRdvrPrdr 0: 59

Thus, the functions vr and Pr are orthogonal to oneanother. The exact integral conditions (58) and (59) can beused to test the accuracy of numerical methods that yieldestimates of the pair correlation function.

B. Behavior of the pair correlation functionnear the origin

It is instructive to determine the behavior of the paircorrelation function g2r for small r. Substitution of thegeneral form (11) for ~hk into the definition of the totalcorrelation function hr as obtained from Eq. (12), andexpanding hr in a Taylor series around r 0 throughsecond order in the radial distance r, yields

hr hr 0 12

2hr2r0

r2 Or4; 60

where

hr 0 2d 2d2Z

Kkd1 ~Qkdk; 61

and the corresponding curvature is

2hr2r0

2dd 2 2d

Z

Kkd1 ~Qkdk: 62

Therefore, from Eq. (61), we see that the pair correlationfunction at the origin is given by


021020-10

g2r 0 1 2d 2d2Z

Kkd1 ~Qkdk: 63

Since g2r must be non-negative for all r, we have thefollowing integral condition on ~Qk:

2d2Z

Kkd1 ~Qkdk 2d 1: 64

Hence, this integral must be positive for

1

2d: 65

We also conclude from Eq. (62) that for hr or g2r tohave positive curvature at the origin, ~Qk must obey theadditional integral condition:

d 2Z

Kkd1 ~Qkdk 1: 66

Finally, we note that when g2r 0 0, the results aboveyield the equality

Z

Kkd1 ~Qkdk 2d 1

2d2; 67

and, because the curvature must be positive in this instance,the inequality (66) must generally be obeyed.The inequality (64), conditional inequality (66), and

conditional equality (67) provide integral conditions to testthe accuracy of numerical methods that yield estimates ofthe structure factor.

VI. EXISTENCE OF STEALTHY DISORDEREDDEGENERATE GROUND STATES

It is noteworthy that any periodic crystal with a finitebasis is a stealthy ground state for all positive up to itscorresponding maximum value max (or minimum value ofthe number density min) determined by its first positiveBragg peak kBragg [minimal positive wave vector for whichSk is positive]. Tables IIV list the pair max,min forsome common periodic patterns in one, two, three, and fourdimensions, respectively, all of which are part of theground-state manifold; see Appendix A for mathematical

definitions. (The crystals denoted by Diad and Kagd are d-dimensional generalizations of the diamond and kagomcrystals, respectively, for d 2 [63].) While the mereexistence of such periodic ground states does not provideany clues about their occurrence probability in someensemble, we will use these results here to show howdisordered degenerate ground states arise as part of theground-state manifold for sufficiently small .At fixed d, the smallest value of min listed in Tables IIV,

which we call min, corresponds to the dual of the densestBravais lattice in direct space and represents the criticaldensity value below which a stealthy ground state does notexist for all k jkBraggj. The fact that min corresponds to thebody-centered-cubic (BCC) lattice for d 3 was initiallyshown analytically in Ref. [19] and subsequently numeri-cally in Ref. [14]. We note that the values of min for thesimple hexagonal lattice and hexagonal close-packed crystalfor d 3 reported in Ref. [19] are incorrect because thosecalculations were based on the erroneous assumption that thestructure factors at the corresponding shortest reciprocallattice vectors have nonvanishing values.Observe that in the case d 1, there is no non-Bravais

lattice (periodic structure with a basis n 2) for which maxis greater than 1=2, implying that the ground-state manifoldis nondegenerate (uniquely the integer lattice) for1=2 < 1. This case is to be contrasted with the casesd 2 where the ground-state manifold must be degenerate[65] for 1=2 < < max and nondegenerate only at thepoint max, as implied by Tables IIIV. Here, max is thelargest possible value of max in some fixed dimension.Lemma.At fixed K, a configuration comprised of the

union (superposition) of m different stealthy ground-stateconfigurations in Rd with 1; 2;; m, respectively, isitself stealthy with a value given by

Xmi1

1i

1; 68

which is the harmonic mean of the i divided by m.Proof.Formula (68) is a direct consequence of the fact

that is inversely proportional to the number density Pmi1 i of the union of the configurations in R

d, where iis the number density associated with the ith configuration,which is inversely proportional to i.This Lemma, together with the fact that any periodic

crystal with a finite basis is a stealthy ground state, can beused to demonstrate rigorously how complex aperiodicpatterns can be ground states, entropically favored or not. Asketch of such a proof would involve the consideration ofthe union of m different periodic structures in Rd withdensities 1; 2;; m, respectively, each of which arerandomly translated and oriented with respect to somecoordinate system such that m is very large but boundedand i j for all i and j. It is clear that the resultingconfiguration will be a highly complex aperiodic structure

TABLE I. Maximum values of and corresponding minimumvalues of for certain periodic stealthy ground states in R withK 1. The configuration with the largest possible value of max(smallest possible value of min) corresponds to the integer lattice.

Structure max min

Integer lattice (Z) 1 1=2 0.15915Periodic with n-particle basis 1=n n=2 0.15915n


021020-11

in Rd that tends toward a disordered stealthy pattern with avalue of that is very small but positive according torelation (68).

VII. PAIR STATISTICS IN THE CANONICALENSEMBLE: PSEUDO HARD SPHERES IN

FOURIER SPACE

The task of formulating an ensemble theory that yieldsanalytical expressions for the pair statistics of stealthydegenerate ground states is highly nontrivial because thedimensionality of the configuration space depends onthe density (or ) and there is a multitude of ways ofsampling the ground-state manifold, each with its ownprobability measure for finding a particular ground-stateconfiguration. Therefore, it is desirable to specialize toequilibrium ensembles with Gibbs measures because the

characterization of the ground states (as well as thecorresponding excited states) would be most tractabletheoretically. In particular, our objective is to deriveanalytical formulas for the pair statistics of stealthy dis-ordered ground states for sufficiently small in thecanonical ensemble as temperature T tends to zero; i.e.,the probability of observing a configuration is proportionalto exprN=kBT in the limit T 0. We show herethat under such circumstances, the pair statistics in thethermodynamic limit can be derived under the ansatz thatstealthy ground states behave remarkably like pseudo-equilibrium hard-sphere systems in Fourier space. Thisansatz enables us to exploit well-known accurate expres-sions for the pair statistics in direct space. As will be shown,agreement with computer simulations is excellent forsufficiently small .

A. Pseudo-hard-sphere ansatz

We have already noted that the step-function contribu-tion to ~hk for stealthy ground states, denoted by ~fk inrelation (52), is identical to the Mayer-f function for anequilibrium hard-sphere system in direct space. Thisimplies that the corresponding contribution to Sk is asimple hard-core step function k K, which can beviewed as an equilibrium hard-sphere system in Fourierspace with spheres of diameter K in the limit that tendsto zero. Why is this the case? Because such a step functionis exactly the same as the pair correlation g2r k of anequilibrium hard-sphere system in direct space in the limit

TABLE III. Maximum values of and corresponding minimum values of for certain periodic stealthy ground states in R3 withK 1. Here, MCC refers to the mean-centered cuboidal lattice, which is a Bravais lattice intermediate between the BCC and FCClattices and has an equivalent dual lattice [64]. The configuration with the largest possible value of max (smallest possible value of min)corresponds to the BCC lattice.

Structure max min

Pyrochlore crystal (Kag3) =412

p 0.2267 2=3 3p 3 0.01241Diamond crystal (Dia3) =2

12

p 0.4534 1=3 3p 3 0.00620Simple hexagonal lattice

3

p=9 0.6045 1=4 3p 3 0.00465

SC lattice (Z3 Z3) 2=9 0.6981 1=83 0.00403HCP crystal 8

6

p=81 0.7600 3 3p =32 2p 3 0.00370

FCC lattice (D3 A3) = 12p 0.9068 1=6 3p 3 0.00310MCC lattice 0.9258 0.00303BCC lattice (D3 D3) 2 2p =9 0.9873 1=8 2p 3 0.00285

TABLE IV. Maximum values of and corresponding minimumvalues of for certain periodic stealthy ground states in R4

with K 1. The configuration with the largest possible value ofmax (smallest possible value of min) corresponds to the four-dimensional checkerboard lattice D4 D4.Structure max min

Kag4 crystal 2=40 0.2467 5=323 0.001640Dia4 crystal 2=16 0.6168 1=163 0.0006416Z4 lattice 2=16 0.6168 1=163 0.0006416D4 lattice 2=8 1.2337 1=323 0.0003208

TABLE II. Maximum values of and corresponding minimum values of for certain periodic stealthy ground states in R2 withK 1. The configuration with the largest possible value of max (smallest possible value of min) corresponds to the triangular lattice.Structure max min

Kagom crystal (Kag2) =312

p 0.3022 3 3p =82 0.06581Honeycomb crystal (Dia2) =2

12

p 0.4534 3p =42 0.04387Square lattice (Z2 Z2) =4 0.7853 1=42 0.02533Triangular lattice (A2 A2) = 12p 0.9068 3p =82 0.02193


021020-12

that tends to zero. That the structure factor must have thebehavior Sk k K in the limit 0 is perfectlyreasonable since a perturbation about the ideal-gas limit[where Sk 1 for all k] in which an infinitesimal fractionof the degrees of freedom are constrained should onlyintroduce an infinitesimal change in Sk of zero inside theexclusion zone (constrained region). We call this theweakly constrained limit, where a step function Sk isexpected on maximum entropy grounds; it corresponds tothe most disordered (decorrelated) form of Sk subjectto the impenetrability condition in Fourier space. We referto this phenomenon as equilibrated pseudo hard spheres inFourier space because there are actually no points in thatspace that have a hard-core repulsion like true hard spheresdo in direct space.On the same maximum entropy grounds, we expect that

a perturbation expansion about the weakly constrained limit 0 will lead to a perturbation expansion in for Skthat can be mapped to the low-density expansion of g2rfor equilibrium hard spheres. More generally, we make theansatz that, in the canonical ensemble as T 0, this hard-sphere analogy continues to hold as is increased fromzero to positive values, provided that is small enough,implying that the collective coordinate variables ~nk(defined in the Introduction) are weakly correlated.Though the pseudo-hard-sphere picture must break downin some intermediate range of , for d 1 and d 2, thishard-sphere mapping is again exact when max, whichcorresponds to the maximal value of the packing fraction in these dimensions (see Tables I and II). This exactcorrespondence with the maximal value of when max does not hold for d 3 or d 4, however. Thus, oneshould only expect that and are proportional to oneanother, even at small values.Under the pseudo-hard-sphere ansatz, the direct-space

pair correlation function gHS2 r; of a disordered hard-sphere system at a packing fraction for sufficiently small can be mapped into the structure factor Sk; for adisordered stealthy ground state derived from the canonicalensemble at fixed for sufficiently small as follows:

Sk; gHS2 r k; : 69

As alluded to above, the parameter can be viewed as aneffective packing fraction for pseudo hard spheres ofdiameter K in reciprocal space that is proportional to , i.e.,

bd; 70

where bd is a d-dependent parameter that is to bedetermined. Let hHSr be the total correlation functionof a disordered equilibrium hard-sphere system in directspace and let us define, for stealthy ground states,

~Hk Sk 1 ~hk: 71

The ansatz is also defined by the alternative mapping

~Hk hHSr k: 72This mapping then enables us to exploit the well-knownstatistical-mechanical theory of equilibrium hard-spheresystems. In particular, we can employ a generalizedOrnstein-Zernike convolution relation that defines theappropriate direct correlation function ~Ck, namely,

~Hk ~Ck ~Hk ~Ck; 73where the symbol denotes the convolution operation inRd. Therefore, in direct space,Hr is given by the relationof the following form:

Hr Cr1 2dCr : 74

For example, for d 1,

~Ck 1 k 1 k1 2 : 75

Inverting this function yields

Cr r sinr rsinr cosr 1r21 2 : 76

For d 2 and d 3, one can use the Percus-Yevickclosure of the Ornstein-Zernike integral equation [54],which is highly accurate for low to intermediate densitiesalong the liquid branch, or when mapped to the stealthyproblem, for low to intermediate values of .It is noteworthy that the exact low-density expansion of

hHSr, for practical purposes, is sufficient to produceaccurate estimates of ~Hk ~hk and its counterparthr for low to intermediate values of or . In particular,using the mapping (72), we obtain, for any dimension d, thefollowing low- expansion of ~hk:

~hk K k1 2dbdk;K O2;77

where bd is the proportionality constant in Eq. (70) andk;K is the scaled intersection volume of two identical d-dimensional spheres of diameter K whose centers areseparated by a distance k [cf. Eq. (20)] [66]. This formulaindicates that Sk develops a peak value at k K (overand above the value of unity due to the step function in thelimit 0) and then monotonically decreases untilk 2K, where it achieves its long-range value of unityfor all k > 2K, which we will see is verified by computersimulations. Fourier inversion of Eq. (77), division by ,and use of (35) yields a corresponding low- expansion of


021020-13

the total correlation function hr through second order in and hence has an error term of order 3.To get an idea of the large-r asymptotic behavior of the

pair correlations, consider the limit 0 for any d. In thislimit, the total correlation function hr for any r obtainedfrom Eq. (77) is given by

hr K2r

d=2Jd=2Kr 0; 78

which for large r is given asymptotically by

hr 1rd1=2

cosr d 1=4 r :79

Thus, the longed-ranged oscillations of hr are controlledby the power law 1=rd1=2. Equation (78) indicates thatin the limits 0 and , hr 0, and therefore,the pair correlation function tends to the ideal gas eventhough the structure factor [Eq. (69)] cannot tend to theideal-gas form because of its stealthy property. This result isin contrast to the situation considered in Fig. 1, where wetake the limit by fixing N and letting vF 0. Inthat case, both g2r and Sk tend to the associated ideal-gas forms, i.e., g2r 1 for all r and Sk 1 for all k.

B. Comparison of theoretical predictions to simulations

In order to test our theoretical results for the pair statisticsof stealthy ground states in the canonical ensemble, wehave carried out computer simulations to generate andsample such configurations, the details of which aredescribed in Appendix B. In all cases, we take K 1,which sets the length scale. Our simulation results revealthat the functional trends for Sk and g2r predicted bythe ansatz of pseudo hard spheres in Fourier space with aneffective packing fraction are remarkably accurate for amoderate range of about 0. Because it is theoreticallyhighly challenging to ascertain the proportionality constantbd in Eq. (70) that arises in Eq. (77), we must rely on thesimulations to guide us in its determination. First, weobserve that for d 1, the mapping between and is oneto one, i.e., b1 1. Second, the simulation data suggestthat, to an excellent approximation, bd for d 2 is givenby assuming that the peak value of Sk or ~hk, achievedat k K for sufficiently small , is invariant with respect tothis peak value as in the one-dimensional case, andconsequently bd K;K2d1.Figure 6 shows that the structure factor Sk, as obtained

from Eqs. (71) and (77), is in excellent agreement with thecorresponding simulated quantities for 0.05, 0.1, and0.143 for d 3. In Fig. 7, we compare our theoreticalresults for the pair correlation function g2r, as obtainedby Fourier inversion of Eq. (77), to corresponding simu-lation results across the first three space dimensions.Again, we see excellent agreement between theory and

k0

0.5

1

1.5

S(k)

TheorySimulation

d=3, =0.05

4k

0

0.5

1

1.5

S(k)

TheorySimulation

d=3, =0.1

0 1 2 3 4 0 1 2 3 0 1 2 3 4k

0

0.5

1

1.5

S(k)

TheorySimulation

d=3, =0.143

FIG. 6. Comparison of theoretical and simulation results for the structure factor Sk for 0.05, 0.1, and 0.143 for d 3.Here, K 1.

0 5 10r

0

0.5

1

1.5

g 2(r) d=1, Simulation

d=2, Simulationd=3, Simulationd=1, Theoryd=2, Theoryd=3, Theory

=0.05

0 5 10r

0

0.5

1

1.5



=0.1

0 5 10r

0

0.5

1

1.5



=0.143

FIG. 7. Comparison of theoretical and simulation results for the pair correlation function g2r for 0.05, 0.1, and 0.143 across thefirst three space dimensions. Here, K 1.


021020-14

simulations, which validates the pseudo-hard-sphereFourier-space ansatz. Figure 8 depicts our theoreticalpredictions for g2r for 0.15 across the first fourspace dimensions. It is seen that increasing dimensionalityincreases short-range correlations.

C. Translational order or disorder metric

We have seen that both short- and long-scale correlationsincrease as increases. A useful scalar positive order metricthat captures the degree to which translational orderincreases with is given by

1Dd

ZRdh2rdr

12dDdZRd

~h2kdk; 80

where we have used Parsevals theorem and D is somecharacteristic length scale [68]. Note that for an ideal gas(spatially uncorrelated Poisson point process), 0because hr 0 for all r. Thus, a deviation of fromzero measures translational order with respect to the fullyuncorrelated case. Because diverges for any perfectcrystal, it is a quantity that is better suited to distinguishthe degree of pair correlations in amorphous systems.In the case of stealthy ground-state configurations, is

given explicitly by the relation

12d2DdZRd

~H2kdk; 81

where ~Hk is given by Eq. (71). Substitution of theleading-order term in the expansion (77) into Eq. (81)yields

4d22dv11

2 O3; 82

where we have taken D K1. Thus, for stealthy groundstates, the order metric grows quadratically with for

small . Since the error is of order 3, we expect that thisquadratic form will be a very good approximation of up tomoderately large values of . Indeed, this is confirmed byour simulations up to 0.25. Note that because stealthydisordered ground states (for sufficiently small ) arepseudo-equilibrium hard-sphere systems in Fourier space,the form of the order metric [Eq. (81)] ensures that it willbehave similarly to for equilibrium hard spheres in directspace for low densities.

VIII. LOCAL NUMBER VARIANCE FORSTEALTHY HYPERUNIFORM DISORDERED

GROUND STATES

Here, we investigate theoretically the local numbervariance for stealthy disordered ground states as a functionof and then use these results to extract an order metric[31] that describes the extent to which large-scale densityfluctuations are suppressed as increases in these hyper-uniform systems (see Sec. II A). The local number variance2R associated with a general statistically homogeneousand isotropic point process in Rd at number density for aspherical window of radius R is determined entirely by paircorrelations [31]:

2R v1R1

ZRdhrr;Rdr

v1R

1

2dZRdSk ~k;Rdk

; 83

where v1R is the d-dimensional volume of a sphericalwindow [cf. Eq. (36)], hr is the total correlation function[cf. Eq. (5)], r;R is the scaled intersection volume oftwo spherical windows of radius R, as given by Eq. (20),and ~k;R is the Fourier transform of r;R, which isexplicitly given by [31]

~k;R 2dd=21 d=2 Jd=2kR2

kd: 84

We have already noted that the stealthy ground statesconsidered in the present paper are hyperuniform, i.e.,Sk 0 as k 0 (see Sec. II A). This means that suchsystems obey the sum rule

RRd hrdr 1 and, because

of the rapid manner in which Sk vanishes in the limitk 0, the number variance has the following large-Rasymptotic behavior [31]:

2R RRd1 ORd3; 85where R is a bounded function that oscillates around anaverage value

limL

1

L

ZL

0

RdR: 86

0 5 10r

0

0.5

1

1.5

g 2(r) d=1

d=2d=3d=4

=0.15

FIG. 8. Theoretical predictions for the pair correlation functiong2r for 0.15 across the first four space dimensions.Here, K 1.


021020-15

The scaling (85) occurs for a broader class of hyperuniformsystems, as specified by relation (10). The parameter isan order metric that quantifies the extent to which large-scale density fluctuations are suppressed in such hyper-uniform systems [31]. To compare different hyperuniformsystems, Torquato and Stillinger used the followingrescaled order metric:

B d1=d

; 87

which is independent of the density, where v11=2.Among all hyperuniform point patterns having the scaling(85), B is minimized (greatest suppression of large-scaledensity fluctuations) for the integer, triangular, BCC, andD4 lattices for d 1, 2, 3, and 4, respectively [31,32].Using the analytical results for g2r or Sk described in

the previous section, we have computed relation (83) for2R versus R for selected values of across the first threedimensions and compared them to our correspondingsimulation results; see Fig. 9. The analytical and numerical

results are in excellent agreement with one another. Table Vlists the order metric B for various values of across thefirst four dimensions for disordered stealthy ground states,as obtained from the analytical estimates of 2R andEq. (87). These results are also compared to the corre-sponding optimal values. As expected, B decreases as increases for fixed d.

IX. NEAREST-NEIGHBOR FUNCTIONS

Here, we obtain theoretical predictions for the nearest-neighbor functions of stealthy disordered ground states.Nearest-neighbor functions describe the probability offinding the nearest point of a point process in Rd at somegiven distance from a reference point in space. Suchstatistical quantities are called void or particle near-est-neighbor functions if the reference point is an arbitrarypoint of space or an actual point of the point process,respectively [71]. Our focus here is on the particle nearest-neighbor functions.The particle nearest-neighbor probability density func-

tion HPr is defined such that HPrdr gives the prob-ability that the nearest point to the arbitrarily chosen pointlies at a distance between r and r dr from this chosenpoint of the point process. The probability that a sphere ofradius r centered at a point does not have other points,called the exclusion probability EPr, is the associatedcomplementary cumulative distribution function and soEPr 1

Rr0 HPxdx and hence HPr EP=r.

The nearest-neighbor functions can be expressed as aninfinite series whose terms are integrals over n-bodycorrelation functions defined in Sec. II [71,72]. In general,an exact evaluation of this infinite series is not possiblebecause the gn are not known accurately for n 3, exceptfor simple cases, such as the Poisson point process.Theoretically, one must either devise approximations orrigorous bounds to estimate nearest-neighbor quantities forgeneral models [71,73].

TABLE V. The order metric B for various values of across thefirst four dimensions for disordered stealthy ground states forselected values of up to 0.25, as obtained from theanalytical estimates of 2R and Eq. (87). Included for com-parison are the structures in each dimension that have the minimalvalues of B, all of which are Bravais lattices [31,32].

d 1 d 2 d 3 d 40.05 2.071 1.452 2.164 4.5600.1 1.051 1.040 1.738 3.8750.143 0.745 0.880 1.558 3.5760.2 0.54 0.755 1.411 3.3270.25 0.439 0.683 1.325 3.179Integer lattice 0.167Triangular lattice 0.508BCC lattice 1.245D4 lattice 2.798

R0

1

2

3

4

2 (R

)/Rd-

1

d=1, Simulationd=2, Simulationd=3, Simulationd=1, Theoryd=2, Theoryd=3, Theory

=0.05

R0

0.5

1

1.5

2

2 (R

)/Rd-

1


=0.1

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10R

0

0.2

0.4

0.6

0.8

1

1.2

2 (R

)/Rd-

1


=0.2

FIG. 9. Comparison of analytical and numerical results for thenumber variance for 0.05, 0.1, and 0.2 across the first threespace dimensions. Here, K 1.


021020-16

Torquato has given rigorous upper and lower bounds onthe so-called canonical n-point correlation function Hn forpoint processes in Rd [72,74]. Since nearest-neighborfunctions are just special cases of Hn, then we also havestrict bounds on them for such models [71,72]. Here, weemploy upper and lower bounds on EPr, which relies onknowledge of the pair correlation function:

EPr 1 Zr; 88

EPr expZr; 89

where Zr is the cumulative coordination number[cf. Eq. (4)]. The upper bound (89) was presentedin Ref. [47].These bounds are evaluated for stealthy ground-state

configurations using the analytical expression for the paircorrelation function given in Sec. VII. Figures 10 and 11compare these bounds to our numerical results for both 0.05 and 0.1 for d 2 and d 3, respectively.We see that the bounds on EPr provide the correctqualitative trends as a function of r, the upper bound beingthe sharper of the two bounds for these cases.The mean nearest-neighbor distance is defined as the

first moment of HPr or, equivalently, zeroth moment ofEPr, i.e.,

Z

0

rHPrdr Z

0

EPrdr: 90

For an ideal gas (Poisson point process) at number density, the mean nearest neighbor can be explicitly given in anydimension [72]:

Ideal 1 1=d

2v11=21=d: 91

Using the upper bound (89) and relation (90), we plot inFig. 12 upper bounds on the mean nearest-neighbordistance , scaled by the corresponding ideal-gas quantityobtained from Eq. (91), as a function of for the first fourspace dimensions. For fixed , the upper bounds on =idealdecrease as the space dimension increases, as expected, andtends to unity in the large-d limit, consistent with the so-called decorrelation principle [47,59].

r

0

0.5

1

1.5

E p(r)

SimulationTheoretical Lower BoundTheoretical Upper Bound

d=2, =0.05

0 1 2 3 0 1 2 3 4 5r

0

0.5

1

1.5

E p(r)


d=2, =0.1

FIG. 10. Comparison of the lower and upper bounds (88) and (89) to our numerical results for the exclusion probability function EPrfor 0.05 and 0.1 for d 2. Here, K 1.

r

0

0.5

1

1.5

E p(r)


d=3, =0.05

0 1 2 3 4 0 1 2 3 4 5r

0

0.5

1

1.5

E p(r)


d=3, =0.1

FIG. 11. Comparison of the lower and upper bounds (88) and (89) to our numerical results for the exclusion probability function EPrfor 0.05 and 0.1 for d 3. Here, K 1.

0 0.05 0.1 0.15 0.2

1

1.1

1.2

1.3

1.4

1.5

/ I

deal

d=1d=2d=3d=4

FIG. 12. Upper bounds on the mean nearest-neighbor distance, scaled by ideal [cf. Eq. (91)], as a function of for the first fourspace dimensions.


021020-17

X. EXCITED STATES: STRUCTURE FACTORAND THERMAL EXPANSION COEFFICIENT

Here, we derive accurate analytical formulas for thestructure factor and thermal expansion coefficient for theexcited states associated with stealthy ground states atsufficiently small temperatures. We see from the compress-ibility relation (3) that if the isothermal compressibility Tis bounded, then S0 must be zero for any ground state,stealthy or not. Recall that for stealthy ground states, T isbounded according to Eq. (42). Now consider excited statesinfinitesimally close to the stealthy ground states, i.e., whentemperature T is positive and infinitesimally small. Underthe highly plausible assumption that the structure of suchexcited states will be infinitesimally near the ground-stateconfigurations for sufficiently small and T, then to anexcellent approximation, the pressure is given by

p T 2

2; 92

where the first term is the ideal-gas contribution and thesecond term is the configurational contribution, which,under the stated conditions, is effectively the same as theground-state expression (41), where we have setkB v0 K 1. Thus, relation (92) yields the isothermalcompressibility T T1, which, when substi-tuted into Eq. (3) for large (small ) and small T, yieldsthat S0 varies linearly with T for such excited states:

S0 CdT; 93

in units kB v0 K 1, where Cd 2d2d=v11is a d-dependent constant.Figure 13 shows that the prediction of relation (93) is in

excellent agreement with our MD simulation results(Appendix B) in the case d 2. It is expected that thispositive value of S0 will be the uniform value of Sk for0 k K for the special case of the step-function power-law potential ~vK [the casem 0 in Eq. (15)] for small .

This behavior of Sk has indeed been verified by oursimulation results in various dimensions. For other stealthy-potential function choices, Skwill no longer be a constantfor 0 k K.An interesting conclusion to be drawn from this analysis

is that, for a system, S0 can be arbitrarily close to zero atpositive temperatures, even if T is itself arbitrarily small.This means that, for all practical purposes, such systems atpositive T are effectively hyperuniform. Perfect hyper-uniformity is not necessarily required in order to achievenovel physical properties in technological applications.Usingthecyclic identity=TpT=pp=T

1 and approximation (92), it immediately follows thatthe thermal expansion coefficient 1=Tp, forsufficiently small and T, is given by

C2d2T: 94We see that the thermal expansion is positive under suchconditions, which is to be contrasted with the anomalousnegative thermal expansion behavior for sufficiently large over a low temperature range demonstrated in our earliernumerical work [15,16].

XI. CONCLUSIONS AND DISCUSSION

Stealthy hyperuniform disordered ground states in Rd areinfinitely degenerate and arise from a class of bounded long-ranged pair potentials with compactly supported Fouriertransforms. Such exotic many-particle states of matter werepreviously studied only numerically. Because the configu-rational dimensionality depends on the density (or ), ahighly unusual situation, and there are an infinite number ofdistinct ways to sample the ground-state manifold, each withits own probability measure, it has been theoretically verychallenging to devise predictive ensemble theories. A newtype of statistical-mechanical theory needed to be invented.This paper has initiated such a theoretical program.Specifically, we have derived general exact relations for the

ground-state energy, pressure, and isothermal compressibilitythat apply to any ensemble as a function of the number density in any dimension d. We demonstrated how disordereddegenerate ground states can arise as part of the ground-statemanifold.We also obtained exact integral conditions that boththe pair correlation function g2r and structure factor Skmust satisfy in any ensemble. Then,we specialized our resultsto the canonical ensemble in the zero-temperature limit byexploiting an ansatz that stealthy states behave like pseudo-equilibrium hard-sphere systems in Fourier space [75]. Theresulting theoretical predictions for g2r and Sk wereshown to be in excellent agreement with computer simu-lations across the first three space dimensions for sufficientlysmall . These results were used to theoretically obtain ordermetrics, local number variance, and nearest-neighbor func-tions across dimensions. We also derived accurate analyticalformulas for the structure factor and thermal expansion

10-7 10-6 10-5 10-4 10-3T

10-6

10-5

10-4

10-3

10-2

S(0)

SimulationTheory

d=2, =0.1

FIG. 13. Comparison of the theoretically predicted structurefactor at the origin S0 versus absolute temperature T, asobtained from Eq. (93), to our corresponding simulation resultsfor a two-dimensional stealthy system at 0.1. Here, we takekB v0 K 1.


021020-18

coefficient for the excited states associated with stealthyground states at sufficiently small temperatures. Our analysesprovide new insights onour fundamental understandingof thenature and formation of low-temperature states of amorphousmatter. Our work also offers challenges to experimentalists tosynthesize stealthy ground states at the molecular level,perhaps with polymers, as suggested in Sec. III.There are many remaining open theoretical problems.

While the pseudo-hard-sphere system picture for the canoni-cal ensemble is almost surely exact in the limit 0, a futurechallenge would be to provide rigorous justification for thispicture for positive but small . One possible avenue thatcould be pursued is the formulation of an exact perturbationtheory for the pair statistics about theweakly constrained limit( 0).As noted in the Introduction,while the configurationspace is fully connected for sufficiently small , quantifyingits topology as a function of up to max is an outstandingopen problem. At some intermediate range of , the topologyof the ground-state manifold undergoes a sequence of one ormore disconnection events, but this process is poorly under-stood and demands future study. In the limit max, thedisconnection becomes complete at the unique crystal groundstate [77].The simple constraint and degrees-of-freedom counting

arguments described earlier lead to definite predictions forthe entropically favored stealthy ground states derived fromthe canonical ensemble in the limit T 0. For between 0and 1=2, the ground states are disordered and possess aconfigurational dimension per particle dC of d1 2; seeEq. (37). At 1=2, the configurational dimensionalityper particle collapses to zero, and there is a concomitantphase transition to a crystal phase. The fact that the pressureis a continuous function of density [cf. Eq. (41)] for all upto max (see Tables IIV) implies that any phase transitioncould be continuous. While this eliminates a first-orderphase transition in which the phase densities are unequal, itdoes not prohibit such a phase transition in which the twodistinct phases possess the same number density. At 0.5, our numerical evidence indicates that the struc-turally distinct fluid and crystal phases have equal freeenergies. This does not necessarily imply that the twophases could coexist side by side within the systemseparated by an interface. This phase diagram is depictedin Fig. 14, which applies to the first four space dimensions.For d 1, the only crystal phase allowed is the integerlattice (see Sec. VI), and hence there can be no phasecoexistence. For d 2, our simulations [1416,76] indi-cate that the crystal phase is the triangular lattice for1=2 max. However, for d 3, it is possible thatthere may be more than one crystal phase. For example,while for sufficiently high up to max, we expect the stablecrystal to be the BCC lattice, our simulations cannoteliminate the possibility that the FCC lattice is a stablephase for some in the range 1=2 0.9068; for > 0.9068, the FCC lattice cannot be a ground state

(see Table III). Since four dimensions is more similar to twodimensions in that the lattice corresponding to max isequivalent to its dual, we would expect that theD4 lattice isthe stable crystal for 1=2 max, but this remains to beconfirmed.All of our previous and current simulations for the first

three space dimensions [1216] strongly suggest that all ofthe energy minima attained were global ones for < 0.5,but when > 0.5, the topography of the energy landscapesuddenly exhibits local minima above the ground-stateenergies. The possible configurations that can arise as partof the ground-state manifold for > 1=2, regardless oftheir probability of occurrence, not only include periodiccrystals for d 2, as discussed in Sec. VI, but alsoaperiodic structures, reflecting the complex nature of theenergy landscape. For example, for d 2, the manifoldincludes generally aperiodic wavy phases, which havebeen shown to arise via numerical energy minimizationsfrom random initial conditions with high probability in arange of where the triangular lattice is entropicallyfavored [12,14]. A deeper understanding of such aspectsof the ground-state manifold would undoubtedly shed lighton the topography of the energy landscape.

ACKNOWLEDGMENTS

This research was supported by the U.S. Departmentof Energy, Office of Basic Energy Sciences, Division ofMaterials Sciences and Engineering, under Grant No. DE-FG02-04-ER46108.

APPENDIX A: COMMON d-DIMENSIONALLATTICES

Common d-dimensional lattices include the hypercubicZd, checkerboard Dd, and root Ad lattices, defined,respectively, by

Zd fx1;; xdxi Zg for d 1; A1

Dd fx1;; xd Zdx1 xd eveng for d 3;A2

Ad fx0; x1;; xd Zd1x0 x1 xd 0gfor d 1; A3

where Z is the set of integers ( 3;2;1; 0; 1; 2; 3);x1;; xd denote the components of a lattice vector of either

max(

min)Disordered Crystal

FIG. 14. Phase diagram for the entropically favored stealthyground states in the canonical ensemble as a function of whichapplies to the first four space dimensions.


021020-19

Zd or Dd; and x0; x1;; xd denote a lattice vector of Ad.The d-dimensional lattices Zd , Dd, and A

d are the corre-

sponding dual lattices. Following Conway and Sloane [78],we say that two lattices are equivalent or similar if onebecomes identical to the other possibly by a rotation,reflection, and change of scale, for which we use thesymbol . The Ad and Dd lattices can be regarded as d-dimensional generalizations of the face-centered-cubic(FCC) lattice defined by A3 D3; however, for d 4,they are no longer equivalent. In two dimensions, A2 A2defines the triangular lattice with a dual lattice that isequivalent. In three dimensions, A3 D3 defines the body-centered-cubic (BCC) lattice. In four dimensions, thecheckerboard lattice and its dual are equivalent, i.e.,D4 D4. The hypercubic lattice Zd Zd and its duallattice are equivalent for all d.

APPENDIX B: SIMULATION PROCEDURE TOGENERATE AND SAMPLE STEALTHY GROUND

STATES IN THE CANONICAL ENSEMBLE

To numerically sample the stealthy ground-statemanifold in the disordered regime in the canonicalensemble in the T 0 limit for d 1, 2, and 3, weperformed molecular dynamics (MD) simulations ata very low dimensionless equilibration temperatureTE kBT=v0Kd, periodically took configurationalsnapshots, and then used these configurations as inputto the L-BFGS optimization algorithm [79] to get thecorresponding ground states. The dimensionless temper-atures that we use are TE 2 104 for d 1, TE 2 106 for d 2, and TE 1 106 for d 3. Theequilibration temperature at a fixed dimension was chosenso that no changes in the pair correlation function areobserved over some range of equilibration temperatures.The MD simulations were first performed in the micro-canonical ensemble using the velocity Verlet algorithm[80]. The time steps were chosen so that the relative energychange every 3000 time steps is less than 108. However, toenforce the desired temperature, we also performed MDsimulations in the canonical ensemble using an Andersonthermostat [80]. We employed fifteen million time steps toequilibrate a system. After that, a snapshot was taken every3000 time steps for further energy minimization. Becausestealthy potentials in direct space are long-ranged, theenergy is most accurately calculated in Fourier space usingEq. (1). The numerical errors in the achieved ground-state energies are extremely small, usually on the order of1020 (in units of v0Kd). The force on the jth particle iscalculated using the gradient of Eq. (1), yielding Fj jrN 1=vFPkk ~vkIm ~nk expik rj. Thenumber of particles in the simulation box, N, is calculatedfrom Eq. (2) for given d, , and MK. We chose MK 50 for d 1, MK 54 for d 2, and MK 39for d 3. The fundamental cell employed is the one

corresponding to the crystal with the largest value ofmax in each dimension (see Tables IIV).Structural characteristics, such as the pair correlation

function g2r, structure factor Sk, number variance2R, and nearest-neighbor function EPr, are obtainedby sampling each generated configuration for a fixed valueof and ensemble averaging over at least 20000 configu-rations. Two power-law potentials (15) were used: one withm 0 and the other with m 2. As expected, bothpotentials produced the same ensemble-averaged structuralproperties to within small numerical errors (as explained inSec. IV B), the agreement of which provides a good test onthe validity of the simulation results. Our simulation resultsfor g2r and Sk also satisfied the exact integral con-ditions presented in Secs. VA and V B. Additional simu-lation details will be described elsewhere [76].

[1] F. H. Stillinger, Phase transitions in the Gaussian coresystem, J. Chem. Phys. 65, 3968 (1976).

[2] D. Frenkel, B. M. Mulder, and J. P. McTague, PhaseDiagram of a System of Hard Ellipsoids, Phys. Rev. Lett.52, 287 (1984).

[3] P. M. Chaikin and T. C. Lubensky, Principles of CondensedMatter Physics (Cambridge University Press, New York,1995).

[4] M. Guenza and K. S. Schweizer, Local and microdomainconcentration fluctuation effects in block copolymer sol-utions, Macromolecules 30, 4205 (1997).

[5] J. C. Crocker, M. T. Valentine, E. R. Weeks, T. Gisler, P. D.Kaplan, A. G. Yodh, and D. A. Weitz, Two-Point Micro-rheology of Inhomogeneous Soft Materials, Phys. Rev. Lett.85, 888 (2000).

[6] M. P. Valignat, O. Theodoly, J. C. Crocker, W. B. Russel,and P. M. Chaikin, Reversible self-assembly and directedassembly of DNA-linked micrometer-sized colloids, Proc.Natl. Acad. Sci. U.S.A. 102, 4225 (2005).

[7] B. M. Mladek, D. Gottwald, G. Kahl, M. Neumann, andC. N. Likos, Formation of Polymorphic Cluster Phases fora Class of Models of Purely Repulsive Soft Spheres, Phys.Rev. Lett. 96, 045701 (2006).

[8] C. E. Zachary, F. H. Stillinger, and S. Torquato, Gaussian-core model phase diagram and pair correlations in highEuclidean dimensions, J. Chem. Phys. 128, 224505 (2008).

[9] B. Capone, I. Coluzza, F. LoVerso, C. N. Likos, and R.Blaak, Telechelic Star Polymers as Self-Assembling Unitsfrom the Molecular to the Macroscopic Scale, Phys. Rev.Lett. 109, 238301 (2012).

[10] C. Hanel, C. N. Likos, and R. Blaak, Effective interactionsbetween multilayered ionic microgels, Materials 7, 7689(2014).

[11] Y. Fan, J. K. Percus, D. K. Stillinger, and F. H. Stillinger,Constraints on collective density variables: One dimension,Phys. Rev. A 44, 2394 (1991).

[12] O. U. Uche, F. H. Stillinger, and S. Torquato, Constraints oncollective density variables: Two dimensions, Phys. Rev. E70, 046122 (2004).


021020-20

[13] O. U. Uche, S. Torquato, and F. H. Stillinger, Collectivecoordinates control of density distributions, Phys. Rev. E74, 031104 (2006).

[14] R. D. Batten, F. H. Stillinger, and S. Torquato, Classicaldisordered ground states: Super-ideal gases, and stealthand equi-luminous materials, J. Appl. Phys. 104, 033504(2008).

[15] R. D. Batten, F. H. Stillinger, and S. Torquato, Novel Low-Temperature Behavior in Classical Many-Particle Systems,Phys. Rev. Lett. 103, 050602 (2009).

[16] R. D. Batten, F. H. Stillinger, and S. Torquato, Interactionsleading to disordered ground states and unusual low-temperature behavior, Phys. Rev. E 80, 031105 (2009).

[17] C. E. Zachary and S. Torquato, Anomalous local co-ordination, density fluctuations, and void statistics indisordered hyperuniform many-particle ground states,Phys. Rev. E 83, 051133 (2011).

[18] S. Martis, . Marcotte, F. H. Stillinger, and S. Torquato,Exotic ground states of directional pair potentials viacollective-density variables, J. Stat. Phys. 150, 414(2013).

[19] A. St, Crystalline Ground States for Classical Particles,Phys. Rev. Lett. 95, 265501 (2005).

[20] Indeed, the number of ground-state degeneracies is un-countably infinite for a finite number of particles, whichdistinguishes it from disordered ground states found inclassical Ising-like spin systems in which the number ofdegeneracies is finite for a finite number of spins [2124].We note in passing that while quantum spin liquids havedisordered ground states, they are effectively unique [21].

[21] L. Balents, Spin liquids in frustrated magnets, Nature(London) 464, 199 (2010).

[22] C. L. Henley, The Coulomb Phase in frustrated systems,Annu. Rev. Condens. Matter Phys. 1, 179 (2010).

[23] R. A. DiStasio, . Marcotte, R. Car, F. H. Stillinger, and S.Torquato, Designer spin systems via inverse statisticalmechanics, Phys. Rev. B 88, 134104 (2013).

[24] . Marcotte, R. A. DiStasio, F. H. Stillinger, and S.Torquato, Designer spin systems via inverse statisticalmechanics II. Ground-state enumeration and classification,Phys. Rev. B 88, 184432 (2013).

[25] M. Florescu, S. Torquato, and P. J. Steinhardt, Designerdisordered materials with large complete photonic bandgaps, Proc. Natl. Acad. Sci. U.S.A. 106, 20658 (2009).

[26] M. Florescu, P. J. Steinhardt, and S. Torquato, Opticalcavities and waveguides in hyperuniform disordered pho-tonic solids, Phys. Rev. B 87, 165116 (2013).

[27] W. Man, M. Florescu, E. P. Williamson, Y. He, S. R.Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato,P. M. Chaikin, and P. J. Steinhardt, Isotropic band gapsand freeform waveguides observed in hyperuniform disor-dered photonic solids, Proc. Natl. Acad. Sci. U.S.A. 110,15886 (2013).

[28] More generally, stealthy configurations can be those groundstates that correspond to minimizing Sk to be zero at othersets of wave vectors, not necessarily in a connected setaround the origin, specific examples of which were inves-tigated in Ref. [14]. We have also used the collective-coordinate technique to target more general forms of thestructure factor for a prescribed set of wave vectors such

that Sk is not minimized to be zero in this set (e.g., power-law forms and positive constants) [13,14,17]. There, theresulting configurations are the ground states of interactingmany-particle systems with 2-, 3-, and 4-body interactions.

[29] Since both the real and imaginary contributions to j ~nkj arezero for each wave vector k in the constrained region orexclusion zone, the total number of independent constraineddegrees of freedom is 2MK. Hence, in the large-systemlimit, 1=2 is the critical value when there are no longerany degrees of freedom that can be independently con-strained to be zero (not 1), according to this simplecounting argument. The reason why we use the definition(2) is that in the more general case when j ~nkj is con-strained to be positive (not zero) for some set of wavevectors [13,14,17], 1 is indeed the critical value whenone runs out of degrees of freedom that can be independ-ently constrained in the large-system limit.

[30] The limit described in this caption does notcommute with the limit of the number density going toinfinity, where is derived from the thermodynamic limitdefined in

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Ensemble Theory for Stealthy Hyperuniform Disordered Ground States