Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations

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1 Gound states of stealthy hypeunifom potentials: I. Entopically favoed configuations G. Zhang and F. H. Stillinge Depatment of Chemisty, Pinceton Univesity, Pinceton, New Jesey 8544, USA space, finding the gound states of stealthy potentials is equivalent to constaining the stuctue facto to be zeo fo wave vectos contained within the suppot of the Fouie tansfomed potential [], as will be summaized in Sec. II. In the case when the constained wave vectos lie in the adial inteval < K, the stealthy gound states fall within the class of hypeunifom states of matte [3] and can be tuned to have vaying degees of disode. Disodeed hypeunifom systems in geneal ae of cuent inteest because they ae chaacteized by an anomalously lage suppession of long-wavelength density fluctuations and can exist as equilibium o nonequilibium states, eithe classically o quantum mechanically [4 37]. Moeove, because disodeed hypeunifom states of matte have chaacteistics that lie between a cystal and a liquid [], they ae endowed with novel physical popeties [8, 9, 38 46]. When a dimensionless paamete χ, invesely popotional to the numbe density ρ and popotional to K d (size of the constained egion) is sufficiently small, the hypeunifom gound states ae infinitely degeneate and counteintuitively disodeed (i.e., isotopic without any Bagg peas) []. Howeve, when χ is lage enough (ρ is sufficiently small), thee is a phase tansition to a egime in which the gound states ae cystalline o highly oaxiv: v [cond-mat.stat-mech] 9 Aug 5 S. Toquato Depatment of Chemisty, Depatment of Physics, Pinceton Institute fo the Science and Technology of Mateials, and Pogam in Applied and Computational Mathematics, Pinceton Univesity, Pinceton, New Jesey 8544, USA Systems of paticles inteacting with stealthy pai potentials have been shown to possess infinitely degeneate disodeed hypeunifom classical gound states with novel physical popeties. Pevious attempts to sample the infinitely degeneate gound states used enegy minimization techniques, intoducing algoithmic dependence that is atificial in natue. Recently, an ensemble theoy of stealthy hypeunifom gound states was fomulated to pedict the stuctue and themodynamics that was shown to be in excellent ageement with coesponding compute simulation esults in the canonical ensemble (in the zeo-tempeatue limit). In this pape, we povide details and justifications of the simulation pocedue, which involves pefoming molecula dynamics simulations at sufficiently low tempeatues and minimizing the enegy of the snapshots fo both the high-density disodeed egime, whee the theoy applies, as well as lowe densities. We also use numeical simulations to extend ou study to the lowe-density egime. We epot esults fo the pai coelation functions, stuctue factos, and Voonoi cell statistics. In the high-density egime, we veify the theoetical ansatz that stealthy disodeed gound states behave lie pseudo disodeed equilibium had-sphee systems in Fouie space. The pai statistics obey cetain exact integal conditions with vey high accuacy. These esults show that as the density deceases fom the high-density limit, the disodeed gound states in the canonical ensemble ae chaacteized by an inceasing degee of shot-ange ode and eventually the system undegoes a phase tansition to cystalline gound states. In the cystalline egime (low densities), thee exist apeiodic stuctues that ae pat of the gound-state manifold, but yet ae not entopically favoed. We also povide numeical evidence suggesting that diffeent foms of stealthy pai potentials poduce the same gound-state ensemble in the zeo-tempeatue limit. Ou techniques may be applied to sample the zeo-tempeatue limit of the canonical ensemble of othe potentials with highly degeneate gound states. I. INTRODUCTION Thee has been long-standing inteest in the phase behavio of many-paticle systems in d-dimensional Euclidean spaces R d in which the paticles inteact with soft, bounded pai potentials [ ]. Consideable attention has been devoted to the detemination of the classical gound states (global enegy minima) of such inteactions [3, 6,, ]. While typical inteactions lead to unique classical gound states, cetain special pai potentials ae chaacteized by degeneate classical gound states a phenomenon that has attacted ecent attention [ ]. One family of such pai inteactions ae the stealthy potentials because thei gound states coespond to configuations that completely suppess single scatteing fo a ange of wave numbes. The Fouie tansfoms of these potentials ae bounded and non-negative and have compact suppot [], and hence they have coesponding diect-space potentials that ae bounded and long anged. Because of thei special constuction in Fouie toquato@electon.pinceton.edu

2 deed [3 5, 9]. Fo each spatial dimension d, thee is a special value of χ, χ max, at which the gound state is unique [47]. The unique gound state is the dual (ecipocal lattice) of the densest Bavais lattice pacing in each dimension []. In two and highe dimensions, as soon as χ dops below χ max, the set of the gound states become uncountably infinite and gadually includes pogessively less odeed stuctues []. Similaly to stealthy potentials, a family of two-, thee-, and fou-body potentials that lead to disodeed gound states has also been defined in Fouie space and studied [7, 8, ]. Due to the complexity of the poblem, almost all pevious investigations of the gound states employed compute simulations. Such numeical studies wee caied out in one, two and thee dimensions [3, 4, 7 9]. The gound states wee sampled by minimization of potential enegy at fixed densities stating fom andom initial conditions in a d-dimensional cubic simulation box unde peiodic bounday conditions. A few optimization techniques wee employed to find the global enegy minima with vey high pecision [4, 7]. Geneally, a numeically obtained gound-state configuation depends on the numbe of paticles N within the fundamental cell, initial paticle configuation, shape of the fundamental cell, and paticula optimization technique used []. Adding to the complexity of the poblem is that the disodeed gound states ae highly degeneate with a configuational dimensionality that depends on the density, and thee ae an infinite numbe of distinct ways to sample this complex gound-state manifold, each with its own pobability measue. These nontivial aspects had made the tas of fomulating a statisticalmechanical theoy of stealthy degeneate gound states a daunting one. Recently, we have fomulated such an ensemble theoy that yields analytical pedictions of the stuctual chaacteistics and othe popeties of stealthy degeneate gound states []. A numbe of exact esults fo the themodynamic and stuctual popeties of these gound states wee deived that applied to geneal ensembles. We then specialized ou esults to the canonical ensemble (in the zeo-tempeatue limit) by exploiting an ansatz that stealthy disodeed gound states (fo sufficiently small χ) behave emaably lie pseudo disodeed equilibium had-sphee systems in Fouie space. Ou theoetical pedictions fo the pai coelation function g () and stuctue facto S() of these entopically favoed disodeed gound states wee shown to be in ageement with coesponding compute simulations acoss the fist thee space dimensions. We also made pedictions fo the coesponding excited states fo sufficiently small tempeatues that wee in ageement with simulations. Because the focus of that pevious investigation was the development of ensemble theoies, few simulation details wee pesented about how the canonical ensemble was sampled to poduce stealthy disodeed gound states. One aim of the pesent pape is to povide a compehensive desciption of the numeical pocedue that we used to poduce the simulation esults in Ref.. Moeove, hee we also extend those esults by applying the simulation pocedue to study numeically the gound states in the canonical ensemble fo all allowable values of χ and thus investigate the entie phase diagam fo the entopically favoed states acoss the fist thee space dimensions. In the second pape of this seies, we will study the exotic apeiodic wavy phases identified in pevious numeical wo [4] (o stacedslide phases, as called in the sequel to this pape [48]), a special pat of the gound-state manifold. An analytical model will enable an even moe detailed study of this phase. As a justification of sampling the canonical ensemble instead of minimizing enegy, we also demonstate hee how a vaiety of diffeent optimization techniques affect the gound states that ae sampled, which was not peviously investigated [4, 7, 8]. This investigation eveals that the pai statistics of the gound-state configuations indeed geneally depend on the algoithm. Moeove, we show hee that the enegy minimization esults depend on the initial conditions as well. We also povide the eason why the simulations in Ref. and this pape employ noncubic, possibly defoming, simulation boxes fo d. Because almost all pevious numeical simulations wee pefomed using some specific fom of stealthy potentials, we show hee that diffeent foms of stealthy potentials poduce identical pai coelation functions, suggesting that the specific choice of the potential fom does not affect the ensemble being sampled. Among ou majo findings, we show that enegy minimizations stating fom andom initial conditions may lead to clusteing of paticles, the degee of which depends on the algoithm fo a finite ange of χ below / acoss the fist thee space dimensions. When minimizing the enegy stating fom configuations equilibated at some tempeatue T E, the gound-state configuations discoveed depend on T E. Howeve, the algoithm dependence diminishes in the T E limit. We also demonstate that the pai statistics [g () and S()] in this limit do not depend on the paticula fom of the stealthy potential. The similaity between the stuctue facto in this limit and the pai coelation function of an equilibium had-sphee system in diect space [] is valid fo χ up to some dimension-dependent values between.5 and.33 in the fist thee space dimensions. Beyond this ange of χ, the had-sphee analogy in Fouie space undegoes modification. As χ inceases futhe (to the value of about.4 in two dimensions, fo example), the fist pea in the stuctue facto diminishes while second pea in the stuctue facto gows and engulfs the fist pea. Ou simulated pai statistics obey cetain exact integal conditions in Ref. with vey high accuacy, indicating the high fidelity of the numeical esults. In the infinite-system-size limit, at χ =.5, the entopically favoed gound states undego a tansition fom disodeed states to cystalline states. Depending on the dimension, this phase tansition can occu when apeiodic stuctues

3 3 still ae pat of the gound state manifold, demonstating that cystalline (odeed) stuctues can have a highe entopy than disodeed stuctues. The est of the pape is oganized as follows: In Sec. II, we biefly summaize the numeical collective-coodinate pocedue and othe details of the simulation that we employ in the pesent pape with justifications. In Sec. III, we study the dependence of the esults on a vaiety of enegy minimization algoithms, initial conditions, and the foms of the stealthy potentials. In Sec. IV, we povide pai coelation function, stuctue facto, Voonoi cellvolume distibution, and configuation snapshots of the stealthy hypeunifom gound states obtained fom the canonical ensemble in the zeo-tempeatue limit. We povide concluding emas and discussion in Sec. V, including suggestions fo sampling the canonical ensemble in the zeo-tempeatue limit of othe potentials with degeneate disodeed gound states. II. MATHEMATICAL RELATIONS AND SIMULATION PROCEDURE As detailed in Sec. II of Ref., we simulate point pocesses in peiodic fundamental cells (i.e. simulation boxes) with a paiwise additive potential v() such that its Fouie tansfom exists. Unde neaest image convention, the total potential enegy can be calculated by summing ove all pais of paticles: Φ( N ) = i<j v( ij ), () whee N is the numbe of paticles, N,,..., N is the locations of the paticles in d-dimensional Euclidean space, and ij = i j. Instead of summing ove all paiwise contibutions in the eal space, the potential enegy can also be epesented in Fouie space: Φ( N ) = v F [ ṽ() ñ() N ] ṽ(), () whee v F is the volume of the fundamental cell, ṽ() = v F v() exp( i )d is the Fouie tansfom of the pai potential, ñ() = N j= exp( i j) is the complex collective density vaiable [with ñ( = ) = N], and both summations ae ove all ecipocal lattice vecto s appopiate to the fundamental cell. Fo evey, ñ() is elated to the stuctue facto, S(), via S() = ñ() N. (3) Given a ṽ(), the coesponding eal-space pai potential is v() = ṽ() exp(i ). (4) v F In a finite-sized system, the eal-space pai potential has the same peiodicity as the fundamental cell. Theefoe, in the infinite-volume limit, the cell peiodicity disappeas. A family of stealthy potentials, which completely suppess single scatteing fo all wave vectos within a specific cutoff in thei gound states, ae defined as [3, 4, 7 ]: { V (), if K, ṽ() = (5), othewise, whee V () is a positive isotopic function and K is a constant. In this pape we always tae K =, which sets the length scale. We will also use V () = unless othewise specified. In the infinite-system-size limit, the isotopic ṽ() coespond to an isotopic eal-space pai potential v() []. Howeve, fo finite systems, the coesponding v() is anisotopic. In Appendix A, we compae the infinite-system-size limit v() with the finite-size v() s in diffeent-shaped simulation boxes and select the simulation box shape to be used in this pape based on which v() is closest to the infinite-size-limit v(). Fom Eqs. () and (5), one can see that a configuation is a stealthy gound state if ñ() = fo all points such that < K. Theefoe, finding a gound state of a stealthy potential is equivalent to constaining ñ() = fo all of those points. Howeve, in a simulation, one does not need to chec all of the constaints. As detailed in Ref., if thee ae (M + ) points within the constained adius, only M of them ae independent and needed to be constained to zeo. Equation () can be simplified as [49]: Φ( N ) = ṽ() ñ() + Φ, (6) v F whee the sum is ove all independent constaints, and Φ = [N(N ) N ṽ()]/(v F ) (7) is a constant independent of the paticle positions N. We now intoduce a paamete χ = M d(n ), (8) which detemines the degee to which the gound states ae constained, and theefoe the degeneacy and disode of the gound states [4]. Note that the constaints depend on K and the fundamental cell but ae independent of the specific shape of ṽ() as long as ṽ() > fo all < K. Theefoe, changing ṽ() does not change the set of the gound states. Howeve, thee is no poof that changing ṽ() does not change the elative sampling weights of the gound states. In this pape we study vaious systems with diffeent χ s and N s. One numeical complication is that these

4 4 numbes cannot be chosen abitaily, since M = χd(n ) must be an intege consistent with the specific shape of the simulation box. (Fo example, a list of the allowed M values fo a two-dimensional squae box is given in Table II of Ref. 4.) This constaint is especially had to meet when simulating multiple systems at the same χ value acoss dimensions. In fact, both χ and N in Table I (see Appendix C) had to be chosen caefully to meet this constaint. Taing the gadient of Eq. (6) yields the foces on paticles: F j = j Φ( N ) = ṽ() Im[ñ() exp(i j )], v F (9) whee the sum is also ove all independent constaints. This equation enables us to pefom both enegy minimizations and molecula dynamics (MD) simulations. In an enegy minimization, a deivative-based algoithm is used. The fist tem on the ight side of Eq. (6) is povided to the algoithm as the objective function and the negative of the foce in Eq. (9) is povided as the deivative. In ode to minimize enegy, we have tied diffeent algoithms including the MINOP algoithm [5], the steepest descent algoithm allowing lage steps [5], the low-stoage BFGS (L-BFGS) algoithm [5 54], the Pola-Ribiee conjugate gadient algoithm [5, 55], and ou local gadient descent algoithm descibed in Appendix B. When χ <.5, the objective function always ends up being vey close to zeo (the minimum). The maximum ending objective function fo diffeent algoithms vaies fom as high as 7 fo a conjugate gadient algoithm to 7 fo the local gadient descent and steepest descent algoithms to fo the L-BFGS algoithm, and to as low as 5 fo the MINOP algoithm. Fom ou pactical point of view, all of these algoithms ae pecise enough, since an eo of 7 o lowe is indiscenible fom any esults pesented below. Because the L-BFGS algoithm is the fastest, we will use it unless othewise specified. The enegy minimizations, if stated fom andom initial configuations, will sample an algoithm-dependent, nonequilibium ensemble. To sample the canonical ensemble at a given equilibium tempeatue T E we use MD simulations. One impotant paamete in MD simulations is the integation time step. Since the optimal choice of the time step depends on the tempeatue, and the latte vaies acoss seveal odes of magnitude in this pape, we desie a systematic way to detemine the optimal time step. Stating fom an enegy minimized configuation and a vey small time step (. in dimensionless units), we epeat the following steps 4 times to equilibate the system and find a suitable time step: Assign a andom velocity fom Boltzmann distibution at T E to each paticle. Calculate the total (inetic and potential) enegy of the system E. Evolve the system 5 time steps using the velocity Velet algoithm [56]. Calculate the total enegy of the system E. If ln E E > 5, then the time step is too lage and eos will build up quicly. Theefoe, we decease the time step by 5%. On the othe hand, if ln E E < 4 6, thee is still some oom to incease the time step. Since inceasing the time step inceases the efficiency of MD simulations, we incease the time step by 5%. Afte the system is equilibated and the time step is chosen, we pefom constant tempeatue MD simulations with paticle velocity esetting [57]. A andomly chosen paticle is assigned a andom velocity, dawn fom Maxwell-Boltzmann distibution, evey steps. We tae a sample configuation evey 3 time steps until we have sampled configuations unless othewise specified. This amounts to an implementation of the geneation of configuations in the canonical ensemble. The above MD pocedue wos well fo χ <.5. Howeve, two new featues aise when it is applied to χ.5 in all dimensions. Fist, the potential enegy suface develops local minima and enegy baies that can tap the system if T E is too small. We addess this poblem by using simulated annealing, employing a themodynamic cooling schedule [58] which stats at T = 3 and ends at 6. Note that, by adopting a cooling schedule, we concede that we may only tae one sample at the end of each MD tajectoy, wheeas a fixed-tempeatue MD tajectoy poduces multiple samples. The second new featue is that the entopically favoed gound states ae cystalline fo χ.5. Unlie disodeed stuctues, a cystalline stuctue has long-ange ode and may not fit in simulation boxes with cetain shapes. To ovecome the second poblem, we simulate an isothemal-isobaic ensemble with a defomable simulation box. Evey MD time steps, Monte Calo tial moves to defom the simulation box ae attempted. The pessue is calculated fom Eq. (4) of Ref.. We employed the Wang-Landau Monte Calo [59] to attempt to detemine the entopically favoed gound states fo χ >.5 in two and thee dimensions. The Wang-Landau Monte Calo is used to calculate the micocanonical entopy S(Φ) as a function of the potential enegy Φ. We limit ou simulations to the enegy ange 3 < Φ Φ < 9 (in dimensionless units), whee Φ is the gound state enegy, by ejecting any tial move that violates this enegy toleance. This enegy ange is evenly divided into bins. Stating fom a pefect cystal stuctue in a simulation box shaped lie a fundamental cell, small petubations ae intoduced so the enegy is within the ange. Afte that, 6 stages of Monte Calo simulations ae pefomed, each stage containing 3 7 tial moves. The modification facto in Ref. [59] is f = exp[5/(n + )], whee n is the numbe of stages.

5 5 III. DEPENDENCE ON ENERGY MINIMIZATION ALGORITHM, MD TEMPERATURE, AND ṽ() In this section, we pesent numeical simulation esults demonstating that: Enegy minimizations stating fom Poisson initial configuations using diffeent algoithms can yield gound states with diffeent pai coelation functions. g () MINOP Steepest Descent L-BFGS Conjugate Gadient Local Gadient Descent Enegy minimizations stating fom MD snapshots at diffeent tempeatues can yield gound states with diffeent pai coelation functions. Fo configuations obtained by minimizing enegy stating fom MD snapshots at sufficiently small tempeatue, pai coelation functions do not depend on the minimization algoithm and the fom of the stealthy potential. These esults motivate the eason why we ultimately study and epot esults in Sec. IV in the canonical ensemble in the zeo-tempeatue limit. Fo conceteness and visual claity, we pesent esults hee in two dimensions. Howeve, we have veified that all of the conclusions hee also apply to one and thee dimensions. We pefomed enegy minimizations stating fom Poisson initial configuations (i.e., T E state at fixed density) using each of the five numeical algoithms mentioned in Sec. II at χ =. and χ =.4. The esults ae shown in Figs. and. At χ =., the pai coelation functions poduced by the MINOP algoithm and the L-BFGS algoithm ae almost identical. Howeve, the pai coelation function poduced by the conjugate gadient algoithm noticeably diffes. The steepest descent algoithm and ou local gadient descent algoithm poduce a significantly diffeent pai coelation function with a much weae pea at =. The pai coelation functions poduced by some algoithms appea to have g () log() divegence nea the oigin. Since this divegence means paticles have a tendency to fom clustes, we call it a clusteing effect. At χ =.4, the clusteing effect disappeas, but the pai statistics poduced by diffeent algoithms still diffes. The fact that diffeent optimization algoithms poduce diffeent pai statistics means that they sample the gound-state manifold with diffeent weights. In othe wods, diffeent optimization algoithms ae sampling diffeent gound-state ensembles. FIG.. (Colo online) Pai coelation function as obtained fom diffeent optimization algoithms (as descibed in the legend) stating fom Poisson initial configuations in two dimensions at χ =.. Each cuve is aveaged ove configuations of 36 paticles each. The left inset zooms in nea the oigin, showing the diffeences between the five algoithms moe clealy. The ight inset uses a semilogaithmic scale to show g () log() nea the oigin. g () MINOP Steepest Descent L-BFGS Conjugate Gadient Local Gadient Descent 5 5 FIG.. (Colo online) As in Fig., except that χ =.4 and each cuve is aveaged ove configuations of 5 paticles each. The inset zooms in nea the fist well, showing the diffeences between the five algoithms moe clealy. In ode to avoid the complexity caused by the details of vaious optimization algoithms, we tun ou inteest to the canonical ensemble in the T limit. To sample this ensemble, we pefom MD simulations at sufficiently small tempeatue T E, peiodically tae snapshots, and then use a minimization algoithm to bing each snapshot to a gound state. To detemine a sufficiently small T E, we calculated the pai coelation functions at vaious T E s and pesent them in Fig. 3. The enegy minimization esult stating fom T E initial configuations clealy display the clusteing effect at χ =.. When T E goes to zeo, the clusteing effect also diminishes. At χ =.4, paticles develop had coes

6 6 [g () = ], theefoe thee is no clusteing even if T E is lage o infinite. Howeve, the pea height of g () becomes dependent on T E at this χ value. Fo both χ values, the pai coelation functions of the two lowest T E s ae almost identical, veifying that the T E limit exists. These esults show that T E = 6 is sufficiently small in two dimensions. Similaly, we have found that T E = 4 and T E = 6 ae sufficiently small in one and thee dimensions, espectively. These tempeatues ae used in geneating all of the esults pesented in Sec. IV A. g () g ().5.5 L-BFGS MD at T E = -, then L-BFGS MD at T E = -4, then L-BFGS MD at T E = -6, then L-BFGS (a) L-BFGS MD at T E = -, then L-BFGS MD at T E = -4, then L-BFGS MD at T E = -6, then L-BFGS 5 5 (b) FIG. 3. (Colo online) Pai coelation function poduced by L-BFGS algoithm stating fom snapshots of MD at diffeent equilibation tempeatues T E, (a) χ =. and (b) χ =.4. Each cuve is aveaged ove configuations of 36 paticles each o 5 paticles each. The enegy minimization esult stating fom Poisson initial configuations diffes fo diffeent algoithms, but the canonical ensemble in the T limit should not depend on any paticula algoithm. Afte finding that T E = 6 is sufficiently small, we confim the disappeaing of algoithmic dependence by calculating the pai coelation function poduced by diffeent enegy minimization algoithms stating fom MD snapshots at T E = 6. Figue 4 shows the esults. The cuves fo all algoithms almost coincide. g ().5.5 MINOP Steepest Descent L-BFGS Conjugate Gadient Local Gadient Descent 5 5 FIG. 4. (Colo online) Pai coelation function poduced by the five diffeent algoithms stating fom snapshots of MD at equilibation tempeatue T E = 6 at χ =.. Each cuve is aveaged ove configuations of 36 paticles each. Last, the function V () in Eq. (5) can have diffeent foms. This pape mainly use V () = but we also want to now if the esults obtained using this fom ae equivalent to those geneated using othe positive isotopic foms of V () as well. In pinciple, stealthy potentials of any fom should have the same set of goundstate configuations, but the fom of the stealthy potential could theoetically affect the cuvatue of the potential enegy suface nea each gound-state configuations and thus also affect thei elative weights. Figue (5) shows the pai coelation function poduced by diffeent V () s. The pai coelation functions fo V () = and V () = ( ) at T E = 6 ae almost identical. Fo V () = ( ) 6, we initially tied T E = 6 but found that the clusteing effect is still noticeable. We futhe loweed the tempeatue to T E = to completely suppess the clusteing effect to poduce a pai coelation function identical to that of V () = and V () = ( ) potentials. This esult suggests that the functional fom of V () does not poduce noticeable diffeences in the gound-state ensembles in the T limit of the canonical ensemble.

7 7 g () T E = -, V()=(-) 6 T E = -6, V()=(-) T E = -6, V()= 5 5 FIG. 5. (Colo online) Pai coelation function poduced by diffeent potentials stating fom snapshots of MD at sufficiently low tempeatue at χ =.. Each cuve is aveaged ove configuations of 36 paticles each. IV. CANONICAL ENSEMBLE IN THE T LIMIT We will show hee that the entopically favoed gound states in the canonical ensemble in the T limit fo the fist thee space dimensions diffe maedly below and above χ =.5. Fo χ <.5, the entopically favoed gound states ae disodeed while fo χ.5 the entopically favoed gound states ae cystalline. Theefoe, we will chaacteize them diffeently. Fo χ <.5, we will epot the pai coelation function, stuctue facto, and Voonoi cell statistics. Fo sufficiently small χ, we will show that the simulation esults agee well with theoy []. Fo χ.5, we will epot the cystal stuctues. The numbes of paticles in all of the systems epoted in this section ae collected in Appendix C. A. χ <.5 egion Repesentative entopically favoed stealthy gound states in the fist thee space dimensions at χ =. and χ =.4 ae shown in Figs As χ inceases fom. to.4, the stealthiness inceases, accompanied with a visually peceptible incease in shot-ange ode. This tend in shot-ange ode is consistent with pevious studies [4, 7, 8]. (a) (b) FIG. 6. χ =.4. (Colo online) Repesentative one-dimensional entopically favoed stealthy gound states at (a) χ =. and (b) (a) (b) FIG. 7. (Colo online) Repesentative two-dimensional entopically favoed stealthy gound states at (a) χ =. and (b) χ =.4.

8 8 (a) (b) FIG. 8. (Colo online) Repesentative thee-dimensional entopically favoed stealthy gound states at (a) χ =. and (b) χ =.4..5 χ=.5.5 χ=..5 χ=.43 S() S().5 d=, Theoy d=, Theoy d=3, Theoy d=, Simulation d=, Simulation d=3, Simulation χ=..5 d=, Theoy d=, Theoy d=3, Theoy d=, Simulation d=, Simulation d=3, Simulation 3 4 S() S() d=, Theoy d=, Theoy d=3, Theoy d=, Simulation d=, Simulation d=3, Simulation 3 4 χ=.5 d= d= d=3 3 4 S() S() d=, Theoy d=, Theoy d=3, Theoy d=, Simulation d=, Simulation d=3, Simulation 3 4 χ=.33 d= d= d= FIG. 9. (Colo online) Stuctue factos fo d 3 fo.5 χ.33 fom simulations and theoy []. The smalle χ simulation esults ae also compaed with the theoetical esults in the infinite-volume limit []. Fo χ., the theoetical and simulation cuves ae almost indistinguishable, and the stuctue facto is almost independent of the space dimension. Howeve, simulated S() in diffeent dimensions become vey diffeent at lage χ. Theoetical esults fo χ.5 ae not pesented because they ae not valid in this egime.

9 9 g () g ().5.5 χ=.5 d=, Theoy d=, Theoy d=3, Theoy d=, Simulation d=, Simulation d=3, Simulation 5.5 χ=..5 d=, Theoy d=, Theoy d=3, Theoy d=, Simulation d=, Simulation d=3, Simulation 5 g () g ().5.5 χ=. d=, Theoy d=, Theoy d=3, Theoy d=, Simulation d=, Simulation d=3, Simulation 5.5 χ=.5.5 d= d= d=3 5 g () g ().5.5 χ=.43 d=, Theoy d=, Theoy d=3, Theoy d=, Simulation d=, Simulation d=3, Simulation 5 χ= d= d= d=3 5 FIG.. (Colo online) Pai coelation functions fo d 3 fo.5 χ.33 fom simulations and theoy []. The smalle χ simulation esults ae also compaed with the theoetical esults in the infinite-volume limit []. Fo χ., the theoetical and simulation cuves ae almost indistinguishable. Theoetical esults fo χ.5 ae not pesented because they ae not valid in this egime. We have calculated the pai coelation functions and the stuctue factos fo vaious χ values. Results fo.5 χ.33 ae shown in Figs. 9 and. The χ <. esults ae in excellent ageement with the pseudo-hadsphee ansatz, which states that the stuctue facto behaves lie pseudo equilibium had-sphee systems in Fouie space []. Howeve, the theoy gadually becomes invalid as χ inceases. The pai coelation functions of the entopically favoed stealthy gound states ae shown in Fig.. When χ., since the stuctue facto is simila to the pai coelation function of the had-sphee system, invesely the pai coelation function is also simila to the stuctue facto of the had-sphee system. As χ gows lage, the pseudo had-sphee ansatz gadually deviates fom the simulation esult. We have checed that these pai statistics ae consistent with fou theoetical integal conditions of the pai statistics in the infinite-volume limit []. The fist thee conditions ae Eqs. (58), (59), and (63) of Ref., which ae P ()d =, () R d and g () = dχ + d χ R d P ()v()d =, () K d Q()d, () whee P () is the invese Fouie tansfom of Θ( ) Q(), Θ(x) is the Heaviside step function, and Q() = S(). The fouth condition is that the pessue calculated fom the viial equation [] has to be eithe nonconvegent o convegent to the pessue calculated fom the enegy oute []. All pai statistics in Figs. 9 and wee geneated using the step-function potential [the V () = case of Eq. (5)], but this potential does not lead to a convegent viial pessue. Howeve, as we have shown ealie, the stealthy gound states that we geneated hee ae also the gound states of othe stealthy functional foms ṽ(). In one dimension, to test ou simulation pocedue, we used the potential fom V () = ( ) to calculate the pessue fom both the viial equation (Eq. (43) of Ref. ) and the enegy equation (Eq. (4) of Ref. ). The pessue fom the viial equation conveges and agees with the exact pessue fom the enegy equation, thus confiming the accuacy of ou numeical esults. These checs involve integals of g () and S() that ae only slowly conveging. Theefoe, passing them demonstates that ou esults have vey high pecision.

10 S() g () =.33 =.35 =.38 =.4 =.43 = =.33 =.35 =.38 =.4 =.43 =.46 in Fig.. In the same dimension, as χ inceases, the distibution of Voonoi cell volumes naows. This is expected because the system becomes moe odeed as χ inceases. Fo the same χ, the distibution also naows as the dimension inceases, consistent with theoetical esults that at fixed χ, the neaest-neighbo distance distibution naows as dimension inceases []. In Fig., we additionally show the Voonoi cell-volume distibution of satuated andom sequential addition (RSA) [65 67] pacings, the sphee pacings geneated by andomly and sequentially placing sphees into a lage volume subject to the nonovelap constaint until no additional sphees can be placed. Satuated RSA pacings ae neithe stealthy no hypeunifom [66, 67]. Howeve, the Voonoi cell-volume distibutions of satuated RSA pacings loo simila to that of the entopically favoed stealthy gound states. This is not unexpected because Voonoi cell statistics ae local chaacteistics, and hence ae not sensitive to the stealthiness, which is a lage-scale popety. FIG.. (Colo online) Stuctue facto and pai coelation function fo d = fo.33 χ.46, as obtained fom simulations. Fo smalle χ values, the maximum of the stuctue facto is at the constaint cutoff = K +. Howeve, fo highe χ values, the maximum of S() is no longe at = +. To pobe this tansition we have calculated the stuctue facto in two dimensions fo.33 χ.46. The esults ae shown in Fig.. As χ inceases, the pea at = + gadually deceases its height, while the subsequent pea gadually gows and engulfs the fist pea. Besides pai statistics, othe widely used chaacteization of point pattens include cetain statistics of the Voonoi cells [4, 6 6]. A Voonoi cell is the egion consisting of all of the points close to a specific paticle than to any othe. We have computed the Voonoi tessellation of the entopically favoed stealthy gound states using the dd Convex Hulls and Delaunay Tiangulations pacage [63] of the Computational Geomety Algoithms Libay [64]. Since the numbe density of the stealthy gound states depends on the dimension and χ, we escaled each configuation to unity density fo compaison of the Voonoi cell volumes. The pobability distibution function p(v c ) of the Voonoi cell volumes (whee v c is the volume of a Voonoi cell) ae shown

11 p(v c ) p(v c ) p(v c ) d= χ=.5 χ=. χ=.43 χ=. χ=.5 RSA v c d= χ=.5 χ=. χ=.43 χ=. χ=.5 RSA v c d=3 χ=.5 χ=. χ=.43 χ=. χ=.5 RSA (a) (b) FIG. 3. (Colo online) (a) Low-tempeatue MD snapshot of a 6-paticle system at χ =.48; the gound-state configuation is cystalline. (b) MD snapshot of a 54-paticle system at the same T E and χ; the system does not cystallize and is indeed disodeed without any Bagg peas. v c FIG.. (Colo online) Voonoi cell-volume distibution fo d 3 fo.5 χ.5. Fo the same dimension, the Voonoi cell-volume distibution becomes naowe when χ inceases. Fo the same χ, the Voonoi cell-volume distibution also becomes naowe when dimension inceases. We also pesent Voonoi cell-volume distibutions of RSA pacings at satuation hee. One inteesting phenomenon is that as χ inceases and appoaches /, systems that ae not sufficiently lage can become cystalline. In Fig. 3, we show two snapshots of MD simulations at χ =.48. The smalle configuation is cystalline. Howeve, systems that ae 4 times lage emain disodeed at the same χ and tempeatue. Theefoe, this stongly indicates that cystallization is a finite-size effect fo χ tending to / fom below. B. χ.5 egion As explained in Sec. II, we pefom MD-based simulated annealing with Monte Calo moves of the simulation box fo χ >.5, since this method wos bette with ough potential enegy suface and can mitigate the finite-size effect. We pefomed this simulation at χ =.55, χ =.73, and χ =.8 in two dimensions. The esults ae shown in Fig. 4. The esulting configuation is always tiangula lattice. Even though the gound-state manifold in this χ egime contains apeiodic wavy phases discoveed peviously [4] [but which ae called staced-slide phases in the sequel to this pape [48], since they ae apeiodic configuations with a high degee of ode in which ows (in two dimensions) o planes (in thee dimensions) of paticles can slide past each othe] as well as cystals othe than the tiangula lattice, the entopically favoed gound state is always a tiangula lattice. This means that the tiangula lattice has a highe entopy than staced-slide phases, although the latte appea to be moe disodeed [68]. Although we will show analytically that cystals ae moe entopically favoed than staced-slide phases in the upcoming pape of this seies, we still need simulation esults to detemine which cystal stuctue has the highest entopy. The esults of MD-based simulated annealing with Monte Calo moves of the simulation box suggest that tiangula lattice has the highest entopy in two dimensions. It seems natual to apply the same technique to thee dimensions to detemine the entopi-

12 cally favoed cystal stuctue. Howeve, we wee unable to cystallize the system in thee dimensions. Even the longest cooling schedule that we tied esulted in stacedslide phases. (a) become vey close to each othe, peventing us fom detemining the entopically favoed gound state at these χ values. S(Φ)-S(Φ + -9 ) S(Φ)-S(Φ + -9 ) χ=.5 χ=.67 (Φ Φ ) Tiangula Squae Tiangula Squae (Φ Φ ) S(Φ)-S(Φ + -9 ) S(Φ)-S(Φ + -9 ) χ= χ=.75 (Φ Φ ) Tiangula Squae Tiangula Squae (Φ Φ ) (b) FIG. 5. (Colo online) Micocanonical entopy as a function of enegy S(Φ) calculated fom Wang-Landau Monte Calo of tiangula lattice and squae lattice at vaious χ s. Hee Φ denotes the gound-state enegy. (c) FIG. 4. (Colo online) MD-based simulated annealing esult at (a) χ =.55, (b) χ =.73, and (c) χ =.8. The ending configuation is tiangula lattice except fo small defomations in the χ =.55 case. Anothe way to find the entopically favoed cystal is to use Wang-Landau Monte Calo to diectly calculate the entopy of diffeent cystal stuctues as a function of the potential enegy. We have pefomed this simulation on two-dimensional tiangula lattice, squae lattice, and thee-dimensional body-centeed cubic (BCC) lattice, face-centeed cubic (FCC) lattice, and simple cubic (SC) lattice. The esults ae shown in Figs. 5 and 6. In all cases the entopy deceases as the enegy deceases. In two dimensions, the entopy of the squae lattice clealy deceases faste than that of the tiangula lattice at evey χ value, confiming that the tiangula lattice is entopically favoed ove the squae lattice in the zeotempeatue limit. In thee dimensions at χ =.58, the entopy of the FCC lattice deceases moe slowly than that of the BCC and SC lattice, suggesting that the entopically favoed gound state in thee dimensions at χ =.58 is the FCC lattice. At highe χ values, the scaling of the entopy of the FCC lattice and the BCC lattice S(Φ)-S(Φ + -9 ) S(Φ)-S(Φ + -9 ) χ=.58 BCC FCC SC χ=.77 (Φ Φ ) BCC FCC (Φ Φ ) S(Φ)-S(Φ + -9 ) S(Φ)-S(Φ + -9 ) χ=.68 BCC FCC χ=.845 (Φ Φ ) BCC FCC (Φ Φ ) FIG. 6. (Colo online) Micocanonical entopy as a function of enegy S(Φ) calculated fom Wang-Landau Monte Calo of BCC lattice, FCC lattice, and SC lattice at vaious χ s. A cuve fo SC lattice is not pesented fo χ.68 because the latte is not a gound state at such high χ values. Hee Φ denotes the gound-state enegy. V. CONCLUSIONS AND DISCUSSION The uncountably infinitely degeneate classical gound states of the stealthy potentials have been sampled peviously using enegy minimizations. We demonstate hee that this way of sampling the gound states to poduce ensembles of configuations intoduces dependencies on the enegy minimization algoithm and the initial configuation. Such atificial dependencies ae avoided in

13 3 studying the canonical ensemble in the T limit. We sample this ensemble by pefoming MD simulations at sufficiently low tempeatues, peiodically taing snapshots, and minimizing the enegy of the snapshots. The configuations in this ensemble become moe odeed as χ inceases and obey cetain theoetical conditions on thei pai statistics [], similaly to pevious enegy minimization esults. Howeve, othe popeties of this ensemble ae unique. Fist, ou numeical esults demonstate that the pai statistics of this ensemble displays no clusteing effect [divegence of g () as ] fo any χ value, and is independent of the functional fom of the stealthy potential. Second, we numeically veify the theoetical ansatz [] that fo sufficiently small χ stealthy disodeed gound states behave lie pseudo disodeed equilibium had-sphee systems in Fouie space, i.e., S() has the same functional fom as the pai coelation function fo equilibium had sphees fo sufficiently small densities. Thid, when χ is above the citical value of.5, ou esults stongly indicate that cystal stuctues ae entopically favoed in both two and thee dimensions in the infinite-volume limit. Ou numeical evidence suggests that the entopically favoed cystal in two dimensions is the tiangula lattice. Howeve, we could not detemine the entopically favoed cystal stuctue in thee dimensions. Fo finite systems, the disodeed-to-cystal phase tansition can happen at a slightly lowe χ. A theoetical explanation of this phenomenon emains an open poblem. Besides gound states of stealthy potentials, othe disodeed degeneate gound states of many-paticle systems have been studied using enegy minimizations. Specifically, pevious eseaches have constained the stuctue facto to have some tageted functional fom othe than zeo fo pescibed wave vectos [7, 8, ]. Finding the configuations coesponding to such tageted stuctue factos amounts to finding the gound states of two-, thee- and fou-body potentials, in contast to the two-body stealthy potential studied in the pesent pape. This situation is the most geneal application of the collective-coodinate appoach. It will be inteesting to study the esulting pai statistics of the gound states fo these moe geneal inteactions in the zeo-tempeatue limit of the canonical ensemble. The collective-coodinate appoach is an independent and fuitful addition to the basic statistical mechanics poblem of connecting local inteactions to macoscopic obsevables. One impotant featue of collectivecoodinate inteactions is that it has uncountably infinitely degeneate classical gound states []. In the case of isotopic pai inteactions, the only othe system that we now with this featue is the had-sphee system. Howeve, thee ae two impotant diffeences between had-sphee systems and collective-coodinate gound states. Fist, while the dimensionality of the configuation space of equilibium had-sphee systems consisting of N paticles within a peiodic box is fixed [simply detemined by the nontivial numbe of degees of feedom, d(n )], the dimensionality of the collectivecoodinate gound-state configuation space deceases as χ inceases and, on a pe paticle basis, eventually vanishes []. The deceased dimensionality of the goundstate configuation space ceates challenges fo accuate sampling of the entopically favoed gound states using numeical simulations and hence the development of bette sampling methods is a fetile gound fo futue eseach. Second, while the pobability measue of the equilibium had-sphee system is unifom ove its entie gound-state manifold, that of the stealthy gound states is not unifom. To illustate this point, imagine a onedimensional enegy landscape that has a double-well potential behavio in a potion of the configuation space, as shown in Fig. 7. Each minimum epesents a degeneate gound state (as we find with stealthy potentials) and theefoe the well depths of the minima ae the same. Let us now conside hamonic appoximations of the two wells in the vicinity of x and x, espectively, and V (x) = a (x x ), V (x) = a (x x ), whee x is the configuational coodinate. At vey low tempeatue, to a good appoximation, the system can only visit the pat of the configuation space with enegy less than ε, and ε as T. Solving V i (x) < ε, whee i =,, one finds the feasible egion of configuation space associated with both wells: and x ε/a < x < x + ε/a, x ε/a < x < x + ε/a. When a a, we see that the feasible egions associated with the two potential wells have diffeent anges. Theefoe, the weights associated with the two minima, i.e., the elative pobabilities fo finding the system in the vicinity of those minima, will also diffe. Similaly, in the stealthy multidimensional configuation space that we ae studying, the magnitude of the eigenvalues of the Hessian matix will detemine the elative weights. Theefoe, the pobability measue of the stealthy gound states is not unifom ove the gound-state manifold, unlie the degeneate gound states of classical had sphees. Ou low-tempeatue MD simulations sample gound states with this nonunifom pobability measue. It would be useful to devise theoies to estimate the weights of diffeent potions of the gound-state manifold. Howeve, a featue that complicates the poblem is that the Hessian matix has zeo eigenvalues. In the associated diections of the eigenvectos of the configuation space, the enegy scales moe slowly than quadatically (hamonically) but we do not now the specific fom.

14 4 Potential Enegy, V(x) x x Configuational Coodinate, x FIG. 7. (Colo online) A model one-dimensional enegy landscape with two wells located at x and x of the same depth but diffeent cuvatues. The feasible egions, i.e., egions whee V (x) < ε, is maed by ed dashed lines. This pape, which investigates the entopically favoed gound states, is the fist of a two-pape seies. In the second pape, we will study aspects of the gound-state manifold with an emphasis on configuations that ae not entopically favoed fo χ above / (the odeed egime). In paticula, we will moe fully investigate the natue of so-called wavy cystals o staced-slide phases, discoveed in Ref. 4. Using an analytical desciption of such states, we will demonstate that they ae pat of the gound state but ae not entopically favoed. Ou analytical model will also demonstate that staced-slide phases exist in thee and highe dimensions. ACKNOWLEDGMENTS G. Z. thans Steven Atinson fo his caeful eading of some pats of the manuscipt. This eseach was suppoted by the U.S. Depatment of Enegy, Office of Basic Enegy Sciences, Division of Mateials Sciences and Engineeing unde Awad No. DE-FG-4-ER468. Appendix A: Real-space potential in finite systems In the infinite-system-size limit, an isotopic ṽ() coespond to an isotopic eal-space pai potential v(). Howeve, fo finite systems, the coesponding v() is anisotopic. To illustate the finite-size effect, we compae the two-dimensional eal-space potential v() in the infinite-system-size limit to coesponding potentials associated with finite-sized fundamental cells of squae and hombic shapes of diffeent volumes in Fig. 8. The ealspace potential in the hombic simulation box with a 6 inteio angle is appeciably moe isotopic than the ealspace potential in a squae simulation box. Theefoe, in this pape, we will hencefoth use hombic fundamental cells in two dimensions. Similaly, in thee dimensions, we always use a simulation box shaped lie a fundamental cell of a body-centeed cubic (BCC) lattice since BCC lattice is the unique gound state at χ max. Appendix B: Local Gadient Descent Algoithm Most optimization algoithms ae designed fo efficiency. They use complex ules to detemine the diection of the next step and tae as lage steps as possible. These featues mae thei path less obvious. To minimize enegy in the path following the gadient vecto, we designed a local gadient descent algoithm with the following steps:. Stat fom an initial guess, x, and find the function value f(x) and deivative f (x).. Stat fom a elatively lage ( 3 times the simulation box side length) step size, s, and calculate the vecto to the next step x = s f f (x). Find the function value at the next step f(x + x). Calculate the change of function value f = f(x+ x) f(x). 3. If we ae following the path of steepest descent accuately, the change of the function value should be close to f (x) x. If the diffeence between f and f (x) x is less than %, we accept this move. Othewise, we abot this move and half the step size s. 4. Repeat the above steps until a minimum is found with enough pecision. Appendix C: Numbe of paticles of evey system in Sec. IV TABLE I. The numbe of paticles N of each systems shown in Figs. 9 and. χ N fo d = N fo d = N fo d = In this appendix we epot the numbe of paticles N in each system in Sec. IV. Both configuations in Fig. 6 consist of 5 paticles. Configuations (a) and (b) in Fig. 7 consist of 7 and 5 paticles, espectively. Those in Fig. 8 consist of 3 and 6 paticles, espectively.

15 5 (a) (b) (c) (d) (e) (f) (g) FIG. 8. (Colo online) A potion of the eal-space potential v() aound the oigin fo the stealthy potential (5) with K = and V () =. (a)-(f) Real-space potential in a peiodic simulation box that is [(a), (c), and (e)] squae o [(b), (d), and (f)] hombic in shape; the latte has a 6 inteio angle. The volumes of the simulation boxes, v F, ae [(a) and (b)], [(c) and (d)] 4, and [(e) and (f)] 385. Panels (a)-(d) use unealistically small simulation boxes and is intended to illustate finite-size effect only. (g) The eal-space potential in the infinite-system-size limit. All potentials ae nomalized by thei espective values at the oigin since scaling does not affect the gound state. Note that, stating fom the cente, the da (ed) egion indicates the highest values of the potential, wheeas towads the edge of the box, the da (blue) egion indicates the lowest values of the potential. The numbe of paticles of each system in Figs. 9,,, and ae shown in Tables I, II, and III, espectively. Each configuation in Figs. 4, 5, and 6 consist of 36, 4, and 343 paticles, espectively. [] F. H. Stillinge, J. Chem. Phys. 65, 3968 (976). [] H. Löwen, Phys. Rep. 37, 49 (994).