Counterintuitive ground states in soft-core models

Counterintuiti ve grou nd states in soft-core models Henry Cohn Micr osoft Resear c h New England, One Memorial Drive, Cambridge, Massac husetts 02142, USA ∗ Abhinav Kumar Department of Ma thematics, Massach usetts Institute of T echno logy , Cambridge , Massachu setts 02139, USA † (Dated: No vember 7, 2008) It is we ll known that statistical mechanics systems exhibit subtle behavior in high dimensions. In this paper , we sho w that certain natural soft-core models, such as the Gaussian core model, have unex pectedly complex ground states ev en i n relatively low dimensions. Specifically , we disprov e a conjecture of T orquato and Still- inger , who predicted t hat dil ute ground states of the Gaussian core model in dimensions 2 t hrough 8 wou ld be Brav ais lattices. W e show that in dimensions 5 and 7, there are in fact l o wer-ener gy non-Brav ais lattices. (The nearest three-dimensional analog is the he xagonal close-p acking, bu t it has high er ener gy than the face-centered cubic lattice.) W e believe these phenomen a are in fact quite widespread, and we relate them to decorrelation in high dimensions. P A CS numbers: 05 .20.-y , 61.50.Ah, 8 2.70.Dd I. INTRODUCTION One of the most natural soft-co re m odels in statistical me- chanics is th e Gaussian co re mod el (in troduced by Stillinger [1]), in w hich identical p articles interact via a repulsive Gaus- sian pair potential. This is not only a beautifu l th eoretical model, but also a re asonable mo del f or the effecti ve interac- tion (via entropic r epulsion) between the ce nters of mass of two polymers, namely , the Flor y-Krigb aum po tential [2, 3 ]. Much work has gone into characterizing the phase diagram and grou nd s tates of the Gaussian core model [4, 5]. W e use the Gaussian core model as a test case fo r study- ing the emergence of long- range structure in classical grou nd states. In tw o or three dimen sions, these gro und states are typically lattices, and ev en Brav ais lattices. T he theo ry behind this phen omenon is poor ly understo od: the Lennard -Jones po- tential in two dimen sions has been rigo rously analyzed by Theil [6], and S ¨ ut ˝ o [7 , 8] has an alyzed potential functio ns whose Fourier transfo rms are no nnegative and have compact support, but for no p urely repulsive soft-core potential in more than one dimension is there a compe lling argumen t for cry stal- lization (let alone a proo f). I n the present p aper, we show the subtlety of th is problem b y e xhibiting counterintuitive gr ound states with different structure than anticipated. Specifically , we study the Gaussian core mo del for di- lute systems in high-d imensional spa ces. Although that m ay sound arcane, such systems play an importan t role in statis- tical p hysics. First, th ey in clude sph ere packin g prob lems as a limiting case. Packing in h igh dimension s is of fu nda- mental importance in communication and in formatio n theory , because (as Shan non d iscovered) finding cod es for efficient commun ication in the presence of noise am ounts to a packing problem in the high-d imensional space of possible signals. Second, such systems provide an intriguing test ca se f or ∗ Electroni c address: cohn@microsof t.com † Electroni c address: abhina v@math.mit.ed u the deco rrelation effect, a funda mental phenome non pred icted by T o rquato and Stillinger [9]: in loose terms, unco nstrained spatial correlatio ns shou ld vanish asymptotically in high di- mensions, and all m ultibody c orrelation s will be red ucible to the pair c orrelation fun ction. Although it seems difficult to justify r igorou sly , decorre lation leads to su rprising con- jectures su ch as the existence of extraordin arily dense disor- dered pa ckings in h igh dimen sions (with importan t implica- tions in info rmation theor y). See also Ref. [10] for a rep lica symmetry- breakin g approach to amorph ous p ackings in hig h dimensions. This line of reasoning suggests that glassy states of matter are intrinsically mo re stable than cr ystals in high dim ensions, which stands in stark co ntrast to intuition d erived fro m most two- o r three-d imensional sy stems. In thre e dimensions, f or example, the low-density ground state for the Gaussian cor e model is the face-centered cubic (fcc) lattice, which has lower energy th an the competin g hexagonally close-packed (h cp) lattice, let alone disordered structures. In the present p aper, we show that the oppo site happ ens in as few as five dimen- sions: relatively exotic non-Bravais lattices imp rove on more familiar stru ctures. T o find t his behavior in such a low dimen - sion is unexpecte d, and w hile we cann ot demo nstrate the full decorre lation effect (for exam ple, with completely amo rpho us packing s), our results show tha t the role o f o rder and structure in ev en low-dimensional g round states is mor e subtle than was previously realized. Our direct mo ti vation is a recent pr ediction by T orquato and Stillinger [11] for the grou nd states of th e Gaussian core model in mod erately high dimensions (u p th rough R 8 and also R 24 ). Specifically , at sufficiently low p article d ensity , they conjecture d that the grou nd st ates are the Bravais lattices cor - respond ing to the de nsest known sphere packing s, and at suf - ficiently hig h particle d ensity they c onjecture d th at the ground states were the reciproca l Brav ais lattices. In the case o f R 2 , R 8 , and R 24 , this agrees with an earlier conjectu re of Cohn a nd Kumar (Conjectu re 9.4 in Ref. [12]). Zachar y , Stillinge r , and T or quato [13] ha ve gi ven strong num erical evidence that these are indeed the tru e gr ound states among known families of 2 Brav ais lattices. Howev er , in this paper we disprove T orqu ato and Stillinger’ s conjectu re by exhibiting non-Bravais lattices with lower energy in the lo w density limit in R 5 and R 7 . These imp roved lattices in fact corr espond to tigh t sphere packing s ( i.e., sphere pack ings that a re not only as dense as possible g lobally , but also lo cally , in th e sense that there are no missing spheres, small g aps, etc.). Conway and Slo ane [14] provided a conjecturally complete list o f tigh t packing s in low dimensions, an d o ur g round states can be found in th eir list. They stand in the same relationship to the optimal Brav ais lattices as the hcp packing stand s to th e fcc p acking in R 3 , b ut the energy c omparison s work out notab ly d ifferently . This is in effect anothe r facet o f decorrelation . Even with in the re- stricti ve class of tigh t p ackings, in hig h dimen sions Brav ais lattices are no lon ger e nergetically fav ored. Instead, some- what less regular structures are preferred. For comp arison to the mathematical liter ature (an d, in par- ticular , Ref. [ 14]), note that mathema ticians use “lattice” to mean “Brav ais lattice” and “ periodic pack ing” to mean “lat- tice with a basis. ” In this paper, we follow the phy sics termi- nology . II. THET A SERIES All of o ur work in th is paper takes place in the low density limit. Because of the scaling inv ariance o f Euclidean space, we can instead fix the particle den sity and rescale the Gaus- sian. Specifically , we u se the po tential func tion V ( r ) = e − α r 2 between tw o p articles at distance r , and we let α tend to infin- ity , which correspon ds to taking the low-density limit. The theta ser ies for a packing P (i.e., a collection of parti- cle loca tions) is a generating fun ction that describ es the aver - age number of particles at a given distance f rom a particle in P . Specifically , Θ P ( q ) = ∑ r N r q r 2 , where the sum is over all distances r between points in the packing , N r denotes the average over all x ∈ P of the n um- ber of y ∈ P such th at | x − y | = r , and q is a f ormal variable. The use of r 2 rather than r in the exp onent is trad itional in mathematics. Note that the th eta series encod es th e same in- formation as the pair co rrelation functio n; we use th is notation since it is con venient for th e Gaussian core model. Under the Gaussian cor e model poten tial fun ction V ( r ) = e − α r 2 , th e average energy per po int in P eq uals ( Θ P  e − α  − 1 ) / 2. (W e subtr act 1 to correct for th e r = 0 term in the th eta series, which would correspo nd to a self-interactio n, a nd we divide by 2 to av oid double coun ting.) Th us, computin g theta series is exactly the same as com puting energy in th e Gaussian core model. The limit as α → ∞ of energy correspon ds to the limit as q → 0 of the theta series. Giv en two pack ings with the same density (i.e., the same number of particles per un it volume in space), we can ea sily compare their behavior in the q → 0 limit. Supp ose their theta series are Θ 1 = 1 + a r 1 q r 2 1 + a r 2 q r 2 2 + · · · and Θ 2 = 1 + b s 1 q s 2 1 + b s 2 q s 2 2 + · · · with r 1 < r 2 < · · · a nd s 1 < s 2 < · · · . T o compare Θ 1 with Θ 2 , we need only conside r the first term at which they differ . If r 1 > s 1 , then Θ 1 < Θ 2 for small q ; if r 1 = s 1 , then the compar- ison am ounts to wh ether a r 1 < b s 1 . If r 1 = s 1 and a r 1 = b s 1 , then we must proceed to the next term. Correspon ding to a ny po int configu ration in R n , we obtain a sphere pa cking b y centering identical spheres at the p oints of the configur ation, with the maximal possible rad ius subject to a voiding overlap. The d ensity of the packing is the frac- tion of space covered. T o a void con fusion, we will disting uish between the particle density (the number of particles per unit volume in space) and the pa cking density (the fr action of space covered b y balls). As pointed o ut above, maximizing packin g density is a c on- sequence of min imizing energy in the G aussian core mo del in the α → ∞ lim it ( with fixed par ticle den sity): the domin ant contribution to the Gaussian energy comes f rom the small- est d istance b etween points, which is large exactly when the packing density is large. In other words, the problem of maxi- mizing the sphere packing density arises naturally as the lo w- density limit of the Gaussian core model. III. TIGHT P A CKINGS In most dim ensions, the spher e p acking pro blem exh ibits high degeneracy , in th e sense that th ere ar e m any geo metri- cally d istinct, optimal solution s (such as in thr ee dim ensions, with the fcc and hcp packing s and their relatives). Conway and Sloane [14] gave a con jectural classification of all th e tight packings in low dim ensions. Her e, tight means roug hly that the glob al den sity is maximized an d fu rthermo re n o lo- cal chan ges can add mo re sph eres. (For example, removing one sphere fro m a dense packing lea ves the global d ensity un- changed , but the r esult is n o lon ger tight.) The p recise defi- nition of tightness in Ref. [14] is problem atic; see Ref. [15] for details on the problem an d better definitions. B ecause they recognized that their definition was only tentative, Con- way and Sloane characterized tightness by articula ting “pos- tulates” th at th ey felt a co rrect de finition sho uld satisfy . These postulates are by no means obvious statements; in stead, they are empirical o bservations fro m Conway and Sloane’ s stud y of the packing prob lem. Conway and Sloane [14] postulate that, in dimensio ns up to 8, e very tight packin g fibe rs over a tight p acking whose dimension is the previous power of 2. T o say that a pack ing P fibers over Q mean s that P c an b e decom posed into pa rallel layers lyin g in dim ( Q ) -dime nsional sub spaces, each of which is a pa cking isometr ic to Q . (In fact, in tig ht pac kings of dimensions u p to 8 it will be a translate of Q . ) The loc ations of th ese par allel sub spaces sho uld themselves be d etermined by anoth er tight pack ing. Although the Conway an d Sloane postulates are on ly conjectu res, they seem likely to be tru e and in th is paper we assume their tr uth (but we note wh ich ones are required for each theorem) . 3 IV . DIMENSIONS UP TO 4 In R 1 , there is exactly one tight packing, namely that giv en by the integers. It is p rovably o ptimal for the Gaussian core model by Proposition 9.6 in Ref. [12]. In R 2 , the triangu lar lattice A 2 is the only tight pack ing. Montgom ery [ 16] sho wed th at it is optimal amo ng all Bravais lattices for the Ga ussian core model, and it was conjectu red in Ref. [12] that it is optimal among all lattices. In R 3 , all tigh t packin gs fiber ov er the triangu lar lattice A 2 . In othe r words, they are formed b y stacking trian gular layers, with the layers nestled together as den sely as p ossible; each additional lay er inv olves a binar y choice for how to place it relativ e to the p revious lay er . These are th e Barlow pack - ings ( i.e., the stacking variants of the fcc an d h cp p ackings). It is n ot hard to c heck that, amon g th ese packing s, the face- centered cub ic la ttice minimizes energy in the Gaussian core model in the low particle-den sity limit. This is consistent with the conjecture in Ref. [11]. In R 4 , there is only one tight p acking, nam ely the D 4 or checkerboar d lattice ( it is shown in Ref. [1 4] th at on ly o ne tight packing fibers over A 2 ). It is defined to be the set of all integral points whose coordinates ha ve e ven sum : D 4 = ( x ∈ Z 4 : 4 ∑ i = 1 x i ≡ 0 ( mod 2 ) ) . The un iqueness of D 4 is remarkab le, c ompared with the div er- sity of tigh t p ackings in R 3 , and the D 4 lattice plays a fu nda- mental role as a building b lock for higher-dimension al struc- tures. It also a ppears that, much like the trian gular lattice, D 4 may be u niversally optimal, in the sense that it is the gro und state of the Gaussian core model at any density . V . DIMENSION 5 In R 5 , every tight packing fibers over D 4 , with the distance between successi ve layers being 1. T o specify such a pa ck- ing, one need only sp ecify how each four-dimensional layer is translated r elativ e to its neig hbors. Th e deep h oles in D 4 (the p oints in space furth est fro m the lattice) are lo cated a t ( 1 , 0 , 0 , 0 ) , ( 1 / 2 , 1 / 2 , 1 / 2 , 1 / 2 ) , and ( 1 / 2 , 1 / 2 , 1 / 2 , − 1 / 2 ) , a s well as of cour se th e tran slates of these points by vectors in D 4 . Each lay er o f a tigh t packing in R 5 must either be a n untranslated copy of D 4 or b e translated b y one of these vec- tors, s o that th e distance between layers is minimized; fu rther- more, adjacent layers must b e tra nslated by d ifferent vectors. In othe r words, the sp heres in each layer m ust be nestled into the gaps in the adjacent layers. If we let a d enote the tr anslation vector ( 0 , 0 , 0 , 0 ) , b denote ( 1 , 0 , 0 , 0 ) , etc., then each layer must be translated by on e of a , b , c , or d , an d no two ad jacent layers can b e translated b y the same vector . In other words, a tigh t packing in R 5 is specified by a four-coloring of the in tegers (if we treat a , b , c , and d as “colors”). For example, the D 5 packing , which is th e Brav ais lattice with th e h ighest pa cking den sity , co rrespond s to the following coloring : . . . . . . ♠ ♠ ♠ ♠ ♠ a b a b a Note that the symm etries of the D 4 lattice arb itrarily permu te a , b , c , and d , so the choice of labeling is irre lev ant. For D 5 , all that matters is that the layers alter nate between tw o colors. Conway and Slo ane fou nd that fou r tight p ackings are un i- form, in the sen se that all spheres play the same r ole (rather than the less symmetric situation of h aving several ineq uiv- alent classes of spheres). I n addition to D 5 = Λ 1 5 , the th ree others correspond to the following p atterns: Λ 2 5 : · · · abc d abcd · · · , Λ 3 5 : · · · abc abc · · · , Λ 4 5 : · · · bac bd cad bacbd cad · · · . These three ad ditional lattices ar e n ot Bra vais lattices, but rather lattices with bases. One ca n calculate that the theta series for Λ 1 5 is 1 + 40 q 2 + 90 q 4 + 240 q 6 + · · · , while the theta series of Λ 2 5 is 1 + 40 q 2 + 88 q 4 + 16 q 5 + · · · . I t follows that Λ 2 5 has lower energy than D 5 in the q → 0 limit, wh ich disproves T or quato and Stillinger’ s conjecture . In fact, the situation is even worse for D 5 , which is no t only sub optimal but in fact the worst tig ht pack ing of all. Theorem 1. Und er P ostula tes 2, 4, an d 5 of R ef. [14], the Bravais lattice D 5 has the highest e ner gy among all the tig ht five-dimen sional lattices, in the q → 0 limit. T o com plete this calcu lation, we requ ire four geometric al facts about D 4 . Spe cifically , each lattice po int h as 24 neigh - boring lattice poin ts at squared distanc e 2, the next closest lattice points are 2 4 more at squ ared distance 4, each deep hole has 8 neighb oring lattice points at sq uared distance 1, and the next closest lattice points to a deep hole are 3 2 points at squared distanc e 3. T hese assertio ns are easily checked by a short calculation. Pr o of. Let Λ be a tight fi ve-dimensional la ttice, obtained by a four-coloring of the integers. W e first observe that every sphere in Λ must h av e 40 n eighbo rs at squared d istance 2, for the following reason. Without loss of gene rality , we may assume that layer 0 is colored a and la yer 1 is colored b (since the d ifferent deep holes a re equiv alent und er the symmetries of D 4 ). Now , layer − 1 cann ot b e colo red a either , so the lay ers 0, 1, and − 1 c ontribute 24 + 8 + 8 = 40 neighbor s of a given sphere in lay er 0. (Every sph ere in D 4 has 24 neighbor s, which accounts f or the 24 f rom lay er 0 , an d each deep h ole in D 4 is at distance 1 f rom 8 points of D 4 .) Th erefore th e theta series of Λ must start with 1 + 40 q 2 + · · · . The ne xt smallest possible squ ared distan ce in Λ is 4 (squared distanc e 3 does n ot occu r in D 4 , and it cannot o ccur between adjacent layers since that would a mount to having a lattice poin t at squared distance 2 from a deep h ole). There are 24 spheres at that distance in D 4 , and 32 in each of laye rs ± 1, for a total o f 88. The only way ther e can be more is if they 4 come fro m layers ± 2, a nd each o f tho se lay ers con tributes one sphere (lying over the orig in) if and on ly if it is co lored the same as laye r 0. Since D 5 correspo nds to the colo ring · · · abab aba · · · , its theta function has the maximum co ntribu- tion to the q 4 term, making it the worst for en ergy as q → 0. Furthermo re, among all tigh t lattices only D 5 maximizes that term, so it is the unique pessimum. The lattice Λ 2 5 turns out to be the best. Theorem 2. Und er P ostula tes 2, 4, and 5 of Ref. [14], the lattice Λ 2 5 has the lowest ener gy amon g all the tight fi ve- dimensiona l lattices, in the q → 0 limit. Pr o of. Th e pr oof is similar to th at of the previous theor em. Let Λ be a tight packing as above, fibered over D 4 . W e m ay assume as before that layer 0 is colored a . The first two terms of the theta series of Λ are 1 an d 40 q 2 . Now , if layer 2 or layer − 2 were color ed a , th en Λ would have a larger q 4 term than Λ 2 5 , making it worse fo r potential energy in the q → 0 limit. Therefo re we may assume neither 2 nor − 2 is colored a . Th e theta series is no w determined up to the q 8 term, and it equals 1 + 40 q 2 + 8 8 q 4 + 1 6 q 5 + 1 92 q 6 + 6 4 q 7 + 1 52 q 8 + · · · . The q 9 term is n ot y et determin ed, since it depend s on lay- ers 3 a nd − 3. Merely b eing three laye rs apart contributes 3 2 to the squared distance, so th ey c ontribute to the q 9 term if and only if th ey are colored a . Thu s, to m inimize energy th ey must not be colored a . In other words, tw o layers of the same color must be separated by at lea st 4. The only way to do this is to co lor the layers · · · abc d abcd · · · , up to permu tations of the four co lors. Sin ce permu ting the f our colors will no t chang e the resu lting lattice (b ecause of th e symm etries of D 4 ), we see that Λ 2 5 is the uniqu e b est lattice among all the tight five- dimensiona l lattices in the q → 0 limit. VI. DIMENSION 6 In R 6 , the way to f orm tigh t packing s is again to fiber over D 4 , and we mu st u se the triangu lar lattice A 2 to arrange the fibers (w ith A 2 normalized so the closest lattice po ints are at distance 1). Thus, we ar e looking fo r f our-colorings of the tri- angular lattice A 2 , where th e colors specify which translation vector to u se for the copy of D 4 . As in th e previous dime nsion, the separation between adjacent layers will be 1. The E 6 lattice, which is the Bravais lattice with the h ighest packing density , is g i ven by the following coloring: ♠ ♠ ♠ ♠ ♠ a c a c a ♠ ♠ ♠ a c a ♠ ♠ ♠ a c a ♠ ♠ ♠ ♠ b d b d ♠ ♠ ♠ ♠ d b d b The theta series of E 6 is 1 + 72 q 2 + 270 q 4 + 936 q 6 + 216 0 q 8 + · · · . As shown b y Conway and Sloan e, the re are th ree other unifor m packin gs, c orrespon ding to the following p ossibilities for the six neighbo rs sur round ing a ce ntral a : Λ 2 6 : b cbd cd , Λ 3 6 : b cbcbc , Λ 4 6 : b cbcbd . In contrast to the fi ve-dimension al case, the Brav ais lattice E 6 is in fact optimal among all tight lattices in the q → 0 limit. Theorem 3. Und er P ostula tes 2, 4, an d 6 of R ef. [14], the Bravais la ttice E 6 has the lo west en er gy a mong all th e tigh t six-dimensiona l lattices, in the q → 0 limit. Pr o of. Let Λ b e a tight packing forme d as ab ove by four- coloring the triangular lattice. Let us assume that the central sphere is colored a . The squared distance s in the A 2 lattice are 1 , 3 , 4 , . . . , so n eighbor s a t squ ared distanc e 2 in Λ can come only fr om th e centr al layer and its six adjacen t layer s. The num ber of these vectors is 24 + 6 · 8 = 7 2, which is in accordan ce with th e theta function of E 6 . The next possible squared d istance is 3. Note that this distance does not o ccur in E 6 , since in the colorin g above, t here are no two spheres at squared distance 3 which have the same color . But in f act, the coloring above is th e only coloring with this property (up to a permutatio n of the colors a , b , c , d , of course, b ut that is irrel- ev ant because of the symmetries o f D 4 ). T o see th is, start with the central sphere colored a , and notice that the six sphere s around it must b e colored bcd bcd (or bd cbd c ) to av oid two spheres of the sam e color being √ 3 units apart. One can then apply th e argument to the six sph eres cen tered aro und o ne o f these six neig hbors and p roceed outw ard, to ar riv e at a unique packing : nam ely , the one above. This shows that E 6 is in - deed the b est f or en ergy in the q → 0 limit, am ong all tigh t lattices. One can also determine the worst tight packing. Theorem 4. Und er P ostula tes 2, 4, an d 6 of R ef. [14], the lattice Λ 3 6 has the high est ener gy among all the tight six- dimensiona l lattices, in the q → 0 limit. W e om it th e details of the proo f. Howe ver , the calculation amounts to showing th at the Λ 3 6 coloring maximizes the num- ber of iden tically colored spher es at squared distan ce 3 in A 2 . In the following p icture of the coloring , th e six bold c ircles are at squared distance 3 from the central circle: ♠ ♠ ❧ ♠ c a b ♠ ❧ ♠ ♠ ♠ ❧ a b c a ♠ ♠ ♠ ♠ ♠ b c a b c ♠ ❧ ♠ ♠ ♠ ❧ a b c a ♠ ♠ ❧ ♠ c a b 5 ♠ a ♠ c ♠ a ♠ c ♠ a ♠ b ♠ d ♠ b ♠ d ♠ d ♠ b ♠ d ♠ b ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ♠ d ♠ b ♠ d ♠ b ♠ ❥ a ♠ c ♠ ❥ a ♠ c ♠ ❥ a ♠ c ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ♠ b ♠ d ♠ b ♠ d ♠ ❥ a ♠ c ♠ ❥ a ♠ c ♠ ❥ a ♠ c ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ r r r ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ r r r ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ FIG. 1: The four-c oloring of the fcc lattice used to obtain E 7 . VII. DIMENSION 7 Finally , in dimen sion 7 the optimal Brav ais lattice E 7 is nei- ther the worst n or the best for energy amon g t ight packings in the low particle- density limit. Accordin g to Ref. [1 4], each tight p acking in R 7 fibers over D 4 , and the f our-dimensional layers are arranged using a tig ht packing in R 3 with adjacent D 4 layers separated by 1. T o spe cify a fou r-coloring of the three-dim ensional packing, w e need o nly specify it on a sin- gle tria ngular layer, since each such layer determ ines the col- ors on both adjacent layers and hence on every layer . W e cannot u se an arbitrary four-colorin g of the triang ular layer, since so me color ings do not extend consistently to the other layer s. Conway and Sloane showed th at the cond ition for extending co nsistently is that the co loring should have “pe- riod 2” in the following sense: the packing sho uld decompose into para llel strings o f ad jacent spher es, so th at in each string the colors alter nate between two possibilities. For example, the E 6 coloring shown in the pre vious section h as this proper ty (the strings lie alo ng ho rizontal lines), while th e Λ 3 6 coloring does not. T o obtain E 7 , we use the face-centered cubic as our tight packing in R 3 , a nd we use the same co loring of a tr iangu- lar layer as was used to constru ct E 6 . W e get the pictur e in Fig. 1, which shows thr ee triangular layers of th e fcc lattice surroun ding a cen tral ball colore d a (the dotted line s show how the layers are aligned, and the different styles o f cir cles are for referen ce in the argument belo w). The theta series of E 7 is 1 + 126 q 2 + 756 q 4 + · · · , and we can see the first n ontrivial term as f ollows. A po int in the D 4 layer cor respond ing to the c entral circle colore d a above h as 24 neighbors at squared distance 2 in the same D 4 layer, 12 · 8 in neighbo ring D 4 layers (8 each from the 12 neighbor s in the face-centered cubic, which have bold circles in Fig. 1), and 6 from non -neighb oring D 4 layers (1 each fro m th e 6 p oints in the face-centere d cubic at squared distance 2, which are shown with two n ested circles in Fig. 1 a nd are each colo red a ). T o improve upon E 7 in the q → 0 limit, we can use the Λ 2 6 coloring of a triang ular layer; the resulting tight p acking ♠ a ♠ c ♠ a ♠ c ♠ a ♠ b ♠ d ♠ b ♠ d ♠ b ♠ d ♠ b ♠ d ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ♠ d ♠ b ♠ d ♠ b ♠ ❥ a ♠ c ♠ ❥ a ♠ a ♠ c ♠ a ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ♠ d ♠ b ♠ d ♠ b ♠ ❥ a ♠ c ♠ ❥ a ♠ a ♠ c ♠ a ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ r r r ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ r r r ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ FIG. 2: The four-c oloring of the fcc lattice used to obtained Λ 3 7 . ♠ a ♠ c ♠ a ♠ c ♠ a ♠ b ♠ d ♠ b ♠ d ♠ d ♠ b ♠ d ♠ b ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ♠ d ♠ b ♠ d ♠ b ♠ ❥ a ♠ c ♠ ❥ a ♠ c ♠ ❥ a ♠ c ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ♠ c ♠ a ♠ c ♠ d ♠ b ♠ b ♠ d ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ r r r ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ FIG. 3: The four-c oloring of the hcp lattice used to obtained Λ 2 7 . is called Λ 3 7 . One can see fro m Fig. 2 th at among the final six points in the c alculation above, o nly fou r shar e the sam e color as the cen tral point. T hus, the theta series o f Λ 3 7 begins 1 + 124 q 2 + · · · , which is an improvement over the E 7 lattice, and the next squared distance is 4. T o con struct a tight packing with higher energy than E 7 in the lo w-density limit, we can use th e hexagonal close packing in R 3 , while using the sam e coloring on a triangular layer as for E 7 (namely , the one a lso used to constru ct E 6 ). The re- sulting colo ring is shown in Fig. 3, and th e packing is called Λ 2 7 . The large tr iangular layer at the bottom of the figure plays the same role as the central layer in the previous figu res. W e have not drawn the layers below it bec ause the hcp packing is mirror symmetric about each layer . The theta series begins 1 + 126 q 2 + · · · for the same reason as above, but the next term is 2 q 8 / 3 , which occurs between nonad jacent triangular layers. S pecifically , each poin t in the hcp packing is at distance p 8 / 3 (i.e., twice the heig ht p 2 / 3 of a regular tetrahe dron w ith edge len gth 1) fr om two points, which are two layers above and below it. The dotted lines in Fig. 3 co nnect such points. Because the c orrespon ding points always hav e the same color, the the ta series of Λ 2 7 beings 1 + 126 q 2 + 2 q 8 / 3 + · · · , an d h ence Λ 2 7 has higher en ergy than E 7 in the q → 0 limit. 6 ♠ a ♠ c ♠ a ♠ c ♠ a ♠ b ♠ d ♠ b ♠ d ♠ b ♠ d ♠ b ♠ d ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ♠ d ♠ b ♠ d ♠ b ♠ ❥ a ♠ c ♠ ❥ a ♠ a ♠ c ♠ a ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ♠ c ♠ a ♠ c ♠ d ♠ b ♠ d ♠ b ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ r r r ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ FIG. 4: The four-c oloring of the hcp lattice used to obtained Λ 4 7 . There is on e further possibility worth analyzing , namely the coloring of the hcp lattice shown in Fig. 4 (which uses the Λ 2 6 coloring on a tr iangular layer and lead s to a tight p acking called Λ 4 7 ). I ts theta series begins 1 + 1 24 q 2 + 2 q 8 / 3 + · · · . The f our tigh t packing s we have analyzed in this section are of course not the only tigh t pack ings, but they are the only un iform ones. Their local con figuration s cover e nough possibilities to determine the lowest- and highest-energy tight packing s. Specifically , there are relati vely f ew p eriod 2 color- ings of a triang ular layer . Observe the large tr iangular layers in the figur es: with out loss of g enerality we can assum e that the middle ho rizontal row in the large triang ular layer is col- ored a caca (b y the period 2 assumption ), as is shown in ea ch figure. Th en th ere are only two variables in the pictures. Th e first is whether the adjacent two h orizontal rows line up with b above b and d above d (as in Fig s. 2 and 4) or whether they are stagg ered (as in the remaining two figur es). The second variable is wheth er the trian gular layers are them selves stag- gered (as in the fcc lattice) or mirror-symmetric (as in the hcp lattice). If the pictures were to be enlarged, more of these choices would ar ise, but within th e scope of what has b een drawn, th ere a re only these four possibilities. It follows that Λ 3 7 has the best local con figuration at each poin t, wh ile Λ 2 7 has the worst at each point. Theorem 5. Under P ostulate s 2, 3, 4 , an d 7 o f Ref. [1 4], th e lattice Λ 3 7 has the lowest energy among all the tight seven- dimensiona l lattices, in the q → 0 limit, an d Λ 2 7 has the high- est. VIII. HIGHER DIMENSIONS In R 8 there is a u nique tight packing, n amely the E 8 lattice, which is almost certainly the gro und state for the Gaussian core m odel. Becau se of the uniqueness of E 8 , the approac h used in R 5 and R 7 does not apply . Above dimen sion 8, th e appr oach of Ref. [14] breaks down, and tigh t p ackings no lon ger fib er n icely . Outside of a handful of exceptional dimensions (certainly 24 and perhap s 12 o r 16), we expect that the grou nd states o f th e Gaussian co re m odel become quite complicated . IX. CONCLUSIONS AND DISCUSSION W e ha ve shown that the gro und states o f the Gaussian core model can b e unexpected ly complex. Specifically , in five and se ven dimension s, th e gr ound states are not Brav ais lattices, which co ntrasts with the mor e familiar b ehavior in two or three dimension s. This be havior is n ot limited to the Gaus- sian core mo del. The non -Brav ais lattices stud ied in this paper are in fact superio r for a wide ran ge of soft-co re models, in- cluding for examp le in verse po wer laws with high exponen ts. (Note th at inverse power laws are scale-fr ee, so in that case our results hold for all densities.) These phenomena are char acteristic of h igh dimensions, and they provide su pport for the T orquato -Stillinger d ecorre- lation principle. As the dimension increases, familiar symme- tries become increasingly likely to be broken. One notew orthy example is the kissing configu rations in five dimensions ( i.e., the spherica l configuration s fo rmed by the p oints of tange ncy with adjacent spheres). Th e D 5 lattice’ s k issing config uration is high ly symm etrical; in suitable co ordinates it is given by the vectors ( ± 1 , ± 1 , 0 , 0 , 0 ) and all vectors obtain ed by per- muting t he coordinates. By contrast, the k issing configur ation of Λ 2 5 is far less symm etrical. T o form it, replac e th e eigh t vectors that ha ve a 1 in the first coordinate with the eight vec- tors ( 1 , ± 1 / 2 , ± 1 / 2 , ± 1 / 2 , ± 1 / 2 ) , where the num ber of mi- nus sign s m ust be even. This clearly breaks th e symmetry , and ind eed the size of th e sym metry grou p is red uced by a factor of 1 0 (f rom 3840 to 3 84). Nevertheless, Λ 2 5 has lower energy than D 5 , and its kissing configuratio n alone has lower energy than that of D 5 as spherical configu rations. Sy mmetry simply does not align with consideratio ns o f energy . Because of th e conn ections betwe en high-d imensional sphere p acking and informatio n theory , these i ssues shed light on co ding theo ry . Computer scientists and engineer s h av e learned throug h lon g exp erience that effi cient error -correc ting codes sho uld be chosen to be pseu do-ran dom ( truly rand om would be even better , but it is genera lly not pr actical). For example, M acKay [17, p. 596] summar izes his coding the- ory advice as fo llows: “The best solution to th e commu ni- cation pro blem is: Combine a simple, pseudo -rando m co de with a message-passing decode r . ” From a naive perspec ti ve, this situation is puzzling , since o ne might expect that h ighly structured codes would offer the most scop e for powerful al- gorithms. Instead , elaborate algebraic structure seems incom- patible with hig h-perf ormanc e cod ing. This is n ot p urely a geometric question, b ecause of the role of alg orithms, b ut it is largely g eometric, an d th e und erlying geo metry in volves the same decorrelation effect ob served in physics. This emph a- sizes the need for a detailed theoretical u nderstand ing of high- dimensiona l pack ing and related statistical mechanics mo dels. One natural are a f or f urther exploration would be non- Euclidean spaces. Introdu cing curvature illuminates the prob- lem of geo metrical frustra tion, in which id eal local config - urations do no t exten d consistently to g lobal arran gements. Specifically , curvature may r eliev e (or intr oduce) fr ustration, and comparing results in different cur vatures clarifies the role of f rustration. See, fo r example , Ref . [18]. Much work has been done in positively cur ved spaces such as sphe res, and 7 Modes and Kamien [19, 20] have recently studied hard-co re models in negativ ely curved two-dimensional space. It wou ld be intriguing to extend this w ork to higher dimension s. Another area for fu ture in vestigation is m ore sophisticated models than the Gaussian core mo del. For example, in the Ziherl-Kamie n th eory of m icellar crystals [21, 22], area- minimizing effects (as in soap fro ths) fru strate the clo se- packing one expects from a hard core . It would be inter esting to study dimensiona l trends in such systems. W e c onclude with a few specific o pen pro blems abo ut th e Gaussian core model. (1) W e have been able to address the low-density limit, but our appro ach says noth ing abo ut th e high-d ensity limit. Are Brav ais lattices optimal for the G aussian core mo del at hig h density in lo w dimen sions, as T orquato and Stillinger [11] conjecture d? W e suspect that Brav ais la ttices may again be suboptimal in as few as five dimensions, but th at is merely a guess. (2) In this paper, we were lucky to be able to construct improved n on-Brav ais lattices essentially by caref ul modifi- cation of the Bravais lattices (mu ch as th e hcp packing can be ob tained by modifying the f cc lattice). It is unlikely tha t this sort of modification will yield a com plete picture o f the Gaussian core mod el’ s gro und states at all den sities. In the absence of new geometrical ideas, it is natu ral to turn to nu- merical simulation s. Un fortun ately , simulations b ecome in- creasingly difficult as th e dimension increases, because of the curse of dimen sionality (the nu mber of particles req uired in- creases exponen tially as a fu nction o f dimension). Can one develop an effi cient enoug h simulator to perfor m useful work in four, five, or even six dimension s? Skoge, Do nev , Still- inger, an d T or quato [23] h av e performe d such simulations to compute jammed hard -core pac kings, but that problem may be somewhat ea sier as there are no long-ran ge interactions. (3) Is the D 4 lattice universally op timal in R 4 ? (In other words, is it the grou nd state of the Gaussian core model at ev- ery d ensity?) All av ailable evidence sug gests that the answer is yes, except for on e observation of Cohn, Conway , Elk ies, and Kumar [24]. They show that the D 4 kissing config ura- tion of 24 poin ts does not form a universally optimal sp herical configur ation, by finding a competing family of configu rations that occasionally beats it. (By contrast, Cohn and Kumar [12 ] proved that the E 8 kissing co nfiguratio n is un iv ersally opti- mal.) Unf ortunately , the spherical comp etitors do not seem to extend to Euclidean p ackings. Because D 4 is such a symmet- rical and beautifu l structu re, it would be inter esting to know more definitiv ely wh ether it is universally o ptimal. Simu la- tions could help resolve this iss ue. Acknowledgments W e than k Salvatore T orq uato and Frank Stillinger for help- ful d iscussions and the referees fo r the ir suggestion s. A.K. was suppor ted in part by Nationa l Science Foundatio n Gr ant No. DMS-0757 765. [1] F . H. Stillinger, J. Chem. Phys. 65 , 3968 (1976 ). [2] P . J. F lory and W . R. Krigbaum, J . Chem. Phys. 18 , 10 86 (1950). [3] A. A. Louis, P . G. Bolhuis, J. P . Hansen, and E. J. Meijer , Phys. Rev . Lett . 85 , 2522 (2000). [4] A. Lang, C . N. Likos, M. W atzla wek, and H. L ¨ owen, J. P hys.: Condens. Matter 12 , 5087 (2000). [5] S. Presti pino, F . S aija, and P . V . Giaquinta, Phys. Re v . E 71 , 050102 (R) (2005). [6] F . Theil, Commun. Math. Phys. 262 , 209 (2006). [7] A. S ¨ ut ˝ o, Phys. Rev . Lett. 95 , 265501 (2005). [8] A. S ¨ ut ˝ o, Phys. Rev . B 74 , 10411 7 (2006). [9] S. T orquato and F . H. Stillinger , Exp. Math. 15 , 307 (2006 ). [10] G. Parisi and F . Zamponi, J. Stat. Mech.: Theory Exp. (2006), P03017. [11] S. T orquato and F . H. Stillinger, Phys. Rev . Lett. 100 , 020602 (2008). [12] H. Cohn and A. Kuma r , J. Am. Math. Soc. 20 , 99 (2007). [13] C. Zachary , F . H. Stillinger , and S. T orquato, J. Chem. Phys. 128 , 22450 5 (2008). [14] J. H. Conway and N. J. A. Sloane, Discrete Comput. Geom. 13 , 383 (1995). [15] G. Kuperb erg, Geom. T opol. 4 , 277 (2000). [16] H. L. Montgomery , Glasg. Math. J. 30 , 75 (1988). [17] D. MacKay , Information T heory , Inferenc e, and Learning Al- gorithms (Cambridge Univ ersity Press, Cambridge, 2003). [18] J. F . S adoc and R. Mosseri, Geometrical F rustr ation (Cam- bridge Univ ersity Press, Cambridge, 1999). [19] C. D. Modes and R . D. Kamien, Phys. Rev . Lett. 99 , 235701 (2007). [20] C. D. Modes and R. D. Kamien, Phys. R e v . E 77 , 04112 5 (2008). [21] P . Ziherl and R. D. Kamien, Phys. Re v . Lett. 85 , 3528 (2000). [22] P . Ziherl and R. D. Kamien, J. Phys. Chem. B 105 , 10147 (2001). [23] M. Skog e, A. Done v , F . H. Stilli nger , and S. T orquato, Phys. Rev . E 74 , 041127 (2006). [24] H. C ohn, J. Conway , N. Elkies, and A. Kumar , Exp. Math. 16 , 313 (2007).