EEffect of periodic arrays of defects on lattice energy minimizers
Laurent B´eterminFaculty of Mathematics, University of Vienna,Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria [email protected] . ORCID id: 0000-0003-4070-3344August 4, 2020
Abstract
We consider interaction energies E f [ L ] between a point O ∈ R d , d ≥
2, and a lattice L containing O , where the interaction potential f is assumed to be radially symmetric anddecaying sufficiently fast at infinity. We investigate the conservation of optimality results for E f when integer sublattices kL are removed (periodic arrays of vacancies) or substituted (periodicarrays of substitutional defects). We consider separately the non-shifted ( O ∈ kL ) and shifted( O (cid:54)∈ kL ) cases and we derive several general conditions ensuring the (non-)optimality of auniversal optimizer among lattices for the new energy including defects. Furthermore, in thecase of inverse power laws and Lennard-Jones type potentials, we give necessary and sufficientconditions on non-shifted periodic vacancies or substitutional defects for the conservation ofminimality results at fixed density. Different examples of applications are presented, includingoptimality results for the Kagome lattice and energy comparisons of certain ionic-like structures. AMS Classification:
Primary 74G65 ; Secondary 82B20.
Keywords:
Lattice energy; Universal optimality; Defects; Theta functions; Epstein zeta functions;Ionic crystals; Kagome lattice.
Contents a r X i v : . [ m a t h - ph ] A ug Introduction, setting and goal of the paper
Mathematical results for identifying the lattice ground states of interacting systems have recentlyattracted a lot of attention. Even though the ‘Crystallization Conjecture’ [16] – the proof ofexistence and uniqueness of a periodic minimizer for systems with free particles – is still open infull generality, many interesting results have been derived in various settings for showing the globalminimality of certain periodic structures including the uniform chain Z , the triangular lattice A ,the square lattice Z , the face-centred cubic lattice D (see Fig. 1), as well as the other bestpackings E and the Leech lattice Λ (see [12, 23] and references therein). Moreover, the samekind of investigation has been made for multi-component systems (e.g. in [10, 29, 30, 35, 36]) wherethe presence of charged particles yield to rich energetically optimal structures. These problems ofoptimal point configurations are known to be at the interface of Mathematical Physics, Chemistry,Cryptography, Geometry, Signal processing, Approximation, Arithmetic, etc. The point of viewadopted in this work is the one of Material Science where the points are thought as particles oratoms.Figure 1: In dimension d = 2, representation of the triangular and square lattices respectively defined by A = λ (cid:2) Z (1 , ⊕ Z (1 / , √ / (cid:3) and Z . In dimension d = 3, representation of the simple cubic and theface-centred cubic lattices respectively defined by Z and D := λ [ Z (1 , , ⊕ Z (0 , , ⊕ Z (1 , , λ , λ are such that the lattices have unit density. In this paper, our general goal is to show mathematically how the presence of periodic arrays ofcharges (called here ‘defects’ in contrast with the initial crystal ‘atoms’) in a perfect crystal affectsthe minimizers of interaction energies when the interaction between species is radially symmetric.Since the structure of crystals are often given by the same kind of lattices, it is an important questionto know the conditions on the added periodic distribution of defects and on the interaction energyin order to have conservation of the initial ground state structure. Furthermore, only very fewrigorous results are available on minimization of charged structures among lattices (see e.g. ourrecent works [9, 10]).We therefore assume the periodicity of our systems, and once we restrict this kind of problem tothe class of (simple) lattices and radially symmetric interaction potentials, an interesting non-trivialproblem is to find the minimizers of a given energy per point among these simple periodic sets ofpoints, with or without a fixed density. In this paper, we keep the same kind of notations we haveused in our previous works (see e.g. [8, 10, 14]). More precisely, for any d ≥ L d theclass of d -dimensional lattices, i.e. discrete co-compact subgroups or R d , L d := (cid:40) L = d (cid:77) i =1 Z u i : { u , ..., u d } is a basis of R d (cid:41) , and, for any V > L d ( V ) ⊂ L d denotes the set of lattices with volume | det( u , ..., u d ) | = V , i.e.2uch that its unit cell Q L defined by Q L := (cid:40) x = d (cid:88) i =1 λ i u i : ∀ i ∈ { , ..., d } , λ i ∈ [0 , (cid:41) , (1.1)has volume | Q L | = V . We will also say that L ∈ L d ( V ) has density V − . The class F d of radiallysymmetric functions we consider is, calling M d the space of Radon measures on R + , F d := (cid:26) f : R + → R : f ( r ) = (cid:90) ∞ e − rt dµ f ( t ) , µ f ∈ M d , | f ( r ) | = O ( r − p f ) as r → ∞ , p f > d/ (cid:27) . When µ f is non-negative, f is a completely monotone function, which is equivalent by Hausdorff-Bernstein-Widder Theorem [3] with the property that for all r > k ∈ N , ( − k f ( k ) ( r ) ≥ F cmd := { f ∈ F d : µ f ≥ } . For any f ∈ F d , we thus defined the f -energy E f [ L ] of a lattice L , which is actually theinteraction energy between the origin O of R d and all the other points of L , by E f [ L ] := (cid:88) p ∈ L \{ } f ( | p | ) . (1.2)Notice that this sum is absolutely convergent as a simple consequence of the definition of F d . Wecould also define E f without such decay assumption by renormalizing the sum using, for instance,a uniform background of opposite charges (see e.g. [34]) or an analytic continuation in case ofparametrized potential such as r − s (see [17]).One can interpret the problem of minimizing E f in L d (or in L d ( V ) for fixed V >
0) as ageometry optimization problem for solid crystals where the potential energy landscape of a systemwith an infinite number of particles is studied in the restricted class of lattice structures. Eventhough the interactions in a solid crystal are very complex (quantum effects, angle-dependentenergies, etc.), it is known that the Born-Oppenheimer adiabatic approximation used to describethe interaction between atoms or ions in a solid by a sum of pairwise contributions (see e.g. [40,p. 33 and p. 945] and [46]) is a good model for ‘classical crystals’ (compared to ‘quantum crystals’[18]), i.e. where the atoms are sufficiently heavy. Moreover, since all the optimality properties weare deriving in this paper are invariant under rotations, all the results will be tacitly considered upto rotations.Furthermore, studying such interacting systems with this periodicity constraint is a goodmethod to keep or exclude possible candidates for a crystallization problem (i.e. with free par-ticles). We are in particular interested in a type of lattice L d that is the unique minimizer of E f in L d ( V ) for any fixed V > f ∈ F cmd . Following Cohn andKumar [21] notion (originally defined among all periodic configurations), we call this property the universal optimality among lattices of L d (or universal optimality in L d (1)).Only few methods are available to carry out the minimization of E f . Historically, the first oneconsists to parametrize all the lattices of L d (1) in an Euclidean fundamental domain D d ⊂ R d ( d +1)2 − (see e.g. [44, Sec. 1.4]) and to study the variations of the energy in D d . It has been done indimension 2 for showing the optimality of the triangular lattice A at fixed density for the Epsteinzeta function [19, 27, 28, 42] and the lattice theta function [38] respectively defined for s > d and α > ζ L ( s ) := (cid:88) p ∈ L \{ } | p | s , and θ L ( α ) := (cid:88) p ∈ L e − πα | p | . (1.3)3n particular, a simple consequence of Montgomery’s result [38] for the lattice theta function isthe universal optimality among lattices of A (see e.g. [4, Prop. 3.1]). Other consequences ofthe universal optimality of A among lattices have been derived for other potentials (including theLennard-Jones one) [4, 7, 14, 15] as well as masses interactions [11]. Furthermore, new interestingand general consequences of universal optimality will be derived in this paper, including a sufficientcondition for the minimality of a universal minimizer at fixed density (see Theorem 2.9).This variational method is also the one we have recently chosen in [9] for showing the maximalityof A in L (1) – and conjectured the same results in dimensions d ∈ { , } for the lattices E andΛ – for the alternating and centered lattice theta function respectively defined, for all α >
0, by θ ± L ( α ) := (cid:88) p ∈ L ϕ ± ( p ) e − πα | p | , and θ cL ( α ) := (cid:88) p ∈ L e − πα | p + c L | , (1.4)where L = (cid:76) di =1 Z u i , { u i } i being a Minkowski (reduced) basis of L (see e.g. [44, Sect. 1.4.2]), ϕ ± ( p ) := (cid:80) di =1 m i for p = (cid:80) di =1 m i u i , m i ∈ Z for all i , and c L = (cid:80) i u i is the center of its unitcell Q L . In particular, the alternate lattice theta function θ ± L ( α ) can be viewed as the Gaussianinteraction energy of a lattice L with an alternating distribution of charges ±
1, which can beitself seen as the energy once we have removed 2 times the union of sublattices ∪ di =1 ( L + u i ) fromthe original lattice L . This result shows another example of universal optimality – we will call it universal maximality – among lattices, i.e. the maximality of A in L (1) for the energies E ± f and E cf defined by E ± f [ L ] := (cid:88) p ∈ L \{ } ϕ ± ( p ) f ( | p | ) , or E cf [ L ] := (cid:88) p ∈ L f ( | p + c L | ) , (1.5)where f ∈ F cmd . This kind of problem was actually our first motivation for investigating the effectsof periodic arrays of defects on lattice energy minimizers, since removing two times the sublattices2 L + u and 2 L + u totally inverse the type of optimality among lattices. Furthermore, thismaximality result will also be used in Theorem 2.4, applied – in the general case of a universalmaximizer L ± d for E ± f in any dimension where this property could be shown – for other potentials F d \F cmd in Theorem 2.11 and compared with other optimality results in Section 3.2.The second method for showing such optimality result is based on the Cohn-Elkies linear pro-gramming bound that was successfully used for showing the best packing results in dimensions 8and 24 for E and Λ in [22, 47], as well as their universal optimality among all periodic configu-rations in [23]. As in the two-dimensional case, many consequences of these optimality results havebeen shown for other potentials [14, 39] and masses interactions [8]. The goal of this work is to investigate the effect on the minimizers of E f when we change, given alattice L ⊂ L d and K ⊂ N \{ } , a certain real number a k (cid:54) = 0 of integer sublattices kL , k ∈ K , inthe original lattice, and where the lattices kL might be shifted by a finite number of lattice points L k := { p i,k } i ∈ I k ⊂ L for some finite set I k . Writing κ := { K, A K , P K } , K ⊂ N \{ } , A K = { a k } k ∈ K ⊂ R ∗ , P K = (cid:91) k ∈ K L k , L k = { p i,k } i ∈ I k ⊂ L, (1.6)the new energy E κf we consider, defined for f ∈ F d and κ as in (1.6) and such that the followingsum on K is absolutely convergent, is given by E κf [ L ] := E f [ L ] − (cid:88) k ∈ K (cid:88) i ∈ I k a k E f [ p i,k + kL ] . (1.7)4n particular, in the non-shifted case, i.e. P K = ∅ , then E κf [ L ] = E f κ [ L ] , where f κ ( r ) := f ( r ) − (cid:88) k ∈ K a k f ( k r ) . (1.8)Since we are interested in the effects of defects on lattice energy ground states, we thereforewant to derive conditions on κ and f such that E f and E κf have the same minimizers in L d or L d ( V )for fixed V >
0. In particular, we also want to know if the universal minimality among latticesof a lattice L d is conserved while removing or substituting integer sublattices. This a naturalstep for investigating the robustness of the optimality results stated in the previous section of thispaper when the interaction potential is completely monotone or, for instance, of Lennard-Jonestype. Furthermore, it is also the opportunity to derive new applications and generalizations of themethods recently developed in [4, 9, 14] for more ‘exotic’ ionic-like structures.Replacing integer sublattices as described above can be interpreted and classified in two relevantcases in Material Science:1. If a k = 1, then removing only once the sublattice kL from L creates a periodic array ofvacancies (also called periodic Schottky defects [45, Sect. 3.4.3]);2. If a k (cid:54) = 1, then ‘removing’ a k times the sublattice kL from L creates a periodic array ofsubstitutional defects (also called impurities), where the original lattice points (initially withcharges +1) are replaced by points with ‘charges’ (or ‘weights’) 1 − a k (cid:54) = 0.In Figure 2, we have constructed three examples of two-dimensional lattices with periodic arraysof defects which certainly do not exist in the real world. In contrast, Figure 3 shows two importantexamples of crystal structures arising in nature: the Kagome lattice and the rock-salt structure.These examples are discussed further in Section 3.Figure 2: Mathematical examples of periodic array of defects performed on a patch of the square lattice Z (left and right) and the triangular lattice A (middle). The cross × represents the origin O of R . The pointsmarked by • are the original points of the lattice whereas the points marked by + and ◦ are substitutionaldefects of charge 1 − a k for some a k ∈ R ∗ \{ } and some k ∈ K : = { , , , } . The missing lattice points arethe vacancy defects. The patch on the right contains two shifted periodic arrays of defects. While the substitutional defects case has different interpretations and applications in terms ofoptimal multi-component (ionic) crystals (see e.g. Section 3.2), the vacancy case is also of interestwhen we look for accelerating the computational time for checking numerically the minimality ofa structure. Indeed, if the minimizer does not change once several periodic arrays of points areremoved from all lattices, then a computer will be faster to check this minimality. This is ofpractical relevance in particular in low dimensions since the computational time of such latticeenergies, which grows exponentially with the dimension, are extremely long in dimension d ≥
8– even with the presence of periodic arrays of vacancies – and shows how important are rigorousminimality results in these cases. 5igure 3:
Two examples of 2d lattices patches with a periodic array of defect arising in nature. The left-hand structure is the Kagome lattice obtained by removing from the triangular lattice A the sublattice2 A + (1 ,
0) + (1 / , √ / Z two times the sublattices 2 Z + (1 , Z + (0 ,
1) in such a way that particles of opposites signs ± • and ◦ correspond respectivelyto charges of signs 1 and − Furthermore, from a Physics point of view, it is well-known (see e.g. [45]) that point defectsplay an important role in crystal properties. As explained in [1]: ‘Crystals are like people, it isthe defects in them which tend to make them interesting’. For instance, they reduce the electricand thermal conductivity in metals and modify the colors of solids and their mechanical strength.We also notice that substitutional defects control the electronic conductivity in semi-conductors,whereas the vacancies control the diffusion and the ionic conductivity in a solid. In particular, thereis no perfect crystal in nature and it is then interesting and physically relevant to study optimalityresults for periodic systems with defects, in particular for models at positive temperature wherethe number of vacancies per unit volume increases exponentially with the temperature (see e.g.[45, Sec. 3.4.3]). Notice that the raise of temperature also creates another kind of defects calledself-interstitial – i.e. the presence of extra atoms out of lattice sites – but they are known to benegligible (at least if they are of the same type than the solid’s atoms) compared to the vacancieswhen disorder appears, excepted for Silicon.
Plan of the paper.
Our main results are presented in Section 2 whereas their proofs arepostponed to Section 4. Many applications of our results are discussed in Section 3, includingexplicit examples of minimality results for the Kagome lattice and other ionic structures.
We start by recalling the notion of universal optimality among lattices as defined by Cohn andKumar in [21].
Definition 2.1 ( Universal optimality among lattices ) . Let d ≥ . We say that L d is univer-sally optimal in L d (1) if L d is a minimizer of E f defined by (1.2) in L d (1) for any f ∈ F cmd . Remark 2.1 (Universally optimal lattices) . We recall again that the only known universally op-timal lattices in dimension d ≥ A (see [38]), E and the Leech lattice Λ (see [23]) indimensions d ∈ { , , } . It is also shown in [43, p. 117] that there is no such universally optimallattice in dimension d = 3. There are also clear indications (see [14, Sect. 6.1]) that the space offunctions for which the minimality at all the scales of L d holds is much larger than F cmd .Before stating our results, notice that all of them are stated in terms of global optimality butcould be rephrased for showing local optimality properties. This is important, in particular in6imensions d = 3 where only local minimality results are available for E f (see e.g. [6]) and can begeneralized for energies of type E κf , ensuring the local stability of certain crystal structures.We now show that the universal optimalities among lattices in dimension d ∈ { , , } provedin [23, 38] are not conserved in the non-shifted case once we only removed a single integer sublatticea positive number a k > a k < Theorem 2.2 ( Conservation of universal optimalities - Non-shifted case ) . Let f be definedby f ( r ) = e − παr , α > . For all d ∈ { , , } , all k ∈ N \{ } , all a k > and κ = { k, a k , ∅} , thereexists α d such that for all α ∈ (0 , α d ) , A , E and the Leech lattice Λ are not minimizers of E κf in L d (1) .Furthermore, for any d ∈ { , , } , for any K ⊂ N \{ } , any A K = { a k } k ∈ K ⊂ R − and κ = { K, A K , ∅} , A , E and the Leech lattice Λ are the unique minimizers of E κf in L d (1) for all α > . Remark 2.3 (Generalization to 4-designs) . The non-optimality result in Theorem 2.2 is obtainedby using the Taylor expansion of the theta function found by Coulangeon and Sch¨urmann in [25,Eq. (21)]. Therefore, the result is actually generalizable to any universal optimal lattice L d suchthat all its layers (or shells) are 4-designs, i.e. such that for all r > { ∂B r ∩ L d } (cid:54) = ∅ , B r being the ball centred at the origin and with radius r , and all polynomial P of degree up to 4 wehave 1 | ∂B r | (cid:90) ∂B r P ( x ) dx = 1 (cid:93) { ∂B r ∩ L d } (cid:88) x ∈ ∂B r ∩ L d P ( x ) . We now present a sufficient condition on P K such that the triangular lattice is universallyoptimal in L (1) for E κf . This result is based on our recent work [9] where we have proven themaximality of A in L (1) for the centred lattice theta functions, i.e. L (cid:55)→ θ L + c L ( α ), where c L isthe center of the unit cell Q L (see also Remark 2.12). Theorem 2.4 ( Conservation of universal optimality - 2d shifted case ) . Let d = 2 and κ = { K, A K , P K } be as in (1.6) where A K ⊂ R + , and be such that ∀ k ∈ K, ∀ i ∈ I k , p i,k k = c L modulo Q L , L = Z u ⊕ Z u , c L := u + u , (2.1) where Q L is the unit cell of L defined by (1.1) with a Minkowski basis { u , u } and its center c L .Then, for all f ∈ F cm , A is the unique minimizer of E κf in L (1) . Example 2.5.
Theorem 2.4 holds in a particularly simple case, when k = 2 and p i, = u + u ∈ L . Remark 2.6 (Conjecture in dimensions d ∈ { , } ) . Theorem 2.4 is based on the fact that A has been shown to be the unique maximizer of E cf defined in (1.5) in L d (1) for any f ∈ F cmd (seealso Remark 2.12). As discussed in [9], we believe that this result still holds in dimensions 8 and24 for E and the Leech lattice Λ , as well as our Theorem 2.4. Remark 2.7 (Phase transition for the minimizer in the Gaussian case - Numerical observation) . In the non-universally optimal case of Theorem 2.2 and the shifted case satisfying (2.1), numericalinvestigations suggest that the minimizer of E κf exhibits a phase transition as the density decreases. Non-shifted case.
Let us consider the example f ( r ) = e − παr given in Theorem 2.2 (i.e. f ( r ) is aGaussian function) and f κ ( r ) = e − παr − . e − παr (defined by (1.8)), κ := { , . , ∅} , correspondingto removing a = 0 . L ( k = 2) from the original lattice L . In dimension d = 2, we numerically observe an interesting phase transition of type ‘triangular-rhombic-square-rectangular’ for the minimizer of E κf in L (1) as α (which plays the role of the inverse density here)increases. Shifted case with a k < . Let us assume that K = { } , A K := { a < } , I = { } and p , = u + u
7n such a way that (2.1) is satisfied. If we consider f ( r ) = e − παr , then for all the negative parameters a we have chosen, the minimizer of E κf [ L ] := θ L ( α ) + | a | θ L + c L ( α ) in L (1) numerically shows thesame phase transition of type ‘triangular-rhombic-square-rectangular’ as α increases.This type of phase transition seems to have a certain universality in dimension 2 since it was alsoobserved for Lennard-Jones energy [5], Morse energy [7], Madelung-like energies [10] and provedfor 3-blocks copolymers [35] and two-component Bose-Einstein condensates [36] by Wei et al.. Remark 2.8 (Optimality of Z d in the orthorhombic case) . Another type of universal optimalityis known in the set of orthorhombic lattices, i.e. the lattice L which can be represented by anorthogonal basis. As proved by Montgomery in [38, Thm. 2], the cubic lattice Z d is universallyminimal among orthorhombic lattices of unit density in any dimension (see also [10, Rmk. 3.1]).The proof of Theorem 2.2 can be easily adapted to show the same optimality result for Z d amongorthorhombic lattices of unit density. Furthermore, it has also been shown (see e.g. [13, Prop. 1.4])that Z d is the unique maximum of L (cid:55)→ E f [ L + c L ] among orthorhombic lattices of fixed density forany f ∈ F cmd . Therefore, the proof of Theorem 2.4 can be also easily adapted in this orthorhombiccase in order to show the universal optimality of Z d in this particular shifted case. Moreover, allthe next results involving any universally optimal lattice can be rephrased for Z d in the space oforthorhombic lattices. Examples of applications of such result will be discussed in Section 3.2.We now give a general criterion that ensures the conservation of an universal optimizer’s mini-mality for E κf . Theorem 2.9 ( General criterion for minimality conservation - Non-shifted case ) . Let d ≥ , κ = { K, A K , ∅} be as in (1.6) (possibly empty) where A K ⊂ R + , and L d be universallyoptimal in L d (1) . Furthermore, let f ∈ F d be such that dµ f ( t ) = ρ f ( t ) dt and f κ be defined by (1.8) .Then:1. For any κ , we have f κ ( r ) = (cid:90) ∞ e − rt dµ f κ ( t ) where dµ f κ ( t ) = ρ f κ ( t ) dt, ρ f κ ( t ) = ρ f ( t ) − (cid:88) k ∈ K a k k ρ f (cid:18) tk (cid:19) .
2. The following equivalence holds: f κ ∈ F cmd if and only if ∀ t > , ρ f ( t ) ≥ (cid:88) k ∈ K a k k ρ f (cid:18) tk (cid:19) ; (2.2)
3. If (2.2) holds, then L d is the unique minimizer of E κf in L d (1) .4. If there exists V > such that for a.e. y ≥ there holds g V ( y ) := ρ f κ (cid:18) πyV d (cid:19) + y d − ρ f κ (cid:32) πV d y (cid:33) ≥ , (2.3) then V d L d is the unique minimizer of E κf in L d ( V ) . The fourth point on Theorem 2.9 generalizes our two-dimensional result [4, Thm. 1.1] to anydimension and with possible periodic arrays of defects. It is an important result since only fewminimality results for E f are available for non-completely monotone potentials f ∈ F d \F cmd , andthis also the first result of this kind for charged lattices (i.e. when the particles are not of the samekind). Condition (2.3) has been used in dimension d = 2 in [4, 7] for proving the optimality of atriangular lattice at fixed density for non-convex sums of inverse power laws, differences of Yukawa8otentials, Lennard-Jones potentials and Morse potentials and we expect the same property tohold in higher dimension. In Theorem 2.17, we will give an example of such application in anydimension d by applying the fourth point of Theorem 2.9 to Lennard-Jones type potentials. Wenow add a very important remark concerning the adaptation of the fourth point of Theorem 2.9 inthe general periodic case, i.e. for crystallographic point packings (see [2, Def. 2.5]). Remark 2.10 ( Crystallization at fixed density as a consequence of Cohn-Kumar Con-jecture ) . When κ = ∅ , i.e. all the particles are present and of the same kind, the proof of point 4.of Theorem 2.9 admits a straightforward adaptation in the periodic case, i.e among all configura-tions C = (cid:83) Ni =1 (Λ + v k ) ∈ S being Λ-periodic of unit density, where Λ ∈ L d , i.e. such that | Λ | = N ,and with a f -energy defined for V > E f [ V d C ] := 1 N N (cid:88) j,k =1 (cid:88) x ∈ Λ \{ v k − v j } f (cid:16) V d | x + v k − v j | (cid:17) . Using again the representation of f as a superposition of Gaussians combined with the Jacobitransformation formula (see the proof of Theorem 2.9), the same condition (2.3) ensures the crys-tallization on L d at fixed density once we know its universal optimality in the set of all periodicconfigurations with fixed density V − . This result is in the same spirit as the one derived by Petra-che and Serfaty in [39] for Coulomb and Riesz interactions. In dimensions d ∈ { , } , (2.3) impliesthe crystallization on E and Λ at fixed density V − as a consequence of [23] whereas in dimension d = 2 it is conjectured by Cohn and Kumar in [21] that the same holds on the triangular lattice.It is in particular true for the Lennard-Jones potential at high density as a simple application ofour Theorem 2.17.Using exactly the same arguments as the fourth point of Theorem 2.9, we show the followingresult which gives a sufficient condition on an interaction potential f for a universal maximizer L ± d of θ ± L ( α ) to be optimal for E ± f , where θ ± L ( α ) := (cid:88) p ∈ L ϕ ± ( p ) e − πα | p | , and E ± f [ L ] := (cid:88) p ∈ L \{ } ϕ ± ( p ) f ( | p | ) , (2.4)with L = (cid:76) di =1 Z u i , { u , ..., u d } being its Minkowski basis, and ϕ ± ( p ) = (cid:80) di =1 m i for p = (cid:80) di =1 m i u i , m i ∈ Z for all i . Remark that E ± f = E κf when κ = { , { , .... } , { u , ..., u d }} , L = (cid:76) di =1 Z u i . Inparticular, it holds for the triangular lattice A as a simple application of our main result in [9]. Theorem 2.11 ( Maximality of a universal maximizer for θ ± L - Shifted case ) . Let d ≥ , V > , κ = { , { , .... } , { u , ..., u d }} , where a generic lattice is written L = (cid:76) di =1 Z u i , { u , ..., u d } being its Minkowski basis, and L ± d be the unique maximizer of θ ± L ( α ) , defined by (2.4) , in L d (1) andfor all α > . If f ∈ F d satisfies (2.3) , then V d L ± d is the unique maximizer of E κf (equivalently of E ± f defined by (2.4) ) in L d ( V ) . Remark 2.12 (Adaptation to shifted f -energy) . We believe that Theorem 2.11 also holds for E and Λ (see [9, Conj. 1.3] and Remark 2.6). Furthermore, the same kind of optimality result couldbe easily derived for any energy shifted energy of type L (cid:55)→ E f [ L + c ] where c ∈ Q L is fixed as afunction of the vectors in the Minkowski basis { u i } and when one knows a universal minimizer ormaximizer for L (cid:55)→ E f [ L + c ], f ∈ F cmd . However, no other result concerning any optimality of alattice for such kind of energy is currently available when c (cid:54)∈ { L, c L } .The rest of our results are all given in the non-shifted case P K = ∅ . It is indeed a rather difficulttask to minimize the sum of shifted and/or non-shifted energies of type E f . Very few results areavailable and the recent work by Luo and Wei [36] has shown the extreme difficulty to obtain any9eneral result for completely monotone function f . Shifting the lattices by a non-lattice point whichis not the center c L appears to be deeply more tricky in terms of energy optimization.We remark that, since F cmd is not stable by difference, it is not totally surprising that Theorem2.2 holds. Furthermore, identifying the largest space of all functions f such that E f is uniquelyminimized by L d in L d (1) seems to be very challenging (see [14]). Therefore a natural question inorder to identify a large class of potentials f such that the minimality of an universal optimizer L d holds for E κf is the following: what are the completely monotone potentials f ∈ F cmd satisfying(2.2), i.e. such that f κ ∈ F cmd ? The following corollary of Theorem 2.9 gives an example of suchpotentials, where we define, for s > A K = { a k } k ∈ K , K ⊂ N \{ } , L ( A K , s ) := (cid:88) k ∈ K a k k s . (2.5)Notice that the notation of (2.5) is inspired by the one of Dirichlet L-series that are generalizingthe Riemann zeta function (see e.g. [20, Chap. 10]). For us, the arithmetic function appearing ina Dirichlet series is simply replaced by A K and can be finite. Corollary 2.13 ( Minimality conservation for special f - Non-shifted case ) . Let d ≥ and f ∈ F cmd be such that dµ f ( t ) = ρ f ( t ) dt and ρ f be an increasing function on R + . Let κ = { K, A K , ∅} be as in (1.6) where A K = { a k } k ∈ K ⊂ R + and be such that L ( A K , s ) defined by (2.5) satisfies L ( A K , ≤ . If L d is universally optimal in L d (1) , then L d is the unique minimizer of E κf in L d (1) . Example 2.14 (Potentials satisfying the assumptions of Corollary 2.13) . There are many examplesof potentials f such that Corollary 2.13 holds. For instance, this is the case for the parametrizedpotential f = f σ,s defined for all r > f σ,s ( r ) = e − σr r s , σ > s >
1, since dµ f σ,s ( t ) = ( t − σ ) s − Γ( s ) [ σ, ∞ ) ( t ) dt and t (cid:55)→ ( t − σ ) s − Γ( s ) [ σ, ∞ ) ( t ) are increasing functions on R + . Notice that theinverse power law f ( r ) = r − s with exponent s > d/ ≥ σ = 0) and the Yukawa potential f ( r ) = e − σr r − with parameter σ > s = 1) are special cases of f σ,s . In this subsection, we restrict our study to combinations of inverse power laws, since they are thebuilding blocks of many interaction potentials used in molecular simulations (see e.g. [33]). Theirhomogeneity simplifies a lot the energy computations and allows us to give a complete picture ofthe periodic arrays of defects effects with respect to the values of L defined by (2.5).In the following result, we show that the values of L ( A K , s ) plays a fundamental role in theminimization of E κf when f is an inverse power law. Theorem 2.15 ( The inverse power law case - Non-shifted case ) . Let d ≥ and f ( r ) = r − s where s > d/ . Let κ = { K, A K , ∅} be as in (1.6) and be such that L ( A K , s ) defined by (2.5) isabsolutely convergent. We have:1. If L ( A K , s ) < , then L is a minimizer of L (cid:55)→ ζ L (2 s ) in L d (1) if and only if L is aminimizer of E κf in L d (1) .2. If L ( A K , s ) > , then L is a minimizer of L (cid:55)→ ζ L (2 s ) in L d (1) if and only if L is amaximizer of E κf in L d (1) .In particular, for any K ⊂ N \{ } , if a k = 1 for all k ∈ K , then L (cid:55)→ ζ L (2 s ) and E κf have the sameminimizers in L d (1) . xamples 2.16 (Minimizers of the Epstein zeta function) . In dimensions d ∈ { , , } , theminimizer L of L (cid:55)→ ζ L (2 s ) in L d (1) is, respectively, A , E and Λ as consequences of [23, 38]. Indimension d = 3, Sarnak and Str¨ombergsson have conjectured in [43, Eq. (44)] that the face-centredcubic lattice D (see Fig. 1) is the unique minimizer of L (cid:55)→ ζ L (2 s ) in L (1) if s > / f ( r ) = c r x − c r x where ( c , c ) ∈ (0 , ∞ ) , x > x > d/ , (2.6)which is a prototypical example of function where µ f is not nonnegative everywhere, and a differenceof completely monotone functions. We discuss the optimality of a universally optimal lattice L d for E κf with respect to the values of L ( A K , x i ), i ∈ { , } as well as the shape of the global minimizerof E κf , i.e. its equivalence class in L d modulo rotation and dilation (as previously defined in [14]). Theorem 2.17 ( The Lennard-Jones case - Non-shifted case ) . Let d ≥ , f be defined by (2.6) and κ = { K, A K , ∅} be as in (1.6) (possibly empty) and be such that L ( A K , x i ) , i ∈ { , } defined by (2.5) are absolutely convergent. Let L d be universally optimal in L d (1) . Then:1. If L ( A K , x ) < L ( A K , x ) < , then for all V > such that V ≤ V κ := π d (cid:18) c (1 − L ( A K , x ))Γ( x ) c (1 − L ( A K , x ))Γ( x ) (cid:19) d x − x , the lattice V d L d is the unique minimizer of E κf in L d ( V ) and there exists V > such thatit is not a minimizer of E κf for V > V . Furthermore, the shape of the minimizer of E f and E κf are the same in L d .2. If L ( A K , x ) > L ( A K , x ) > , then E κf does not have any minimizer in L d and for all V < V κ , V d L d is the unique maximizer of E κf in L d ( V ) .3. If L ( A K , x ) > > L ( A K , x ) , then E κf does not have any minimizer in L d but V d L d is theunique minimizer of E κf in L d ( V ) for all V > . Remark 2.18 (Increasing of the threshold value V κ ) . The fact that 1 − L ( A K , x ) > − L ( A K , x )implies that the threshold value V κ is larger in the κ (cid:54) = ∅ case than in the case without defect κ = ∅ .The same is expected to be true for any non-convex sum of inverse power law with a positive mainterm as r → κ = ∅ ). It isalso totally straightforward to show that V κ → V ∅ as min K tend to + ∞ . Remark 2.19 (Global minimality of A among lattices for Lennard-Jones type potentials) . Indimension d = 2, the triangular lattice L = A has been shown in [4, Thm. 1.2.2] to be the shapeof the global minimizer of E f in L when π − x Γ( x ) x < π − x Γ( x ) x . Point 1. of Theorem 2.17implies that the same holds when L ( A K , x ) < L ( A K , x ) < .3 Conclusion From all our results, we conclude that is possible to remove or substitute several infinite periodicsets of points from all the lattices (i.e. an integer sublattices) and to conserve the already existingminimality properties, but only in a certain class of potentials or sublattices. Physically, it meansthat adding point defects to a crystal can be without any effect on its ground state if we assume theinteraction between atoms to be well-approximate by a pairwise potential (Born model [46]) andthe sublattices to satisfy some simple properties. We give several examples in Section 3 and ourresult are the first known general results giving global optimality of ionic crystals. In particular,the Kagome lattice (see Figure 3) is shown to be the global minimizer for the interaction energiesdiscussed in this paper in the class of (potentially shifted) lattices L \ L where L ∈ L (1). This is,as far as we know, the first results of this kind for the Kagome lattice. We also believe that theresults and techniques derived in this paper can be applied to other ionic crystals and other generalperiodic systems.Furthermore, this paper also shows the possibility to check the optimality of a structure while’forgetting’ many points which, in a certain sense, do not play any role (vacancy case). This allow tosimplify both numerical investigations – leading to a shorter computational time – and mathematicalestimates for these energies. We voluntarily did not explore further this fact since it is only relevantin low dimensions because the computational time of such lattice sums is exponentially growingand gives unreachable durations in dimension d ≥ d ∈ { , } where our global optimality results are applicable.In dimension d = 3, i.e. where the everyday life real crystals exist, our results only apply –combined with the one from [6] – to the conservation of local minimality in the cubic lattices cases( Z , D and D ∗ ) for the Epstein zeta function, the lattice theta function and the Lennard-Jonestype energies. We believe that our result will find other very interesting applications in dimension3 once global optimality properties will be shown for the lattice theta functions and the Epsteinzeta functions (Sarnak-Str¨ombergsson conjectures [43]).Even though the inverse power laws and Lennard-Jones cases have been completely solved here,we still ignore what is the optimal result that holds for ensuring the robustness of the universaloptimality among lattices. An interesting problem would be to find a necessary condition for thisrobustness. Furthermore, we can also ask the following question: is it enough to study this kind ofminimization problem in a (small) ball centred at the origin? In other words: can we remove all thepoints that are far enough from O and conserving the minimality results? Numerical investigationsand Figure 5 tend to confirm this fact, and a rigorous proof of such property would deeply simplifythe analysis of such lattice energies. We now give several examples of applications of our results. In particular, we identify interestingstructures that are minimizers of E f in classes of sparse and charged lattices. Being the vertices of a trihexagonal tiling, this structure – wich is actually not a lattice as wedefined it in this paper – that we will write K := A \ A is the difference of two triangular latticesof scale ratio 2 (see Fig. 4). Some minerals – which display novel physical properties connectedwith geometrically frustrated magnetism – like jarosites and herbertsmithite contain layers havingthis structure (see [37] and references therein). We can therefore apply our results of Section 2 with κ = { , , ∅} or κ = { , , u + u } . The following optimality results for E f in the class of lattices12f the form L \ L (or L \ (2 L + u + u ) in the shifted case) are simple consequences of our resultscombined with the universal optimality of A among lattices proved by Montgomery in [38]:1. Universal optimality of K . Applying Theorem 2.4 to κ = { , , u + u } , it follows that for all f ∈ F cm , the shifted Kagome lattice K + (1 / , −√ /
2) (see Fig. 4) is the unique minimizerof E f among lattices of the form L \ (2 L + u + u ), where L = Z u ⊕ Z u ∈ L (1).2. Minimality of K at all densities for certain completely monotone potentials. A direct conse-quence of Theorem 2.9 is the following. For any completely monotone function f ∈ F cm suchthat dµ f ( t ) = ρ f ( t ) dt and ρ f is an increasing function, the Kagome lattice K is the uniqueminimizer of E f among all the two-dimensional sparse lattices L \ L where L ∈ L (1). Thisis the case for instance for f = f σ,s defined in Example 2.14, including the inverse power lawsand the Yukawa potential.3. Optimality at high density for Lennard-Jones interactions.
Applying Theorem 2.17, we obtainits optimality at high density: if f ( r ) = c r − x − c r − x , x > x > E f at high density among all the two-dimensionalsparse lattices L \ L , where L has fixed density, has the shape of K .4. Global optimality for Lennard-Jones interactions with small exponents.
Furthermore, usingTheorem 2.17 and [4, Thm. 1.2.2] (see also Remark 2.19), we obtain the following interestingresult in the Lennard-Jones potential case: if π − x Γ( x ) x < π − x Γ( x ) x , then the uniqueglobal minimizer of E f among all the possible sparse lattices L \ L has the shape of K .Figure 4: Two patches of the Kagome lattice. On the left, the origin O does not belong to K and is thecenter of one of the hexagons. On the right, O belongs to a shifted version K + (1 / , −√ / These are the first minimality results for K in a class of periodic configurations. We recall thata non-optimality result has also been derived by Grivopoulos in [31] for Lennard-Jones potentialin the case of free particles, and different attempts have been made for obtaining numerically orexperimentally a Kagome structure as an energy ground state (see e.g. [26, 32, 41]). Remark 3.1 (The honeycomb lattice) . We notice that the honeycomb lattice H := A \√ A , alsoconstructed from the triangular lattice, does not belong to the set of sparse lattices L \ kL , k ∈ N .That is why no optimality result for H is included in this paper. We recall that, in [9], we have shown with Faulhuber the universal optimality of the triangularlattice among lattices with alternating charges, i.e. the fact that A uniquely maximizes L (cid:55)→ θ ± L ( α ) := (cid:88) p ∈ L ϕ ± ( p ) e − πα | p | and ζ ± L ( s ) := (cid:88) p ∈ L \{ } ϕ ± ( p ) | p | s , L = Z u ⊕ Z u , (3.1)13n L (1), where, for all p = mu + nu , ϕ ± ( p ) := m + n . Notice that the maximality result at allscales for the alternating lattice theta function is equivalent with the fact that A maximizes L (cid:55)→ E κf [ L ] := E f [ L ] − E f [2 L + u ] − E f [2 L + u ] , where κ := { , { , } , { u , u }} in L (1) for any f ∈ F cm . It has been also proven in [13, Thm. 1.4] that Z d is the unique maximizerof the d -dimensional generalization of the two lattice energies θ ± L ( α ) and ζ ± L ( s ) among d -dimensionalorthorhombic (rectangular) lattices of fixed unit density, whereas it is a minimizer of the latticetheta functions and the Epstein zeta functions defined in (1.3). Furthermore, applying Theorem2.9 in dimension d = 2 (resp. any d ), we see that A (resp. Z d ) minimizes in L (1) (resp. amongthe orthorhombic lattices of unit density) the energy E κf [ L ] := ζ L ( s ) − ζ kL ( s ) , f ( r ) = r − s , K = { k } , a k = 2 , (3.2)for all s > d/
2. We remark that Z d , d ∈ { , } is also a saddle point (see [6, 38]) of E κf in L d (1).It is then interesting to see how the array of substitutional defects with charges − Three periodic arrays of defects on Z . Blue points • are points with charges +1 and red points ◦ are with charges −
1. For the inverse power laws energies, the left-hand configuration is the unique maximizeramong rectangular lattices of fixed density with alternation of charges whereas the centred configuration isits unique minimizer with this distribution of charges among rectangular lattices. However, the configurationon the right is a saddle point of any energy on the form E f , f ∈ F cm in this class of charged configurations.For the two structures on the left, the same is true in higher dimension while generalizing the ionic-likedistribution on orthorhombic lattices. We first show Theorem 2.2, i.e. the non-robustness of universal optimality results under non-shiftedperiodic arrays of defects.
Proof of Theorem 2.2 . Let Λ ∈ { A , E , Λ } . We consider the potential f ( r ) := e − παr where α >
0. For all k ∈ N \{ } , all a k > L ∈ L d (1), we have, using the fact that θ kL ( α ) = θ L ( k α ), E κf [ L ] = θ L ( α ) − a k θ ( k α ) . Let us show that there exists α d such that for all 0 < α < α d , Λ does not minimize E κf in L d (1).Indeed, we have the following equivalence: for all L ∈ L d (1) \{ Λ } , E κf [ L ] > E κf [Λ] if and only ifinf L ∈L d (1) L (cid:54) =Λ θ L ( α ) − θ Λ ( α ) θ L ( k α ) − θ Λ ( k α ) > a k . (4.1)14igure 6: Three periodic arrays of defects on a patch of A . Blue points • are points with charges +1 andred points ◦ are with charges −
1. On the left, the triangular alternate configuration maximizes ζ ± L ( s ) in L (1) with this alternation of charges, while the configuration in the middle minimizes the inverse power lawenergy in this class of charged lattices. The configuration on the right minimizes any energy on the form E f , f ∈ F cm in this class of charged configurations. Let us show that (4.1) does not hold for small α , and in particular that the left term tends to 0 as α →
0. We use Coulangeon and Sch¨urmann’s work [25, Eq. (21)], in the lattice case, who derivedthe Taylor expansion of the theta function as L (cid:55)→ Λ in L d (1). We then obtainlim L (cid:55)→ Λ L (cid:54) =Λ θ L ( α ) − θ Λ ( α ) θ L ( k α ) − θ Λ ( k α ) = (cid:88) p ∈ Λ \{ } πα | p | (cid:0) πα | p | − (cid:1) e − πα | p | (cid:88) p ∈ Λ \{ } παk | p | (cid:0) παk | p | − (cid:1) e − παk | p | = k − (cid:88) p ∈ Λ \{ } πα | p | e − πα | p | − (cid:88) p ∈ Λ \{ } | p | e − πα | p | (cid:88) p ∈ Λ \{ } παk | p | e − παk | p | − (cid:88) p ∈ Λ \{ } | p | e − παk | p | . By absolute convergence, the first term of both numerator and denominator are vanishing as α → α → lim L (cid:55)→ Λ L (cid:54) =Λ θ L ( α ) − θ Λ ( α ) θ L ( k α ) − θ Λ ( k α ) = lim α → k − (cid:88) p ∈ Λ \{ } | p | e − πα | p | (cid:88) p ∈ Λ \{ } | p | e − παk | p | = 0 , by comparing the convergence rate of these two exponential sums that are going to + ∞ as α → α < α d where α d depends on d , k and a k , and the proof ofthe first part of the theorem is completed.The second part of the theorem is a simple consequence of the fact that f κ defined by (1.8)belongs to F cmd if f ∈ F cmd and a k < k ∈ K .The proof of our second result, namely Theorem 2.4, is a direct and simple consequence of ourwork [9]. Proof of Theorem 2.4 . If p i,k /k = c L modulo Q L for all k ∈ K and all i ∈ I k , we obtain E κf [ L ] = E f [ L ] − (cid:88) k ∈ K a k (cid:88) i ∈ I k (cid:88) p ∈ L \{ } f (cid:18) k (cid:12)(cid:12)(cid:12) p i,k k + p (cid:12)(cid:12)(cid:12) (cid:19) = E f [ L ] − (cid:88) k ∈ K a k (cid:93)L k E f ( k · ) [ L + c L ] .
15s proved in [9], for any f ∈ F cm , A is the unique maximizer of L (cid:55)→ E f [ L + c L ] in L (1). Itfollows that A , which uniquely minimizes E f in L (1) is the unique minimizer of E κf in L (1) since a k > k ∈ K .We now show Theorem 2.9 which gives a simple criterion for the conservation of the minimalityof a universal optimizer. Proof of Theorem 2.9 . In order to show the three first points, it is sufficient to show the firstpoint of our theorem, i.e. the fact that dµ f κ ( t ) = (cid:0) ρ f ( t ) − (cid:80) k ∈ K a k k − ρ f (cid:0) tk (cid:1)(cid:1) dt . We remark that ρ f is the inverse Laplace transform of f , i.e. ρ f ( t ) = L − [ f ]( t ). By linearity, it follows that dµ f κ ( t ) = ρ f κ ( t ) dt, where ρ f κ ( t ) = ρ f ( t ) − (cid:88) k ∈ K a k L − [ f ( k · )]( t ) . By the basic properties of the inverse Laplace transform, we obtain that, for all t > L − [ f ( k · )]( t ) = k − L − [ f ]( k − t ) = k − ρ f ( k − t ) , and our result follows by the universal optimality of L d in L d (1) and the definition of completelymonotone function.To show the last point of our theorem, we adapt [4, Thm 1.1]. Let L ∈ L d (1) and V >
0, thenwe have E κf [ V d L ] = (cid:88) p ∈ L \{ } f κ (cid:16) V d | p | (cid:17) = (cid:90) ∞ (cid:34) θ L (cid:32) V d tπ (cid:33) − (cid:35) ρ f κ ( t ) dt = πV d (cid:90) ∞ [ θ L ( y ) − ρ f κ (cid:18) πyV d (cid:19) dy = πV d (cid:90) [ θ L ( y ) − ρ f κ (cid:18) πyV d (cid:19) dy + πV d (cid:90) ∞ [ θ L ( y ) − ρ f κ (cid:18) πyV d (cid:19) dy = πV d (cid:90) ∞ (cid:20) θ L (cid:18) y (cid:19) − (cid:21) ρ f κ (cid:32) πyV d (cid:33) y − dy + πV d (cid:90) ∞ [ θ L ( y ) − ρ f κ (cid:18) πyV d (cid:19) dy. (4.2)A simple consequence of the Poisson summation formula is the well-known identity (see e.g. [24,Eq. (43)]) ∀ y > , θ L (cid:18) y (cid:19) = y d θ L ∗ ( y ) . (4.3)From (4.3), we see that if L d is the unique minimizer of L (cid:55)→ θ L ( α ) for all α > , L ∈ L d (1) then L ∗ d = L d . From (4.2) and (4.3), for all V > , L ∈ L d (1), we have E κf [ V d L ] = πV d (cid:90) ∞ (cid:104) y d θ L ∗ ( y ) − (cid:105) ρ f κ (cid:32) πyV d (cid:33) y − dy + πV d (cid:90) ∞ [ θ L ( y ) − ρ f κ (cid:18) πyV d (cid:19) dy. (4.4)and E κf [ V d L ] − E κf [ V d L d ] = πV d (cid:90) ∞ [ θ L ∗ ( y ) − θ L d ( y )] ρ f κ (cid:32) πyV d (cid:33) y d − dy + πV d (cid:90) ∞ [ θ L ( y ) − θ L d ( y )] ρ f κ (cid:18) πyV d (cid:19) dy. (4.5)16y (4.5) and the definition of g V , if V is such that g V ( y ) ≥ y ≥ E κf [ V d L ] − E κf [ V d L d ] + E κf [ V d L ∗ ] − E κf [ V d L d ]= πV d (cid:90) ∞ [ θ L ∗ ( y ) − θ L d ( y )] g V ( y ) dy + πV d (cid:90) ∞ [ θ L ( y ) − θ L d ( y )] g V ( y ) dy ≥ πV d (cid:90) ∞ m L ( y ) g V ( y ) dy, (4.6)where m L ( y ) := min { θ L ∗ ( y ) − θ L d ( y ) , θ L ( y ) − θ L d ( y ) } . Since m L ( y ) ≥ L ∈ L d (1) , y > L = L d , and g V ( y ) ≥ y ∈ [1 , ∞ ), we get from (4.6) that E κf [ V d L ] + E κf [ V d L ∗ ] ≥ E κf [ V d L d ] , with equality if and only if L = L d . It follows that L d is the unique minimizer of L (cid:55)→ E κf [ V d L ] on L d (1), or equivalently that V d L d isthe unique minimizer of E κf in L d ( V ), and the result is proved.The previous proof contains the main ingredients for showing Theorem 2.11. Proof of Theorem 2.11 . Following exactly the same sequence of arguments as in the proof ofthe fourth point of Theorem 2.9, we obtain the maximality result of V d L ± d at fixed density for E ± f .Indeed, (4.3) is replaced by θ ± L ( α ) = y d θ L ∗ + c L ∗ ( α ) , and, by using the maximality of L ± d for L (cid:55)→ θ ± L ( α ) and L (cid:55)→ θ L + c L ( α ) for all α >
0, we obtain E ± f [ V d L ] − E ± f [ V d L d ] + E ± f [ V d L ∗ ] − E ± f [ V d L d ]= πV d (cid:90) ∞ (cid:20) θ L ∗ + c L ∗ ( y ) − θ L ± d + c L ± d ( y ) (cid:21) g V ( y ) dy + πV d (cid:90) ∞ (cid:20) θ ± L ( y ) − θ ± L ± d ( y ) (cid:21) g V ( y ) dy ≤ πV d (cid:90) ∞ m ± L ( y ) g V ( y ) dy, (4.7)where m ± L ( y ) := max { θ L ∗ + c L ∗ ( y ) − θ L ± d + c L ± d ( y ) , θ ± L ( y ) − θ ± L ± d ( y ) } . We again remark that m ± L ( y ) ≤ L ∈ L d (1), y > L = L ± d .Therefore, the positivity of g V as well as the universal maximality of L ± d implies in the same waythat V d L ± d is the unique maximizer of E ± f in L d ( V ).The proof of Corollary 2.13 is a straightforward consequence of Theorem 2.9. Proof of Corollary 2.13 . Let A K := { a k } k ∈ K ⊂ R + be such that L ( A K , ≤
1. Since µ f ≥ ρ f is positive, and furthermore ρ f is increasing by assumption. Therefore, we have,for all t > (cid:88) k ∈ K a k k ρ f (cid:18) tα (cid:19) ≤ (cid:88) k ∈ K a k k ρ f ( t ) = L ( A K , ρ f ( t ) ≤ ρ f ( t ) , where the first inequality is obtained from the monotonicity of ρ f and the last one from its positivityand the fact that L ( A K , ≤
1. The proof is completed by applying Theorem 2.9.We now show Theorem 2.15 which is a simple consequence of the homogeneity of the Epsteinzeta function and a property of the Riemann zeta function.17 roof of Theorem 2.15 . Using the homogeneity of the Epstein zeta function, we obtain E κf [ L ] = (cid:88) p ∈ L \{ } | p | s − (cid:88) k ∈ K (cid:88) p ∈ L \{ } a k k s | p | s = (1 − L ( A K , s )) ζ L (2 s ) , the exchange of sums being ensured by their absolute summability. If L ( A K , s ) <
1, then L (cid:55)→ ζ L (2 s ) and E κf have exactly the same minimizer. If L ( A K , s ) >
1, then the optimality are reversedand the proof is complete.Furthermore, if a k = 1 for all k ∈ K , then we have L ( A K , s ) = (cid:88) k ∈ K k s ≤ ζ (2 s ) − , where ζ ( s ) := (cid:80) n ∈ N n − s is the Riemann zeta function. Since ζ ( x ) < , ∞ ) if and only if x > x ≈ .
73, it follows that ζ (2 s ) − < s > x / ≈ .
865 which is true for all s > d/ d ≥
2. We thus have L ( A K , s ) < Proposition 4.1 ( Optimality at high density for Lennard-Jones type potentials ) . Let f ( r ) = b r x − b r x where b , b ∈ (0 , ∞ ) and x > x > d/ , and let L d be universally optimal in L d (1) . If V ≤ π d (cid:18) b Γ( x ) b Γ( x ) (cid:19) d x − x , then V d L d is the unique minimizer of E f in L d ( V ) . Proof of Proposition 4.1 . We follow the lines of [4, Prop. 6.10] and we apply point 4. ofTheorem 2.9. For i ∈ { , } , let β i := b i π xi − Γ( x i ) and α := V d , then g V ( y ) = y d − x − α x − ˜ g V ( y ) where g V is given by (2.3) and˜ g V ( y ) := β α x − x y x − d − β y x + x − d − β y x − x + β α x − x . We therefore compute ˜ g (cid:48) V ( y ) = y x − x − u V ( y ) where u V ( y ) := β (cid:18) x − d (cid:19) y x + x − d α x − x − β (cid:18) x + x − d (cid:19) y x − d − β ( x − x ) . Differentiating again, we obtain u (cid:48) V ( y ) = (cid:18) x + x − d (cid:19) y x − d − (cid:18) β (cid:18) x − d (cid:19) y x − x α x − x − β (cid:18) x − d (cid:19)(cid:19) , and we have that u (cid:48) V ( y ) ≥ y ≥ (cid:18) β (2 x − d ) β (2 x − d ) (cid:19) x − x α . By assumption, we know that α ≤ π (cid:18) a Γ( x ) a Γ( x ) (cid:19) x − x = (cid:18) β β (cid:19) x − x < (cid:32) β (2 x − d ) β (2 x − d ) (cid:33) x − x , u (cid:48) V ( y ) ≥ y ≥
1. We now remark that u V (1) = (cid:18) x − d (cid:19) (cid:18) β α x − x − β (cid:19) ≥ , by assumption, since p > d/ > d/ α ≤ π (cid:18) b Γ( x ) b Γ( x ) (cid:19) x − x ⇐⇒ β α x − x − β ≥ . (4.8)It follows that g (cid:48) V ( y ) ≥ y ≥
1. Since g V (1) = 2 (cid:18) β α x − − β α x − (cid:19) ≥ g V ( y ) ≥ y ≥ Proof of Theorem 2.17 . Let A K = { a k } k ∈ K for some K ⊂ N \{ } and f ( r ) = c r − x − c r − x ,then we have, using the homogeneity of the Epstein zeta function, E κf [ L ] = c ζ L (2 x ) − c ζ L (2 x ) − (cid:88) k ∈ K a k ( c ζ kL (2 x ) − c ζ kL (2 x ))= c (1 − L ( A K , x )) ζ L (2 x ) − c (1 − L ( A K , x ))) ζ L (2 x ) . We now assume that L ( A K , x ) < L ( A K , x ) <
1. Therefore, the first part of point 1. is a simpleconsequence of Prop.4.1 applied for the coefficients b i = c i (1 − L ( A K , x i )) > i ∈ { , } .The fact that E κf is not minimized by L d for V large enough is a direct application of [14, Thm.1.5(1)] since µ f is negative on (0 , r ) for some r depending on the parameters c , c , x , x , A K .Furthermore, the fact that the shape of the minimizers are the same follows from [14, Thm. 1.11]where it is shown that the minimizer of the Lennard-Jones type lattice energies does not dependon the coefficients b , b but only on the exponents x , x , which are the same for f and f κ .If L ( A K , x ) > L ( A K , x ) >
1, then f κ ( r ) = − b r − x + b r − x where b i := c i ( L ( A K , x i ) − > i ∈ { , } . If follows that f κ ( r ) tends to −∞ as r →
0, which implies the same for E κf [ L ] as L has its lengths going to 0 and + ∞ , i.e. when L collapses. This means that E κf does not have aminimizer in L d ( V ) and in L d . Furthermore, combining point 1. with the fact that the signs ofthe coefficients are switched, we obtain the maximality of V /d L d at high density (i.e. low volume V < V κ ).If L ( A K , x ) > > L ( A K , x ), then f κ ( r ) = b r − x + b r − x where b := c ( L ( A K , x ) − > b := c (1 − L ( A K , x )) >
0. Therefore f κ ∈ F cmd , which implies the optimality of V /d L d in L d ( V ) for all fixed V > E κf [ L ] tends to 0 as all the points are sent to infinity,i.e. E κf does not have a minimizer in L d . Acknowledgement:
I am grateful for the support of the WWTF research project ”VariationalModeling of Carbon Nanostructures” (no. MA14-009) and the (partial) financial support from theAustrian Science Fund (FWF) project F65. I also thank Mircea Petrache for our discussions aboutthe crystallization at fixed density as a consequence of Cohn-Kumar conjecture stated in Remark2.10. 19
Bibliography [1] G. Allaire.
Shape optimization by the homogenization method , volume 146 of
Applied MathematicalSciences . Springer New York, 2002.[2] M. Baake and U. Grimm.
Aperiodic Order, Volume 1. A Mathematical invitation . Cambridge UniversityPress, 2013.[3] S. Bernstein. Sur les fonctions absolument monotones.
Acta Math. , 52:1–66, 1929.[4] L. B´etermin. Two-dimensional Theta Functions and Crystallization among Bravais Lattices.
SIAM J.Math. Anal. , 48(5):3236–3269, 2016.[5] L. B´etermin. Local variational study of 2d lattice energies and application to Lennard-Jones typeinteractions.
Nonlinearity , 31(9):3973–4005, 2018.[6] L. B´etermin. Local optimality of cubic lattices for interaction energies.
Anal. Math. Phys. , 9(1):403–426,2019.[7] L. B´etermin. Minimizing lattice structures for Morse potential energy in two and three dimensions.
J.Math. Phys. , 60(10):102901, 2019.[8] L. B´etermin. Minimal Soft Lattice Theta Functions.
Constr. Approx. , 52(1):115–138, 2020.[9] L. B´etermin and M. Faulhuber. Maximal theta functions - Universal optimality of the hexagonal latticefor Madelung-like lattice energies.
Preprint. arXiv:2007.15977 , 2020.[10] L. B´etermin, M. Faulhuber, and H. Kn¨upfer. On the optimality of the rock-salt structure among latticesand change distributions.
Preprint. arXiv:2004.04553 , 2020.[11] L. B´etermin and H. Kn¨upfer. Optimal lattice configurations for interacting spatially extended particles.
Lett. Math. Phys. , 108(10):2213–2228, 2018.[12] L. B´etermin, L. De Luca, and M. Petrache. Crystallization to the square lattice for a two-body potential.
Preprint. arXiv:1907:06105 , 2019.[13] L. B´etermin and M. Petrache. Dimension reduction techniques for the minimization of theta functionson lattices.
J. Math. Phys. , 58:071902, 2017.[14] L. B´etermin and M. Petrache. Optimal and non-optimal lattices for non-completely monotone interac-tion potentials.
Anal. Math. Phys. , 9(4):2033–2073, 2019.[15] L. B´etermin and P. Zhang. Minimization of energy per particle among Bravais lattices in R : Lennard-Jones and Thomas-Fermi cases. Commun. Contemp. Math. , 17(6):1450049, 2015.[16] X. Blanc and M. Lewin. The Crystallization Conjecture: A Review.
EMS Surv. in Math. Sci. , 2:255–306,2015.[17] J. M. Borwein, M. L. McPhedran, R. C. Wan, and I. J. Zucker.
Lattice sums: then and now . volume150 of Encyclopedia of Mathematics, 2013.[18] M. Buchanan. Quantum crystals.
Nature Physics , 13:925, 2017.[19] J.W.S. Cassels. On a Problem of Rankin about the Epstein Zeta-Function.
Proceedings of the GlasgowMathematical Association , 4:73–80, 7 1959.[20] H. Cohen.
Number Theory II: Analytic and Modern Methods . Springer, 2007.[21] H. Cohn and A. Kumar. Universally optimal distribution of points on spheres.
J. Amer. Math. Soc. ,20(1):99–148, 2007.[22] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska. The sphere packing problem indimension 24.
Ann. of Math. , 185(3):1017–1033, 2017.[23] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska. Universal optimality of the E andLeech lattices and interpolation formulas. Preprint. arXiv:1902:05438 , 2019.[24] J. H. Conway and N. J. A. Sloane.
Sphere Packings, Lattices and Groups , volume 290. Springer, 1999.[25] R. Coulangeon and A. Sch¨urmann. Energy Minimization, Periodic Sets and Spherical Designs.
Int.Math. Res. Not. IMRN , pages 829–848, 2012.[26] G. Zhang D. Chen and S. Torquato. Inverse Design of Colloidal Crystals via Optimized Patchy Inter-actions.
The journal of Physical Chemistry B , 122:8462–8468, 2018.[27] P. H. Diananda. Notes on Two Lemmas concerning the Epstein Zeta-Function.
Proceedings of theGlasgow Mathematical Association , 6:202–204, 7 1964.[28] V. Ennola. A Lemma about the Epstein Zeta-Function.
Proceedings of The Glasgow MathematicalAssociation , 6:198–201, 1964.[29] M. Friedrich and L. Kreutz. Crystallization in the hexagonal lattice for ionic dimers.
Math. ModelsMethods Appl. Sci. , 29(10):1853–1900, 2019.
30] M. Friedrich and L. Kreutz. Finite crystallization and Wulff shape emergence for ionic compounds inthe square lattice.
Preprint. arXiv:arXiv:1903:00331 , 2019.[31] S. Grivopoulos. No crystallization to honeycomb or Kagom´e in free space.
J. Phys. A: Math. Theor. ,42(11):1–10, 2009.[32] S. Hyun and S. Torquato. Optimal and Manufacturable Two-dimensional, Kagom´e-like Cellular Solids.
Journal of Materials Research , 17(1):137–144, 2002.[33] I. G. Kaplan.
Intermolecular Interactions : Physical Picture, Computational Methods, Model Potentials .John Wiley and Sons Ltd, 2006.[34] M. Lewin, E. H. Lieb, and R. Seiringer. Floating Wigner crystal with no boundary charge fluctuations.
Phys. Rev. B , 100(3):035127, 2019.[35] S. Luo, X. Ren, and J. Wei. Non-hexagonal lattices from a two species interacting system.
SIAM J.Math. Anal. , 52(2):1903–1942, 2020.
Preprint. arXiv:1902.09611 .[36] S. Luo and J. Wei. On minima of sum of theta functions and Mueller-Ho Conjecture.
Preprint.arXiv:2004.13882 , 2020.[37] M. Mekata. Kagome: The story of the basketweave lattice.
Physics Today , 56(2):12–13, 2003.[38] H. L. Montgomery. Minimal Theta Functions.
Glasg. Math. J. , 30(1):75–85, 1988.[39] M. Petrache and S. Serfaty. Crystallization for Coulomb and Riesz Interactions as a Consequence ofthe Cohn-Kumar Conjecture.
Proceedings of the American Mathematical Society , 148:3047–3057, 2020.[40] C. Poole.
Encyclopedic Dictionary of Condensed Matter Physics . Elsevier, 1st edition edition, 2004.[41] S. Bae Q. Chen and S. Granick. Directed self-assembly of a colloidal kagome lattice.
Nature , 469:381–384, 2011.[42] R. A. Rankin. A Minimum Problem for the Epstein Zeta-Function.
Proceedings of The GlasgowMathematical Association , 1:149–158, 1953.[43] P. Sarnak and A. Str¨ombergsson. Minima of Epstein’s Zeta Function and Heights of Flat Tori.
Invent.Math. , 165:115–151, 2006.[44] A. Terras.
Harmonic Analysis on Symmetric Spaces and Applications II . Springer New York, 1988.[45] R. J .D. Tilley.
Understanding Solids: The Science of Materials . John Wiley & Sons, 2004.[46] M.P. Tosi. Cohesion if ionic solids in the Born model.
Solid State Physics , 16:1–120, 1964.[47] M. Viazovska. The sphere packing problem in dimension 8.
Ann. of Math. , 185(3):991–1015, 2017., 185(3):991–1015, 2017.