MMinimal Soft Lattice Theta Functions
Laurent B´eterminFaculty of Mathematics, University of Vienna,Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria [email protected] . ORCID id: 0000-0003-4070-3344November 13, 2019
Abstract
We study the minimality properties of a new type of “soft” theta functions. For a lattice L ⊂ R d , a L -periodic distribution of mass µ L and an other mass ν z centred at z ∈ R d , we define,for all scaling parameter α >
0, the translated lattice theta function θ µ L + ν z ( α ) as the Gaussianinteraction energy between ν z and µ L . We show that any strict local or global minimality resultthat is true in the point case µ = ν = δ also holds for L (cid:55)→ θ µ L + ν ( α ) and z (cid:55)→ θ µ L + ν z ( α ) whenthe measures are radially symmetric with respect to the points of L ∪{ z } and sufficiently rescaledaround them (i.e. at a low scale). The minimality at all scales is also proved when the radiallysymmetric measures are generated by a completely monotone kernel. The method is based ona generalized Jacobi transformation formula, some standard integral representations for latticeenergies and an approximation argument. Furthermore, for the honeycomb lattice H , the centerof any primitive honeycomb is shown to minimize z (cid:55)→ θ µ H + ν z ( α ) and many applications arestated for other particular physically relevant lattices including the triangular, square, cubic,orthorhombic, body-centred-cubic and face-centred-cubic lattices. AMS Classification:
Primary 74G65 ; Secondary 82B20,
Keywords:
Theta functions, Lattice energies, Crystal, Defects, Calculus of variations.
The mathematical justification of crystal’s shape and formation is a very difficult problem whichhas been actively studied (see [19] and references therein). Indeed, the analytic or numericalinvestigation of static many-particle Hamiltonian’s ground states plays a central role in the designof materials (see e.g. [59]), but the large number of critical points as well as the nonlinearitiesemerging from the corresponding systems make this mathematical investigation very challenging.Thus, a first natural step is to study systems that are already in a periodic order and wherethe interaction between points is given by a radially symmetric potential. These potentials arisein physics models of matter (see e.g. [47, 49]) in the case of the Born-Oppenheimer adiabaticapproximation: the electrons effects are neglected, the energy is reduced to the nuclei interactions(see [56, p. 33]), and two-body potentials are the simplest way to express the total potential energyof the system (see [56, p. 945]).This problem of minimizing a potential energy per point of the form E f [ L ] := (cid:88) p ∈ L f ( | p | ) , where L is a Bravais lattice (see also Definition 2.4), which is the most simple possible periodicconfiguration of points, has received a lot of attention, especially in the following cases: Lennard-Jones type potentials [7, 8, 9, 18], Morse potential [10], two-dimensional Thomas-Fermi model for1 a r X i v : . [ m a t h - ph ] N ov olid [16], Coulombian renormalized energy [15, 61, 62], completely monotone interaction potentials[7, 26, 28], Bose-Einstein Condensates [2, 53], diblocks and 3-blocks copolymer interactions [24, 50],vortices in quantum ferrofluids [51], inverse power laws (Epstein zeta function) [23, 31, 32, 34, 35, 36,58], and also in more general settings [14, 33]. An important mathematical object, which appearsto be central in this theory (see e.g. [7, 26]), is the lattice theta function. Given a d -dimensionalBravais lattice L , a scaling parameter α > z ∈ R d , we define the lattice theta functionand the translated lattice theta function by θ L ( α ) := (cid:88) p ∈ L e − πα | p | and θ L + z ( α ) := (cid:88) p ∈ L e − πα | p + z | . (1.1)Physically, θ L ( α ) can be viewed as the Gaussian self-interaction of L and θ L + z ( α ) as the Gaussianinteraction between point z and lattice L . They actually are the energies per point of the so-calledGaussian Core Model (GCM) restricted to lattices. This model was initially introduced by Stillinger[65] and motivated by the Flory-Krigbaum potential between the centers-of-mass of two polymerchains in an athermal solvent [42]. The phase diagram of the three-dimensional GCM has beennumerically investigated for example in [69]. Furthermore, two different problems concerning (1.1)appear to be quite natural once α > • the minimization of L (cid:55)→ θ L ( α ) among d -dimensional Bravais lattices with the same density; • the minimization of z (cid:55)→ θ L + z ( α ) among vectors z ∈ R d , where L is fixed.These minimization problems have been studied by many authors, see e.g. [3, 9, 11, 13, 25, 26,28, 33, 34, 37, 38, 39, 40, 41, 52, 54, 60, 64], and two of the most significant results are due toMontgomery [52] – who proved the minimality of the triangular lattice for L (cid:55)→ θ L ( α ) amongtwo-dimensional Bravais lattices of any fixed density – and Cohn, Kumar, Miller, Radchenko andViazovska [28] – who recently proved the minimality among all periodic configurations with thesame density of E and the Leech lattice in dimensions d ∈ { , } . Furthermore, in [11], theminimizer of the translated theta function z (cid:55)→ θ L + z ( α ), for a fixed lattice L , has been also provedto be connected to the optimal electrostatic interaction (and more general long-range weightedinteraction energies) between periodic distributions of charges located on L , solving a conjecturestated by Born in [21] about the optimality of the rock-salt structure.Since polymer chains can be seen as soft interpenetrable spheres with an extent of the orderof their radius of gyration (see [49, Sect. 3]), we propose a generalization of the periodic GCM tomass interactions, in the same spirit as it has been done in dimension d = 2 in [12] with Kn¨upfer.We want to study the minimality properties of lattices L and points z for a kind of “soft GCM”(SGCM), where the objects are smeared out. Furthermore, these mass interaction energies canalso be viewed as the expectation values of the lattice theta function and translated lattice thetafunction defined by (1.1) where the position of the lattice points (resp. the position of z ) follow aradially symmetric probability distribution µ L (resp. ν z ). We therefore define (see also Definition2.8) the translated soft lattice theta function by θ µ L + ν z ( α ) := E µ,ν [ θ L + z ( α )] = (cid:88) p ∈ L (cid:90) (cid:90) R d × R d e − πα | x + p − z − y | dµ ( x ) dν ( y ) . The main goal of this paper is to derive some minimality properties of lattices L and points z for ( L, z ) (cid:55)→ θ µ L + ν z ( α ), where α > L, z ) (cid:55)→ θ L + z ( α ) with those of ( L, z ) (cid:55)→ θ µ L + ν z ( α ).Minimizing L (cid:55)→ θ µ L + ν ( α ) and z (cid:55)→ θ µ L + ν z ( α ) can be interpreted in terms of ”defects” in theperiodic SGCM. Thus, we want to understand at which scales the type (i.e. the profile of µ, ν ) or2he size (i.e. the size of the ball containing almost all the mass of the measures) of the defects do notplay any role in these minimization problems. Indeed, many defects appear in perfect crystals andthey give generally to the material its properties (corrosion resistance, softness, thermal expansion,etc.). In Solid-State Physics, two kinds of point defects are important (see e.g. Kaxiras [48, Chap.9]):1. the extrinsic defects, such as a substitutional impurity, corresponding to z = 0 in our model.An atom in a perfect crystal is substituted by another one of the same kind ( µ = ν ) or ofa different kind ( µ (cid:54) = ν ). This impurity is usually chemically similar to the crystal’s atom,with a similar size. We are looking for the minimizer of L (cid:55)→ θ µ L + ν ( α ), i.e. the Gaussianinteraction between the mass ν centred at the origin and all the masses µ L centred at latticesites, among a class of lattices with the same density. This also includes the energy per pointof the perfect crystal itself when µ = ν (see the top line of Figure 1);2. the intrinsic defects, such as an interstitial defect, corresponding to z (cid:54) = 0 in our model. Anadditional atom is located somewhere in the unit cell of the crystal (but not at a lattice site)and is generally smaller and chemically different than the crystal’s atoms. We are looking forthe minimizer of z (cid:55)→ θ µ L + ν z ( α ), i.e. the Gaussian interaction between the mass ν z centredat z and all the masses µ L centred at lattice sites (see the bottom line of Figure 1).Figure 1: Example of different kinds of defects. The lattice is L = Z and the masses are allGaussian, with different variances. We have chosen to represent only the primitive cell Q Z . Onthe top: the case without defect µ = ν and z = 0 (left), and the case of an extrinsic defect µ (cid:54) = ν and z = 0 (right). On the bottom: the case of a intrinsic defect µ (cid:54) = ν and z (cid:54) = 0.Using the methods developed in [12] extended to the d -dimensional case and to the translatedlattice theta function, we show that for any measures µ, ν and any α >
0, we can rescale themeasures around the points by sufficiently small factors ε and δ , getting two new measures µ ε and ν δ , such that any strict local or global minimality result which is true for ( L, z ) (cid:55)→ θ L + z ( α ) is also3rue for ( L, z ) (cid:55)→ θ µ εL + ν δz ( α ). We will say that the minimality is true at a low scale. In other words,there is a difference of scale between the lattice spacing and the radius of the balls where most ofthe masses are concentrated such that the measures (or the “defects”) do not play any role in termsof strict local or global minima. Furthermore, if µ and ν have densities of the form x (cid:55)→ ρ ( | x | )where ρ is the Laplace transform of a nonnegative Borel measure ( ρ is a completely monotonefunction), then the optimality occurs at all scales, i.e. for all such measures. As in [12], our resultsare purely qualitative: we do not give any value of ε and δ such that the properties hold. However,it is interesting to see that, once ( ε, δ ) are below some threshold values ( ε , δ ), then the desiredminimality occurs whatever the quotient ε/δ is. Furthermore, a numerical investigation have beenmade in [12, Remark 14] for the triangular lattice case with uniform measures on disks, showingthat the strict local minimality of the triangular lattice proved for small values of the parameters( ε, δ ) certainly does not hold when these parameters are too large.We notice that, as recalled in [12], this type of minimization problem involving smeared outparticles appears in many physical and biological systems as condensed matter theory [46], quantumphysics models [17], diblock copolymer systems in the low volume fraction limit [55], magnetizeddisks interactions [44] and swarming or flocking related models [4, 5, 22, 67].We have also devoted a complete section of the paper to the applications of our results. Theymainly are corollaries of minimality results obtained in previous works in the point case µ = ν = δ ,i.e. for the lattice theta function and the translated lattice theta functions defined by (1.1), for someparticular physically relevant lattices. We thought it was a good opportunity to review all thoseresults in this paper in order to know what are the main open problems associated to these softtheta functions. Furthermore, we show that the minimum of z (cid:55)→ θ H + z ( α ), where H is a honeycomblattice (see (2.6) and Figure 4), is the centre of an honeycomb of H , and the same is therefore truefor the smeared out cases previously stated.In terms of generalization, it appears to be straightforward that all the results in this paper canbe proved for more general energies of the form E f [ µ L + ν z ] := (cid:88) p ∈ L (cid:90) (cid:90) R d × R d f ( | x + p − z − y | ) dµ ( x ) dν ( y ) , where f is a L completely monotone summable function, as the one we have studied in [12]. Sinceour original goal was to study a new kind of theta functions that could have other applications inNumber Theory and Mathematical Physics, we did not extend our results to these types of energies.The reader can refer to [12] for details. Plan of the paper.
After giving the definition of lattices, energies, measures and minimality ata low scale and at all scales in Section 2, we show some preliminary results in Section 3, includingthe generalized Jacobi transformation formula. Our main results are stated and proved in Section4 and many applications are given in Section 5, where the minimality of the primitive honeycomb’scenter in the honeycomb lattice case is also proved.
We start by defining the space of Bravais lattices with a fixed density as well as their unit cells andthe notion of dual lattice. We call ( e i ) ≤ i ≤ d the orthonormal basis of R d , | . | the euclidean norm on R d , u · v the associated scalar product of u, v ∈ R d and B r the closed ball of radius r > M d ( R ) the space of n × n matrices with real coefficients. Definition 2.1 (Bravais lattice) . Let d ≥ . We call L ◦ d the space of d -dimensional Bravais latticesof the form L = (cid:76) di =1 Z u i with basis ( u , ..., u d ) ⊂ R d and covolume , i.e. det( u , ...., u d ) = 1 . he unit cell (of volume 1) of such Bravais lattice L is defined by Q L := (cid:40) x = d (cid:88) i =1 λ i u i ∈ R d , λ i ∈ [0 , (cid:41) . Furthermore, the dual lattice of L ∈ L ◦ d is defined by L ∗ := { x ∈ R d : ∀ p ∈ L, x · p ∈ Z } ∈ L ◦ d . We also recall the definitions of the following important lattices belonging to L ◦ d (see also Figure2): The triangular lattice Λ := (cid:115) √ (cid:34) Z (1 , ⊕ Z (cid:32) , √ (cid:33)(cid:35) ; (2.1)The (simple) cubic lattices Z d , d ≥
1; (2.2)The orthorhombic lattices Z da = d (cid:77) i =1 Z ( a i e i ) , ∀ i, a i > , d (cid:89) i =1 a i = 1; (2.3)The Face-Centred-Cubic (FCC) lattice D := 2 − [ Z (1 , , ⊕ Z (0 , , ⊕ Z (1 , , D ∗ := 2 (cid:20) Z (1 , , ⊕ Z (0 , , ⊕ Z (cid:18) , , (cid:19)(cid:21) . (2.5)Figure 2: Representation of the triangular and square lattices Λ , Z (first line), the simple cubic,FCC and BCC lattices Z , D , D ∗ (second line), and the orthorhombic lattice Z a (third line).These lattices are physically relevant because they correspond to the main crystal structuresthat exist in nature (see [48, p. 8-9]) which are Bravais lattices. Notice for instance that amongthe 118 known elements, 21 (resp. 26) have a BCC (resp. FCC) structure. We will also considerthe honeycomb lattice H defined by H := Λ ∪ (Λ + u ) , u := (cid:115) √ (cid:18) , √ (cid:19) , (2.6)5nd we call H one of its primitive hexagons (see Figure 4). This is a typical example of periodicconfiguration, i.e. an union of translated Bravais lattices, that arises in Physics (e.g. as the structureof a graphene sheet). Furthermore, we define D , E , D + d and the Leech lattice as in [30]. It turnsout that these lattices have many interesting properties related to energy minimization or packing(see for instance [27, 28, 33, 34, 64, 68]).As explained in [66, Sect. 1.4] (see also [14, p. 14]), any lattice L ∈ L ◦ d can be parametrized by apoint ¯ L = ( x , ..., x n d ) ∈ R n d , where n d := d ( d +1)2 −
1, in a fundamental domain D d where each latticeof L ◦ d appears only once. Thus, as in [12], the metric on D d is chosen as the euclidean metric on R n d , d ( L, Λ) denotes the euclidean distance between two lattices L, Λ ∈ D d and B r ( L ) denotes the openball of radius r centred at L . We write E ∈ C k ( D d ) if E : D k → R is k -times differentiable withrespect to the variables ( x , ..., x n d ). We denote the gradient of E by ∇ E [ L ] := ( ∂ x E, ..., ∂ x nd E )[ L ].The Hessian D E ∈ M n d ( R ) is defined as the n d × n d real matrix of second derivatives with respectto x , ..., x n d . D E is correspondingly the tensor of all third derivatives. The notions of strict localminimizer and critical point in D d is defined as follows: Definition 2.2.
Let d ≥ and E : D d → R . We say that L is a strict local minimizer of E in D d if there is η > such that E [ L ] < E [ ˜ L ] for all ˜ L ∈ B η ( L ) . Furthermore, L is a critical point of E in D d if ∇ E [ L ] = (0 , ..., . Let us now focus on the interaction potentials we want to consider throughout this paper. Eventhough the main interaction potential will be the Gaussian one, it turns out that we will use manyproperties of more general lattice energies. We define two classes of functions that can be writtenas the Laplace transform of a measure.
Definition 2.3.
Let d ≥ . We say that f ∈ F d if | f ( r ) | = O ( r − d − σ ) as r → + ∞ for some σ > and if f can be represented as the Laplace transform of a Radon measure µ f , i.e. f ( r ) = (cid:90) + ∞ e − rt dµ f ( t ) . Furthermore we say that f ∈ CM d if f ∈ F d and µ f is nonnegative. Furthermore, the energy per point of any L ∈ D d interacting through a radial potential f ∈ F d is defined by an absolutely convergent sum as follows. Definition 2.4 (Energy per point) . Let d ≥ , L ∈ D d and f ∈ F d , then we define E f [ L ] := (cid:88) p ∈ L f ( | p | ) . (2.7)Because our goal is to study masses interactions, we need to specify what kind of measureswe are working with. We note P ( R d ) the space of probability measures on R d and P r ( R d ) thespace of probability measures on R d that are rotationally symmetric with respect to the origin.Furthermore, we define the following subspace of P r ( R d ): P cmr ( R d ) := (cid:110) µ ∈ P r ( R d ) : dµ ( x ) = ρ ( | x | ) dx, ρ ∈ CM d (cid:111) . Remark 2.5 (Completely monotone functions) . The notations CM d and “ cm ” mean that ρ (or f in Definition 2.3) is a completely monotone function, i.e. ( − k ρ ( k ) ( t ) ≥ t > k ∈ N , which is indeed equivalent, by Hausdorff-Bernstein-Widder Theorem [6], for ρ to be theLaplace transform of a nonnegative Radon measure µ ρ .We now define the periodic measure µ L that corresponds to the union of measures µ centred atall the points of a Bravais lattice L . 6 efinition 2.6 (Periodized measure) . For any L ∈ D d and any µ ∈ P ( R d ) , the periodized measure µ L is defined by µ L := (cid:88) p ∈ L µ p , where, for any z ∈ R d , µ z := µ ( · − z ) . (2.8)In Figure 3, we have represented two different kinds of measures µ L for µ being in P r ( R ) and P cmr ( R ).Figure 3: Two kinds of periodic measures µ L , where L = Z . The left-side one (resp. right-sideone) is such that µ ∈ P cmr ( R ) (resp. P r ( R ) \P cmr ( R )).Furthermore, because we want to show the optimality of the previously defined lattices wherethe masses are sufficiently concentrated around the lattice sites, we define the rescaled measure asfollows. Definition 2.7 (Rescaled measure) . Let d ≥ . For any µ ∈ P ( R d ) and any ε > , the rescaledmeasure µ ε is defined, for any measurable set F ⊂ R d , by µ ε ( F ) := µ ( εF ) . We write µ εL and µ εz the corresponding rescaled measures of µ L and µ z defined by (2.8) . We finally define the main energies that we are studying in this paper: the translated soft latticetheta function. It corresponds to the Gaussian interaction energy of the measure µ L with itself orwith an other measure ν z located at z ∈ R d , and are therefore a generalization of the lattice andtranslated lattice theta functions θ L ( α ) and θ L + z ( α ) defined by (1.1). Definition 2.8 (Translated soft lattice theta functions) . Let d ≥ . For any L ∈ D d , any µ, ν ∈P ( R d ) , any z ∈ R d and any α > , we define the translated soft lattice theta function of µ L by themeasure ν z by θ µ L + ν z ( α ) := (cid:88) p ∈ L (cid:90) (cid:90) R d × R d e − πα | x + p − z − y | dµ ( x ) dν ( y ) . Remark 2.9 (The honeycomb lattice case) . Since the honeycomb lattice H = Λ ∪ (Λ + u ) isa union of two translated triangular lattice, it is easy to compute its Gaussian energy per point.More precisely, we have θ µ H + ν z ( α ) = 12 (cid:16) θ µ Λ1 + ν z ( α ) + θ µ Λ1 + ν z + u ( α ) (cid:17) . Remark 2.10 (Special cases) . We notice that: 7
For any µ ∈ P ( R d ), any L ∈ D d (or any periodic configuration) and any α >
0, it turns outthat the self-interaction of µ L is θ µ L + µ ( α ) (i.e. when ν = µ and z = 0). • If µ = ν = δ , then θ µ L + ν ( α ) = θ L ( α ) and θ µ L + ν z ( α ) = θ L + z ( α ) as defined in (1.1). • The translated soft lattice theta function for the rescaled measures µ ε and ν δ is given by θ µ εL + ν δz ( α ) = (cid:88) p ∈ L (cid:90) (cid:90) R d × R d e − πα | x + p − z − y | dµ ε ( x ) dν δ ( y ) . The goal of this work is to study minimization problems for L (cid:55)→ θ µ L + ν ( α ) in D d – whichincludes the self-interaction of µ L – and z (cid:55)→ θ µ L + ν z ( α ) in Q L for fixed L , at different scales. Wetherefore precise what we mean by being critical or minimal at a low scale and at all scales. Definition 2.11 (Criticality and minimality at all scales and at a low scale) . Let d ≥ and µ, ν ∈ P ( R d ) , then:1. We say that L ∈ D d is a critical point or a (strict local) minimum of L (cid:55)→ θ µ L + ν ( α ) at allscales on D d if, for any α > , L is a critical point or a (strict local) minimum on D d of L (cid:55)→ θ µ L + ν ( α ) .2. Let α > be fixed. We say that L ∈ D d is a critical point or a (strict local) minimumof L (cid:55)→ θ µ L + ν ( α ) at a low scale on D d if there exist ε > and δ > such that, forany ε ∈ [0 , ε ) and any δ ∈ [0 , δ ) , L is a critical point or a (strict local) minimum of L (cid:55)→ θ µ εL + ν δ ( α ) on D d .Furthermore, let L be fixed, then we define the same notions of critical point and (strict local)minimality for z (cid:55)→ θ µ L + ν z ( α ) of z in Q L at all scales or at a low scale. It is well-known that the lattice theta function satisfies the following identity, called Jacobi Trans-formation Formula (see e.g. [45, Thm. A] or [20] for a general formula involving harmonic polyno-mials): for any L ∈ D d , any z ∈ R d and any α > θ L + z ( α ) := (cid:88) p ∈ L e − πα | p + z | = 1 α d (cid:88) q ∈ L ∗ e − π | q | α e iπq · z . (3.1)We generalize this formula to mass interaction in the following result. Proposition 3.1 (Generalized Jacobi Transformation Formula) . For any d ≥ , any L ∈ D d , any µ, ν ∈ P r ( R d ) , any z ∈ R d and any α > , we have θ µ L + ν z ( α ) = Γ (cid:0) d (cid:1) α d (cid:88) q ∈ L ∗ e − π | q | α g µ ( | q | ) g ν ( | q | ) | q | d − e iπq · z , (3.2) where, for any m ∈ P r ( R d ) , g m ( | q | ) := (cid:90) ∞ J d − (4 πs | q | ) s − d dψ m ( s ) , and ψ m is the Lebesgue-Stieltjes measure of t (cid:55)→ m ( B t ) , i.e. ψ m ([ r , r )) = m ( B r ) − m ( B r ) and J β is the Bessel function of the first kind. roof. By the classical Jacobi Transformation Formula (3.1) and Fubini’s Theorem, we get θ µ L + ν z ( α ) = (cid:90) (cid:90) R × R (cid:88) p ∈ L e − πα | p + x − z − y | dµ ( x ) dν ( y )= 1 α d (cid:90) (cid:90) R × R (cid:88) q ∈ L ∗ e iπq · ( x − y − z ) e − π | q | α dµ ( x ) dν ( y )= 1 α d (cid:88) q ∈ L ∗ e − π | q | α e − iπq · z (cid:18)(cid:90) R e iπq · x dµ ( x ) (cid:19) (cid:18)(cid:90) R e − iπq · y dν ( y ) (cid:19) = 1 α d (cid:88) q ∈ L ∗ e − π | q | α e − iπq · z ˆ µ (2 q )ˆ ν (2 q ) , where ˆ m is the notation for the Fourier transform of a measure m ∈ P r ( R d ). We now recall thatˆ m is given by the Hankel-Stieltjes transform (see e.g. [29, Section 2]), i.e. for any x ∈ R d ,ˆ m ( x ) = (cid:90) R e − iπx · y dm ( y ) = 2 d − Γ (cid:0) d (cid:1) | x | d − (cid:90) ∞ J d − (2 πs | x | ) s − d dψ m ( s ) , where ψ m is the Lebesgue-Stieltjes measure of t (cid:55)→ m ( B t ), and the proof is completed. Remark 3.2 (Value at the origin and connection with our previous work) . The value for q = 0 iswell-defined. Indeed, using the Taylor expansion of J d − (see e.g. [1, Eq. (9.1.30)]) and the factthat ψ m ∈ P ( R + ), we obtainlim q → Γ (cid:18) d (cid:19) e − π | q | α g µ ( | q | ) g ν ( | q | ) | q | d − e iπq · z = 1 . We furthermore notice that for d = 2, z = 0 and µ = ν , we recover the formula proved in [12, Prop.7], up to a factor 2 in the argument of J d − that we have corrected and which does not influencethe final result.We now give some basic facts that will be used in the proofs of our main results. Lemma 3.3.
Let d ≥ . If f, g ∈ F d , then f g ∈ F d .Proof. If f and g have both this representation as the Laplace transform of a measure, the formula L − [ f ] ∗ L − [ g ] = L − [ f g ] (see e.g. [57, Thm. 5.3.11]) shows that f g has also this representation.Moreover, we naturally have | f ( r ) g ( r ) | = O ( r − d/ − σ ) for some σ > r → + ∞ , because it is thecase for both functions.The next lemma is a simple consequence of the Jacobi Transformation Formula (3.1). Lemma 3.4.
Let d ≥ and α > , then L is a critical point or a (strict local) minimizer of L (cid:55)→ θ L ( α ) on D d if and only if L ∗ is a critical point or a (strict local) minimizer of L (cid:55)→ θ L (1 /α ) on D d . Furthermore, if L is the unique global minimizer of L (cid:55)→ θ L ( α ) on D d for all α > , then L = L ∗ . The following results come from the simple fact that, for any f ∈ F d , any L ∈ D d and any z ∈ R d , E f [ L + z ] := (cid:88) p ∈ L f ( | p + z | ) = (cid:90) + ∞ θ L + z (cid:18) tπ (cid:19) dµ f ( t ) , as explained e.g. in [7, Sect. 3.1]. 9 emma 3.5. Let d ≥ . If L is a critical point of L (cid:55)→ θ L ( α ) in D d for all α > , then, for any f ∈ F d , L and L ∗ are critical points of L (cid:55)→ E f [ L ] in D d . Lemma 3.6.
Let d ≥ . If L is a (strict local) minimizer in D d of L (cid:55)→ θ L ( α ) for all α > , then,for any f ∈ CM d , L is a (strict local) minimizer of L (cid:55)→ E f [ L ] in D d .Let L ∈ D d . If z is a (strict local) minimizer of z (cid:55)→ θ L + z ( α ) in Q L for all α > , then, for any f ∈ CM d , z is a (strict local) minimizer of z (cid:55)→ E f [ L + z ] in Q L . In this part, we generalize the results of [12] about the (strict local) minimality of a lattice for L (cid:55)→ θ µ L + ν ( α ), for a given α >
0, to any dimension. Furthermore, using the same strategy basedon an approximation of the quantity we sum by a completely monotone potential, we show thesame kind of results for z (cid:55)→ θ µ L + ν z ( α ) where L is fixed.It is important to notice that, by the generalized Jacobi Transformation Formula (3.2), if z = 0,then θ µ L + ν ( α ) is the sum of a radial potential over L ∗ , i.e. an energy of type E f [ L ∗ ] for some f , as defined in Definition 2.4. Therefore, as shown for instance in [8, 9, 33], any lattice L suchthat L ∗ has enough symmetries is a critical point of the soft lattice theta function, e.g. Λ , D , D ∗ and Z d , d ≥
2. It turns out that, as proved in [10, Thm. 3.2] these lattices are the only one indimensions d ∈ { , } being “volume-stationary” for an energy of type E f , i.e. they can be criticalpoint of E f in the space of Bravais lattices of fixed density in an open interval of densities. Thefollowing result gives some sufficient conditions for a lattice or a point, which are already criticalfor the theta functions (1.1), to be critical for the soft lattice theta functions. Proposition 4.1 (Criticality) . Let d ≥ and µ, ν ∈ P r ( R d ) then the following hold:1. If L is a critical point of L (cid:55)→ θ L ( α ) in D d for all α > , then L is a critical point of L (cid:55)→ θ µ L + ν ( α ) in D d for all α > .2. Let L ∈ D d . If z is a critical point of z (cid:55)→ θ L + z ( α ) in Q L for all α > , then z is a criticalpoint of z (cid:55)→ θ µ L + ν z ( α ) in Q L for all α > .Proof. Let us prove the first part of the proposition. By Lemma 3.5, if L is a critical point of L (cid:55)→ θ L ( α ) in D d for all α >
0, then L and L ∗ are critical points of E f in D d where f ∈ F d . We claimthat there exists a Radon measure µ h such that the function h defined by h ( r ) := e − πrα g µ ( √ r ) g ν ( √ r ) r d − belongs to the set F d , i.e. h = L [ µ h ]. It is clear by Lemma 3.3 because h is a product of fourfunctions belonging to F d , which completes the proof because L is hence a critical point of E h in D d .Let us prove the second part of the proposition. Since z is a critical point of z (cid:55)→ θ L + z ( β ) forall β >
0, we obtain, using Jacobi transformation formula (3.1) and computing the derivative withrespect to z i , ∀ i ∈ { , ..., d } , (cid:88) q ∈ L ∗ e − π | q | β q i sin(2 πq · z ) = 0 . (4.1)We recall that, by Proposition 3.1, the soft lattice theta function can be written as θ µ L + ν z ( α ) = Γ (cid:0) d (cid:1) α d (cid:88) q ∈ L ∗ h ( | q | ) cos(2 πq · z ) , h ( r ) := e − πrα g µ ( √ r ) g ν ( √ r ) r d − .
10e now use the fact that there exists a Borel measure µ h such that h = L [ µ h ] ∈ F d as explainedabove. Therefore, integrating (4.1) against µ h , where t = π/β is the variable of integration, gives ∀ i ∈ { , ..., d } , (cid:88) q ∈ L ∗ h ( | q | ) q i sin(2 πq · z ) = 0 , for any α >
0. It follows that ∂ z i θ µ L + ν z ( α ) = 0 for all i ∈ { , ..., d } and all α >
0, i.e. z is acritical point of z (cid:55)→ θ µ L + ν z ( α ) in Q L for all α > Remark 4.2 (Critical points of L (cid:55)→ θ L ( α ) for all α > . It turns out that the only lattices L ∈ D d that can be critical point of L (cid:55)→ θ L ( α ) for all α > d ∈ { , } are Z , Λ , Z , D and D ∗ , as proved in [10, Section 3].The next result generalizes [12, Prop. 11] to L (cid:55)→ θ µ L + ν ( α ) and z (cid:55)→ θ µ L + ν z ( α ) by giving asufficient condition for the strict local minimality of a lattice or a point, which are already minimalfor the theta functions (1.1), in the case of masses interactions. Theorem 4.3 (Strict local minimality) . Let d ≥ and µ, ν ∈ P r ( R d ) . Then we have:1. If L is a critical point of L (cid:55)→ θ L ( α ) for all α > and a strict local minimizer of L (cid:55)→ θ L ( α ) in D d for some α > , then L is a strict local minimizer of L (cid:55)→ θ µ L + ν ( α ) in D d at a lowscale.2. Let L ∈ D d . If z is a critical point of z (cid:55)→ θ L + z ( α ) in Q L for any α > and a strictlocal minimizer of z (cid:55)→ θ L + z ( α ) for some α > , then z is a strict local minimizer of z (cid:55)→ θ µ L + ν z ( α ) on Q L at a low scale.The strict local minimality of L and z also holds at all scales if µ, ν ∈ P cmr ( R d ) .Proof. Let us prove the first part of the theorem. The proof is actually a straightforward general-ization of [12, Prop. 11]. According to (3.2), it is equivalent to show the strict local minimality of L for E h ε,δ [ L ] := (cid:88) p ∈ L ∗ h ε,δ ( | p | ) , h ε,δ ( r ) = g µ ε ( √ r ) g ν δ ( √ r )( εδr ) d − e − πrα , (4.2)where, for any measure m ∈ P r ( R d ), g m ε ( √ r ) = (cid:90) ∞ J d − (4 πsε √ r ) s − d dψ m ( s ) . First, because L is a critical point of L (cid:55)→ θ L ( α ) in D d for all α >
0, it implies, by Proposition 4.1,that L is a critical point of E h ε,δ in D d . Second, we also know that D E h , [ L ] = D θ L ( α ) ispositive definite because L is a strict local minimizer of L (cid:55)→ θ L ( α ) in D d . Furthermore, all thecoefficients of D E h ε,δ [ L ] are expressed in terms of Bessel functions J m (see e.g. [1, Eq. (9.1.30)])and it is easy to check that D E h ε,δ [ L ] = D E h , [ L ] + A ε,δ [ L ], A ε,δ [ L ] ∈ M d ( R ), by the Taylorexpansion of J m (see e.g. [1, Eq. (9.1.10)]) where (cid:107) A ε,δ [ L ] (cid:107) → ε, δ ) → (0 ,
0) for any chosennorm (cid:107) . (cid:107) on M d ( R ). Indeed, this Taylor expansion gives J d − (4 πsε | q | ) J d − (4 πtδ | q | ) = | q | d − (1 + εj ( s, t, | q | ) + δj ( s, t, | q | ) + j ε,δ ( s, t, | q | )) , where j , j are independent of ε, δ and j ε,δ is at least of order min { ε, δ } . Since ψ µ and ψ ν areboth probability measures, the expansion of the second derivative is straightforward. Therefore, theresult follows by continuity of ( ε, δ ) (cid:55)→ A ε,δ [ L ] and by the fact – following from the boundedness ofthe Bessel functions J m – that L (cid:55)→ D E h ε,δ [ L ] is bounded on any ball centred at L , independentlyof ( ε, δ ) ∈ [0 , . 11et us prove the second part of the theorem. Since z is a critical point of z (cid:55)→ θ L + z ( α ) in Q L for all α >
0, we have, by Proposition 4.1, that z is a critical point of z (cid:55)→ θ µ L + ν z ( α ). ByProposition 3.1, we can write, as above, θ µ εL + ν δz ( α ) = Γ (cid:0) d (cid:1) α d (cid:88) q ∈ L ∗ h ε,δ ( | q | ) cos(2 πq · z ) , h ε,δ ( r ) = g µ ε ( √ r ) g ν δ ( √ r )( εδr ) d − e − πrα . As explained in the proof of the first part of the theorem, by the Taylor expansion of J d/ − , it isstraightforward to prove that D z θ µ εL + ν δz ( α ) = D z θ L + z ( α ) + A ε,δ,z [ L ] , where A ε,δ,z [ L ] ∈ M d ( R ) and sup z ∈ Q L (cid:107) A ε,δ,z [ L ] (cid:107) → ε, δ ) → (0 , (cid:107) . (cid:107) on M d ( R ). Since z is a strict local minimizer of z (cid:55)→ θ L + z ( α ), it follows that D z θ µ εL + ν δz ( α ) ispositive definite for ε and δ sufficiently small, i.e. there exists ε , δ such that for any 0 ≤ ε < ε and any 0 ≤ δ < δ , z is a strict local minimizer of z (cid:55)→ θ µ εL + ν δz ( α ). Remark 4.4.
We notice that if L (resp. z ) is a (strict local) minimizer of L (cid:55)→ θ µ L + ν ( α ) (resp. z (cid:55)→ θ µ L + ν z ( α )) for any α belonging to a set of values S , therefore ε and δ only depend on themaximum of these values.The next result is a generalization of [12, Thm. 2 and 3] in arbitrary dimension to our translatedlattice theta function and gives a sufficient condition for the global minimality of a lattice in D d ora point in Q L . Theorem 4.5 (Global minimality) . Let d ≥ and µ, ν ∈ P r ( R d ) , then we have:1. If L is the unique global minimizer and a strict local minimizer of L (cid:55)→ θ L ( α ) in D d for all α > , then, for any α > , L is the unique global minimizer of L (cid:55)→ θ µ L + ν ( α ) in D d ata low scale.2. Let L ∈ D d . If z is a global minimizer and a strict local minimizer of z (cid:55)→ θ L + z ( α ) in Q L for all α > , then, for any α > , z is a global minimizer of z (cid:55)→ θ µ L + ν z ( α ) in Q L at alow scale.The global minimality of L and z also hold at all scales if µ, ν ∈ P cmr ( R d ) .Proof. Let us prove the first part of the theorem and let us start with the case µ, ν ∈ P cmr ( R d ). If L is the unique global minimizer of L (cid:55)→ θ L ( α ) for all α >
0, then, by Lemmas 3.4 and 3.6, L = L ∗ is the unique global minimizer of any lattice energy of the form E f where f ∈ CM d . Furthermore,it has been proved in [12, Lem. 10] that µ ∈ P cmr ( R d ) ⇐⇒ ˆ µ ∈ P cmr ( R d ). Therefore, as the setof completely monotone functions is stable by product, we have ˆ µ ˆ ν ∈ P cmr ( R d ) and it follows fromthe complete monotonicity of t (cid:55)→ e − βt for any β > h : t (cid:55)→ e − πt/α t − d/ g µ ( √ t ) g ν ( √ t ) iscompletely monotone. Therefore, L is the unique global minimizer of L (cid:55)→ θ µ L + ν ( α ) for anyfixed α > µ, ν ∈ P r ( R d ), the proof is again a generalization of [12, Thm. 2]. We first remark thatany minimizer of L (cid:55)→ θ µ εL + ν δ ( α ) belongs to a ball of center L with a finite radius. Indeed, thisfact is proved in [12, Lemma 12] in dimension d = 2 and is directly generalizable to any dimensionby bounding below | p | , p ∈ L , by | p (cid:48) | where p (cid:48) ∈ L λ = λ − Z ⊕ Λ for some λ ≥ L , once the lattice is parametrized by n d parameters as explained in Section 2, andΛ ∈ L ◦ d − . We therefore get 12 µ εL + ν δ ( α ) ≥ θ µ εLλ + ν δ ( α ) ≥ (cid:90) (cid:90) R d × R d (cid:88) m ∈ Z e − πα | λ − ( m, ,..., x − y | dµ ε ( x ) dν δ ( y ) → + ∞ as λ → + ∞ . The same can be done in all the unbounded directions of the fundamental domain D d , proving that the global minimizer necessarily belongs to a compact subset of D d , for examplea closed ball containing L that we will note K .In the following, we write, as in (4.2), E h ε,δ [ L ] := (cid:88) p ∈ L h ε,δ ( | p | ) , h ε,δ ( r ) = g µ ε ( √ r ) g ν δ ( √ r )( εδr ) d − e − πrα , and we recall that minimizing L (cid:55)→ θ µ εL + ν δ ( α ) is equivalent with minimizing L (cid:55)→ E h ε,δ because,by Lemma 3.4, L = L ∗ . As in [12], we claim there exists (cid:101) h ε,δ such that, for any L ∈ K , (cid:12)(cid:12)(cid:12) E (cid:101) h ε,δ [ L ] − E h ε,δ [ L ] (cid:12)(cid:12)(cid:12) ≤ C max { ε, δ } , as ε, δ → , (4.3) E (cid:101) h ε,δ [ L ] − E (cid:101) h ε,δ [ L ] ≥ Cd ( L, L ) , (4.4)for some constant C > ε, δ . If (4.3)-(4.4) hold, then, for any L ∈ K , E h ε,δ [ L ] − E h ε,δ [ L ] ≥ Cd ( L, L ) − C max { ε, δ } . Thus, for any L ∈ K such that E h ε,δ [ L ] ≤ E h ε,δ [ L ], this implies that d ( L, L ) ≤ C max { ε, δ } whichcontradicts the strict local minimality of L for sufficiently small ε and δ . To construct ˜ h ε,δ , theidea is to approximate k ε,δ ( r ) := h ε,δ ( r ) e πrα = k ε ( r ) k δ ( r ) , k ε ( r ) := g µ ε ( √ r )( ε √ r ) d − , k δ ( r ) := g ν δ ( √ r )( δ √ r ) d − , by a bounded completely monotone function (cid:101) k ε,δ = L [ m ε,δ ] such that m ε,δ is a positive measure witha compact support, (cid:101) k ε,δ (0) = 1 and (cid:107) (cid:101) k (cid:48) ε,δ (cid:107) ≤ C . It is indeed sufficient to apply the method describedin [12, proof of Thm. 2] to k ε and k δ that are then approximated respectively by (cid:101) k ε = L [ m ε ] and (cid:101) k ε = L [ m δ ] and where m ε,δ = m ε ∗ m δ . We therefore get, for any r > (cid:12)(cid:12)(cid:12) k ε,δ ( r ) − (cid:101) k ε,δ ( r ) (cid:12)(cid:12)(cid:12) ≤ C min { max { ε, δ } r, } . Defining ˜ h ε,δ ( r ) := ˜ k ε,δ ( r ) e − πrα , we then have (cid:12)(cid:12)(cid:12) E (cid:101) h ε,δ [ L ] − E h ε,δ [ L ] (cid:12)(cid:12)(cid:12) ≤ (cid:88) q ∈ L ∗ e − π | q | α (cid:12)(cid:12)(cid:12) k ε,δ ( | q | ) − (cid:101) k ε,δ ( | q | ) (cid:12)(cid:12)(cid:12) ≤ C (cid:88) q ∈ L ∗ min { max { ε, δ } | q | , } e − π | q | α ≤ C max { ε, δ } , by the exponential decay of the lattice theta function and where C does not depend on the lattice L ∈ K . Therefore, (4.3) is proved. For the second inequality, we have E (cid:101) h ε,δ [ L ] − E (cid:101) h ε,δ [ L ] ≥ Cd ( L, L ) , C independent of ε and δ . Indeed, it is a consequence of the fact that ˜ h ε,δ = L [ m ε,δ ]and m ε,δ has a compact support and then follows by the strict local minimality of L for L (cid:55)→ θ L ( α )for all α >
0, which implies the same strict local minimality for any E f where f ∈ CM d (see Lemma3.6), and in particular for E (cid:101) h ε,δ (see [12, Section 2.2] for details).Let us prove the second part of the theorem and let us begin again with µ, ν ∈ P cmr ( R d ).The proof uses the same ingredient as the previous one. Indeed, if z is a global minimizer of z (cid:55)→ θ L + z ( α ) in Q L for all α >
0, then by Lemma 3.6 the same holds for z (cid:55)→ E f [ L + z ] = (cid:88) p ∈ L f ( | p + z | ) , where f ∈ CM d . Let α >
0, then by the generalized Jacobi Transformation Formula (3.2), wehave, for some constant C α ,d > θ µ L + ν z ( α ) = C α ,d (cid:88) q ∈ L ∗ h ( | q | ) e iπq · z , for a completely monotone function h ∈ CM d , as explained previously. We therefore obtain, byPoisson Summation Formula (see e.g. [45, Appendix A]), θ µ L + ν z ( α ) = ˜ C α ,d (cid:88) p ∈ L ˆ h ( | p + z | ) . Since h is completely monotone, then ˆ h is completely monotone by [12, Lem. 10] and therefore z is a global minimizer of z (cid:55)→ (cid:80) p ∈ L ˆ h ( | p + z | ) by Lemma 3.6, which concludes the proof.Let us now consider µ, ν ∈ P r ( R d ). For convenience, we define, for fixed α > L ∈ D d , F h ε,δ ( z ) := (cid:88) q ∈ L ∗ h ε,δ ( | q | ) cos(2 πq · z ) , where h ε,δ is defined by (4.2). We again recall that minimizing z (cid:55)→ θ µ L + ν z ( α ) is equivalent withminimizing F h ε,δ . We remark that, for (cid:101) h ε,δ defined above, we have, for any z ∈ Q L , (cid:12)(cid:12)(cid:12) F (cid:101) h ε,δ ( z ) − F h ε,δ ( z ) (cid:12)(cid:12)(cid:12) ≤ C max { ε, δ } , as ε, δ → F (cid:101) h ε,δ ( z ) − F (cid:101) h ε,δ ( z ) ≥ C | z − z | , for some constant C >
0. The second inequality follows from the strict local minimality of z for z (cid:55)→ θ L + z ( α ). Therefore, for any z ∈ Q L , F h ε,δ ( z ) − F h ε,δ ( z ) ≥ C | z − z | − C max { ε, δ } . Thus, for any z ∈ Q L such that F h ε,δ ( z ) ≤ F h ε,δ ( z ), this implies | z − z | ≤ C max { ε, δ } which isnot true for sufficiently small ε and δ by the strict local optimality of z in Q L previously shown inTheorem 4.3. Remark 4.6 (Difference between ε and δ ) . An interesting property is the fact that ε and δ can bechosen with different scales – for instance ε = 1 /n and δ = 1 / √ n – as long as they are below thecritical values ε and δ . Remark 4.7 (Global minimizer of the lattice theta function) . Notice that the minimizer of L (cid:55)→ θ L ( α ) in D d is known to be the same for all α >
0, so far, only in dimension d ∈ { , , } as provedin [28, 52]. In these dimensions, the minimizers are, respectively, the triangular lattice Λ , E andthe Leech lattice. 14 emark 4.8 (Global minimizer of the translated lattice theta function) . Few results are alreadyknown concerning the minimization of z (cid:55)→ θ L + z ( α ), where L ∈ D d is fixed:1. In dimension d = 2, if L = Λ , then Baernstein [3] proved that the barycenters z = (cid:113) √ (cid:16) , √ (cid:17) and z = (cid:113) √ (cid:16) , √ (cid:17) of the primitive triangle composing Q Λ are the uniqueglobal minimizers of z (cid:55)→ θ Λ + z ( α ) for all α > d ≥
1, we proved in [13, Prop. 1.3] (see also [11, Prop. 3.7]) that the center c a = ( a , ..., a d ) of the primitive cell Q Z da of the orthorhombic lattice Z da defined by (2.3) isthe unique global minimizer of z (cid:55)→ θ Z da + z ( α ) for all α > z is a global minimizer of z (cid:55)→ θ L + z ( α ) for all α >
0, then it is the case as α → + ∞ and it turns out that z is necessarily a deep hole of L , i.e. a solution of the following optimizationproblem: max z ∈ R d min p ∈ L | z − p | , (4.5)as we proved in [13, Thm 1.5]. Therefore, since this property does not necessarily hold as α → d = 2 in [13, Thm 1.6] for asymmetric lattices), the global minimizer of z (cid:55)→ θ L + z ( α ) is not necessarily the same for all α . We finally apply the previous results to some particular lattices defined by (2.1)-(2.4) as well as D , E , the Leech lattice and D + d . This is the perfect opportunity to recall the main results thatare currently known about the local and global minima of the (translated) lattice theta functions,which are now generalized to the masses interactions case.Montgomery [52] proved the minimality of Λ in D for L (cid:55)→ θ L ( α ) for all α >
0. Furthermore,as recalled in the previous section, Baernstein [3] proved the minimality of the two barycenters ofthe primitive triangles composing Q Λ for z (cid:55)→ θ Λ + z ( α ) also for all α >
0. Therefore, applyingMontgomery’s theorem, Baernstein’s theorem, and our main results, we extend the minimality ofΛ and its barycenters to Gaussian masses interactions. Notice that the µ = ν, z = 0 case has beenalready proved in [12]. Corollary 5.1 (The triangular lattice) . Let d = 2 and Λ be defined by (2.1) . We then have:1. For any µ, ν ∈ P r ( R ) , Λ is a critical point of L (cid:55)→ θ µ L + ν ( α ) for all α > . Moreover, forall α > , Λ is a global minimizer of L (cid:55)→ θ µ L + ν ( α ) in D at a low scale. Furthermore, if µ, ν ∈ P cmr ( R ) , then the global minimality holds at all scales.2. Let z = (cid:113) √ (cid:16) , √ (cid:17) and z = (cid:113) √ (cid:16) , √ (cid:17) be the barycenters of the primitive trianglesof Q Λ , then, for any µ, ν ∈ P r ( R ) , z and z are critical points of z (cid:55)→ θ µ Λ1 + ν z ( α ) in Q Λ at all scales. Furthermore, for any α > , z and z are the unique global minimizers of z (cid:55)→ θ µ Λ1 + ν z ( α ) in Q Λ at a low scale. Moreover, if µ, ν ∈ P cmr ( R ) , then the minimality of z and z holds at all scales. Remark 5.2 (Importance of the triangular lattice) . The triangular lattice arises in many physicalmodels such as Bose-Einstein Condensates [2], Superconductivity [62], Coulomb Gases [63] or di-block copolymer interaction [24]. We also recall that Λ is conjectured (see Cohn and Kumar [26,15onjecture 9.4]) to be the unique minimizer of the lattice theta function among periodic configu-rations (not only among Bravais lattices) of fixed density. Furthermore, Λ can also be viewed as alayer of a FCC or BCC lattice potentially shifted by a vector parallel to z or z , as explained in [13,Sect. II.2]. Then, as already discussed in [9], Corollary 5.1 is of great interest for the understandingof BCC and FCC stability in the smeared out particle case, using dimension reduction techniquesas in [13].We now show the following result, using Baernstein’s theorem [3, Thm. 1], for the honeycomblattice H . Proposition 5.3 (The honeycomb lattice) . Let d = 2 , and H be defined by (2.6) with the primitivehexagon H containing the center z = (cid:113) √ (cid:16) , √ (cid:17) . Then, for any µ, ν ∈ P r ( R ) , z is a criticalpoint of z (cid:55)→ θ µ H + ν z ( α ) in H at all scales. Furthermore, z is the unique global minimizer of z (cid:55)→ θ µ H + ν z ( α ) in H at a low scale. Moreover, if µ, ν ∈ P cmr ( R ) , then the minimality holds at allscales.Proof. It is sufficient to show the µ = ν = δ case and to apply the second part of Theorem 4.5which remains true in this non-Bravais case by simply following the lines of its proof and using thefact that θ H + z ( α ) = 12 ( θ Λ + z ( α ) + θ Λ + u + z ( α )) . Since z (cid:55)→ θ Λ + z ( α ) have two global minimizers in Q Λ that are the barycenters of the primitivetriangle of Q Λ , we obtain that, for any z ∈ H , θ Λ + z ( α ) ≥ θ Λ + z ( α ) . Furthermore, by symmetry, z + u = z is the second gobal minimizer of z (cid:55)→ θ Λ + z ( α ) and we get,for any z ∈ H , θ Λ + u + z ( α ) ≥ θ Λ + z ( α ) = θ Λ + u + z ( α ) . We therefore have proved that θ H + z ( α ) ≥ θ H + z ( α ) , and z is the unique global minimizer of z (cid:55)→ θ H + z ( α ) in H for all α >
0, by symmetry.Figure 4: Minimizer z of z (cid:55)→ θ H + z ( α ) in a primitive hexagon H . The black dots represent Λ andthe grey one represent Λ + u . 16 emark 5.4 (Gaussian interaction between Lithium atom and Graphene sheet) . The Gaussianinteraction between a mass and a honeycomb structure is physically relevant. For instance, in[43], the authors have been designed a Gaussian approximation potential modelling the interactionenergy between a Lithium atom and a Graphene sheet structure. This has been done by applyingMachine Learning to Density Functional Theory reference data. Our result gives the exact locationof the Lithium atom minimizing the Gaussian interaction with a Graphene sheet.Because of the particular role of the cubic lattices Z d and their orthorhombic deformations Z da defined by (2.3), we have summarized all the results related to them in the following corollary. Thefirst point is an easy consequence of [9, Prop. 5.1], the second point is based on [52, Thm 2] andthe third point follows from [13, Prop. 1.3]. Corollary 5.5 (Cubic and Orthorhombic lattices) . Let d ≥ , then1. For any µ, ν ∈ P r ( R d ) , Z d is a critical point of L (cid:55)→ θ µ L + ν ( α ) in D d at all scales.2. For any µ, ν ∈ P r ( R d ) and any α > , a = (1 , , ..., is the unique minimizer of a (cid:55)→ θ µ Z da + ν ( α ) in { ( a , ..., a d ) ∈ (0 , + ∞ ) : (cid:81) di =1 a i = 1 } at a low scale, i.e. Z d is the uniqueminimizer of L (cid:55)→ θ µ L + ν ( α ) among orthorhombic lattices at a low scale. Moreover, if µ, ν ∈ P cmr ( R d ) , the minimality holds at all scales.3. Let ( a , ..., a d ) ∈ (0 , + ∞ ) be such that (cid:81) di =1 a i = 1 and c a := ( a , ..., a d ) be the center of Q Z da . Then, for any µ, ν ∈ P r ( R d ) , c a is a critical point of z (cid:55)→ θ µ Z da + ν z ( α ) in Q Z da at allscales. Furthermore, for any α > , c a is the unique global minimizer of z (cid:55)→ θ µ Z da + ν z ( α ) in Q Z da at a low scale. Moreover, if µ, ν ∈ P cmr ( R d ) , the minimality holds at all scales. Remark 5.6 (Octahedral site of the cubic lattice) . In the cubic case ( a , ..., a d ) = (1 , ..., c = (1 / , ..., / Z d , is indeed the preferred location to addan extra atom in order to create, for example, an ion (like the CsCl which has a BCC structure).In [13], it has been proved, using a computer assisted method, that D ∗ (resp. D ) is a strictlocal minimizer of L (cid:55)→ θ L ( α ) for any α ∈ A (resp. α ∈ B ) defined by A := { . k : k ∈ N , ≤ k ≤ } , B := { /x : x ∈ A} . (5.1)We also proved in [9, Thm. 1.3] that the same result holds for extremal values of α , i.e. if α > α (resp. 0 < α < α − ) for D (resp. D ∗ ), for some α >
1. Moreover,we also add that D and D ∗ appear to be saddle points for the lattice theta function respectively for α < α − and α > α .Applying Theorem 4.3 to these specific α , we get the strict local minimality of these lattices formeasures that are sufficiently concentrated around the lattice points, where the threshold values ε and δ only depend on µ, ν , the maximum of A and B or on α (see Remark 4.4). Corollary 5.7 (The FCC and BCC lattices) . Let µ, ν ∈ P r ( R ) , we have:1. The lattices D and D ∗ are critical points of L (cid:55)→ θ µ L + ν ( α ) in D at all scales.2. Let A , B be defined by (5.1) . For any α ∈ A (resp. α ∈ B ), D ∗ (resp. D ) is a strict localminimizer of L (cid:55)→ θ µ L + ν ( α ) on D at a low scale. Furthermore, ε and δ only depend on µ, ν , a = max A and b = max B .3. There exists α > such that for any α ∈ ( α , + ∞ ) (resp. α ∈ (0 , α − ) ), D (resp. D ∗ )is a strict local minimizer of L (cid:55)→ θ µ L + ν ( α ) at a low scale. In the D ∗ case, ε and δ onlydepend on µ, ν and α . emark 5.8 (Conjecture for D and D ∗ in the soft case) . We also notice that, according to Sarnakand Str¨ombergsson [64, Eq. (43)], D (resp. D ∗ ) is expected to be a global minimizer of L (cid:55)→ θ L ( α )in D for any α ≥ α ≤ Z , D and D ∗ (see [10, Section 3]) supports this conjecture forthe soft lattice theta function. Remark 5.9 (Minimality of D and D ∗ among body-centred-orthorhombic lattices) . Based on [13,Thm 1.7], it is also straightforward to prove the global minimality of D ∗ (respectively D ) amongbody-centred-orthorhombic lattices (resp. their dual lattices), which are the anisotropic dilationsof the BCC lattice (based on the unit cube) along the coordinate axes by √ y, / √ y and t . Theyare defined by L y,t := (cid:91) k ∈ Z (cid:18) Z ( √ y, ⊕ Z (cid:18) , √ y (cid:19) + ( √ y/ , / √ y, Z ( k ) + k (0 , , t/ (cid:19) , and D ∗ (resp. D ) is the unique minimizer of L (cid:55)→ θ µ L + ν ( α ) in this class of lattices where α ∈ A (resp. B ) and t = 1, at a low scale. Remark 5.10 (Deep holes and BCC/FCC lattices) . For L ∈ { D , D ∗ } , the global minimizers of z (cid:55)→ θ L + z ( α ) in Q L are expected to be the deep holes of L , solution of (4.5), i.e. z = 2 − (1 , , D and z = 2 − (1 , ,
0) for D ∗ as well as all their images in Q L by symmetry. These locationsare the usual one where a different atom can be added to the structure, in order to create, forexample, an ion. They are also called octahedral sites, as for the cubic lattice (see Remark 5.6).Finally, the same kind of local results for L (cid:55)→ θ µ L + ν ( α ) can be stated for some other speciallattices, based on [33, 34, 64]. Corollary 5.11 (Dimensions d ≥ . We have that:1. D , E and the Leech lattice are strict local minimizers of L (cid:55)→ θ µ L + ν ( α ) on D d , for all α > , at a low scale if µ, ν ∈ P r ( R d ) and at all scales if µ, ν ∈ P cmr ( R d ) , for the correspondingdimensions d ∈ { , , } .2. For all odd integer d ≥ , there exists α d such that for any α > α d , D + d is a strict localminimizer of L (cid:55)→ θ µ L + ν ( α ) on D d at a low scale if µ, ν ∈ P r ( R d ) .3. E and the Leech lattice are global minimizers of L (cid:55)→ θ µ L + ν ( α ) on D d , for any α > ,at a low scale if µ, ν ∈ P r ( R d ) and at all scales if µ, ν ∈ P cmr ( R d ) , for the correspondingdimensions d ∈ { , } Acknowledgement.
I acknowledge support from VILLUM FONDEN via the QMATH Centreof Excellence (grant No. 10059). Furthermore, I would like to thank the anonymous referee forher/his useful comments.
References [1] M. Abramowitz and I. A. Stegun.
Handbook of mathematical functions with formulas, graphs, andmathematical tables . U.S. Government Printing Office, Washington D.C., 1964.[2] A. Aftalion, X. Blanc, and F. Nier. Lowest Landau level functional and Bargmann spaces for Bose–Einstein condensates.
J. Funct. Anal. , 241(2):661–702, 2006.[3] A. Baernstein II. A minimum problem for heat kernels of flat tori.
Contemporary Mathematics , 201:227–243, 1997.
4] D. Balagu´e, J. A. Carrillo, T. Laurent, and G. Raoul. Nonlocal interactions by repulsive-attractivepotentials: radial ins/stability.
Phys. D , 260:5–25, 2013.[5] A. J. Bernoff and C. M. Topaz. A primer of swarm equilibria.
SIAM J. Appl. Dyn. Syst. , 10(1):212–250,2011.[6] S. Bernstein. Sur les fonctions absolument monotones.
Acta Math. , 52:1–66, 1929.[7] L. B´etermin. Two-dimensional Theta Functions and Crystallization among Bravais Lattices.
SIAM J.Math. Anal. , 48(5):3236–3269, 2016.[8] L. B´etermin. Local variational study of 2d lattice energies and application to lennard-jones type inter-actions.
Nonlinearity , 31(9):3973–4005, 2018.[9] L. B´etermin. Local optimality of cubic lattices for interaction energies.
Anal. Math. Phys. , 9(1):403–426,2019.[10] L. B´etermin. Minimizing lattice structures for Morse potential energy in two and three dimensions.
Journal of Mathematical Physics , 60(10):102901, 2019.[11] L. B´etermin and H. Kn¨upfer. On Born’s conjecture about optimal distribution of charges for an infiniteionic crystal.
J. Nonlinear Sci. , 28(5):1629–1656, 2018.[12] L. B´etermin and H. Kn¨upfer. Optimal lattice configurations for interacting spatially extended particles.
Lett Math Phys , 108(10):2213–2228, 2018.[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. , (online first) DOI:10.1007/s13324-019-00299-6:1–41, 2019.[15] L. B´etermin and E. Sandier. Renormalized Energy and Asymptotic Expansion of Optimal LogarithmicEnergy on the Sphere.
Constr. Approx. , Special Issue: Approximation and Statistical Physics - Part I ,47(1):39–74, 2018.[16] 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.[17] X. Blanc. Geometry optimization for crystals in Thomas-Fermi type theories of solids.
Comm. PartialDifferential Equations , 26(3-4):651–696, 2001.[18] X. Blanc, C. Le Bris, and B. H. Yedder. A Numerical Investigation of the 2-Dimensional CrystalProblem.
Preprint du laboratoire J.-L. Lions, Universit´e de Paris 6 , 2003.[19] X. Blanc and M. Lewin. The Crystallization Conjecture: A Review.
EMS Surv. in Math. Sci. , 2(2):255–306, 2015.[20] S. Bochner. Theta relations with spherical harmonics.
Proc. Natl. Acad. Sci. U.S.A. , 37(12):804–808,1951.[21] M. Born. ¨Uber elektrostatische Gitterpotentiale.
Zeit. f. Physik , 7:124–140, 1921.[22] J. A. Carrillo, S. Martin, and V. Panferov. A new interaction potential for swarming models.
Phys. D ,260:112–126, 2013.[23] J.W.S. Cassels. On a Problem of Rankin about the Epstein Zeta-Function.
Proceedings of the GlasgowMathematical Association , 4(2):73–80, 1959.[24] X. Chen and Y. Oshita. An application of the modular function in nonlocal variational problems.
Arch.Ration. Mech. Anal. , 186(1):109–132, 2007.
25] H. Cohn and M. de Courcy-Ireland. The Gaussian core model in high dimensions.
Duke Math. J. ,online first. doi:10.1215/00127094-2018-0018:1–39, 2018.[26] H. Cohn and A. Kumar. Universally optimal distribution of points on spheres.
J. Amer. Math. Soc. ,20(1):99–148, 2007.[27] 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.[28] 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.[29] W.C. Connett and A.L. Schwartz. Fourier Analysis Off Groups.
Contemp. Math. , 137:169–176, 1992.[30] J. H. Conway and N. J. A. Sloane.
Sphere Packings, Lattices and Groups , volume 290. Springer, 1999.[31] R. Coulangeon. Spherical Designs and Zeta Functions of Lattices.
Int. Math. Res. Not. , 2006:49620,2006.[32] R. Coulangeon and G. Lazzarini. Spherical Designs and Heights of Euclidean Lattices.
Journal ofNumber Theory , 141:288–315, 2014.[33] R. Coulangeon and A. Sch¨urmann. Energy Minimization, Periodic Sets and Spherical Designs.
Int.Math. Res. Not. , 2012:829–848, 2012.[34] R. Coulangeon and A. Sch¨urmann. Local energy optimality of periodic sets.
Preprint. arXiv:1802.02072 ,2018.[35] P. H. Diananda. Notes on Two Lemmas concerning the Epstein Zeta-Function.
Proceedings of theGlasgow Mathematical Association , 6(4):202–204, 1964.[36] V. Ennola. A Lemma about the Epstein Zeta-Function.
Proceedings of The Glasgow MathematicalAssociation , 6(4):198–201, 1964.[37] M. Faulhuber. Extremal Determinants of Laplace-Beltrami Operators for Rectangular Tori.
Preprint.arXiv:1709.06006 , 2017.[38] M. Faulhuber. Minimal frame operator norms via minimal theta functions.
Journal of Fourier Analysisand Applications , 24(2):545–559, 2018.[39] M. Faulhuber. Some Curious Results Related to a Conjecture of Strohmer and Beaver.
Preprint. , 2018.[40] M. Faulhuber and S. Steinerberger. Optimal gabor frame bounds for separable lattices and estimatesfor jacobi theta functions.
Journal of Mathematical Analysis and Applications , 445(1):407–422, 2017.[41] M. Faulhuber and S. Steinerberger. An extremal property of the hexagonal lattice. to appear in Journalof Statistical Physics , Preprint. arXiv:1903:06856, 2019.[42] P. J. Flory and W. R. Krigbaum. Statistical Mechanics of Dilute Polymer Solutions. II.
J. Chem. Phys. ,18(8):1086, 1950.[43] S. Fujikake, V.L. Deringer, T.H. Lee, M. Krynski, S.R. Elliott, and G. Csanyi. Gaussian approximationpotential modeling of lithium intercalation in carbon nanostructures.
The Journal of Chemical Physics ,148(24):241714, 2018.[44] B. A. Grzybowski, H. A. Stone, and G. M. Whitesides. Dynamics of self assembly of magnetized disksrotating at the liquid–air interface.
Proc. Natl. Acad. Sci. USA , 99(7):4147–4151, 2002.[45] D. P. Hardin, E. B. Saff, and B. Simanek. Periodic Discrete Energy for Long-Range Potentials.
J. Math.Phys. , 55(12):123509, 2014.[46] D. M. Heyes and A. C. Branka. Lattice summations for spread out particles: Applications to neutraland charged systems.
J. Chem. Phys. , 138(3):034504, 2013.
47] I. G. Kaplan.
Intermolecular Interactions : Physical Picture, Computational Methods, Model Potentials .John Wiley and Sons Ltd, 2006.[48] E. Kaxiras.
Atomic and electronic structure of solids . Cambridge University Press, 2010.[49] C. N. Likos. Effective interactions in soft condensed matter physics.
Physics Reports , 348(4–5):267–439,2011.[50] S. Luo, X. Ren, and J. Wei. Non-hexagonal lattices from a two species interacting system.
Preprint.arXiv:1902.09611 , 2019.[51] A. M. Martin, N. G. Marchant, D. H. J. O’Dell, and N. G. Parker. Vortices and vortex lattices inquantum ferrofluids.
J. Phys.: Condens. Matter , 29(2017):103004, 2017.[52] H. L. Montgomery. Minimal Theta Functions.
Glasg. Math. J. , 30(1):75–85, 1988.[53] E. J. Mueller and T.-L. Ho. Two-Component Bose-Einstein Condensates with a Large Number ofVortices.
Physical Review Letters , 88(18), 2002.[54] F. Nier. A propos des fonctions Thˆeta et des r´eseaux d’Abrikosov. In
S´eminaire EDP-Ecole Polytech-nique , 2006-2007.[55] T. Ohta and K. Kawasaki. Equilibrium morphology of block copolymer melts.
Macromolecules ,19(10):2621–2632, 1986.[56] C. Poole.
Encyclopedic Dictionary of Condensed Matter Physics . Elsevier, 1st edition edition, 2004.[57] A. D. Poularikas.
The Transforms and Applications Handbook . CRC Press, 1996.[58] R. A. Rankin. A Minimum Problem for the Epstein Zeta-Function.
Proceedings of The GlasgowMathematical Association , 1(4):149–158, 1953.[59] M. C. Rechtsman, F. H. Stillinger, and S. Torquato. Designed Interaction Potentials via Inverse Methodsfor Self-Assembly.
Physical Review E , 73:011406, 2006.[60] O. Regev and N. Stephens-Davidowitz. An Inequality for Gaussians on Lattices.
SIAM J. DiscreteMath. , 31(2):749–757, 2017.[61] N. Rougerie and S. Serfaty. Higher dimensional coulomb gases and renormalized energy functionals.
Communications on Pure and Applied Mathematics , 69(3):519–605, 2016.[62] E. Sandier and S. Serfaty. From the Ginzburg-Landau Model to Vortex Lattice Problems.
Comm.Math. Phys. , 313(3):635–743, 2012.[63] E. Sandier and S. Serfaty. 2d Coulomb gases and the renormalized energy.
Ann. Probab. , 43(4):2026–2083, 2015.[64] P. Sarnak and A. Str¨ombergsson. Minima of Epstein’s Zeta Function and Heights of Flat Tori.
Invent.Math. , 165:115–151, 2006.[65] F. H. Stillinger. Phase transitions in the Gaussian core system.
J. Chem. Phys. , 65, 1976.[66] A. Terras.
Harmonic Analysis on Symmetric Spaces and Applications II . Springer New York, 1988.[67] C. M. Topaz and A. L. Bertozzi. Swarming patterns in a two-dimensional kinematic model for biologicalgroups.
SIAM J. Appl. Math. , 65(1):152–174, 2004.[68] M. Viazovska. The sphere packing problem in dimension 8.
Ann. of Math , 185(3):991–1015, 2017.[69] C. Z. Zachary, F. H. Stillinger, and S. Torquato. Gaussian core model phase diagram and pair correla-tions in high Euclidean dimensions.
The Journal of Chemical Physics , 128(22):224505, 2008., 128(22):224505, 2008.