A Proof of Finite Crystallization via Stratification

We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body short-ranged particle interactions and three-body angular potentials favoring bond-angles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality in ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^1$$\end{document}. As a byproduct, we also prove a fluctuation estimate for minimizers of the configurational energy, known as the n3/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{3/4}$$\end{document}-law.


Introduction
At low temperature, atoms and molecules typically arrange themselves into crystalline order.Tackling this phenomenon by using mathematical models consists in proving or disproving that ground states of particle systems for certain configurational energies with interatomic interactions exhibit crystalline order.This issue, referred to as the crystallization problem [5], has attracted a great deal of attention in the physics and mathematics community.By now, various mathematically rigorous crystallization results are available both for systems with a fixed, finite number of atoms, and in the so-called thermodynamic limit dealing with the infinite particle limit.The reader is referred to [5,24] for a general overview and also to Communicated by Alessandro Giuliani.[33] for a detailed account of available results.The goal of this paper is to revisit the problem of finite crystallization in dimension two, and to present a novel and substantially different proof strategy.
We consider a model where configurations are identified with the respective positions of atoms {x 1 , . . ., x n } in the plane with an associated configurational energy E({x 1 , . . ., x n }) comprising classical interaction potentials.More specifically, E = E 2 + E 3 decomposes into E 2 and E 3 describing two-and three-body interactions, respectively.The two-body interaction potential E 2 is short-ranged and attractive-repulsive favoring atoms sitting at some specific reference distance.For E 3 ≡ 0 and for a specific choice of E 2 , namely the so-called sticky disc potential, crystallization in the triangular lattice has been proved by Heitmann and Radin [29] (see also [36,42] for generalizations) and recently revisited in [13], via an approach from discrete differential geometry.If instead E 3 = 0, under specific quantitative assumptions, optimal geometries can be identified as the square or the hexagonal lattice [33,35], depending on whether E 3 favors triples of particles forming angles which are multiples of π 2 or 2π 3 , respectively.Besides crystallization, fine characterizations of ground-state geometries are available by proving the emergence of hexagonal or square macroscopic Wulff shapes for growing particle numbers [2,9,10,22].We also refer to related rigorous crystallization results for particle systems involving different types of atoms [4,7,20,21,23,38] and to [25,26,37,41] for a nonexhaustive list of results in dimension one.
Although the exact realization of the proof of each result is different depending on the used potentials and the underlying optimal geometry, all proofs follow the very same strategy, originally devised in [28,29].First of all, due to range of the two-body interaction, one can naturally associate a planar graph to the configuration where vertices and edges correspond to particles and bonds, respectively.The graph is then separated into the boundary and bulk atoms.The boundary energy (roughly, the number of bonds at the boundary of the configuration) is carefully estimated by geometric arguments involving the angles between atoms and relying on the sum of interior angles in planar polygons.Moreover, by means of Euler's formula for planar graphs a connection between the number of bonds and atoms in the configuration is derived.Then, the essential idea of the proof lies in an induction argument over the number of particles: one removes a bond graph layer, i.e., the boundary atoms of the configuration, and by induction hypothesis one uses information of the remaining configuration consisting of less atoms.The approach in [13] is different in the sense that it endows the bond graph with a suitable notion of discrete combinatorial curvature and uses a discrete version of the Gauss-Bonnet theorem from differential geometry.However, it still vitally hinges on specific geometric arguments and the induction method over bond graph layers.
It appears to be challenging to generalize this strategy to problems beyond the setting described above.On the one hand, it is hardly conceivable to extend the delicate estimates on the boundary energy to particle systems in three dimensions where surfaces have a much richer structure.On the other hand, the induction method over bond graph layers is often not flexible enough to handle more general situations such as particles systems with two types of atoms with prescribed ratio since this ratio might not be preserved by removing a bond graph layer.
In this work, we propose a new strategy to tackle finite crystallization problems which does not use the induction method over bond graph layers and comes along without arguments from the theory of planar graphs and discrete differential geometry such as Euler's formula or Gauss-Bonnet.It relies on an idea that we call stratification.In this paper, we present our technique for the model by Mainini et al. [33] and reprove finite crystallization in the square lattice, see Theorem 2.1.We are confident, however, that the strategy carries over to other lattices as well, such as the triangular [13,29] and the hexagonal [35] lattice.
As observed in [33], ground states correspond to configurations minimizing a specific edge perimeter of the configuration, essentially counting the number of missing bonds of atoms having less than four bonds.For ground-state competitors, the bond graph can be locally interpreted as a deformed version of Z 2 , apart from possible defects in the lattice, see Definition 3.1.Therefore, we can identify chains of atoms in the bond graph where the angle between three successive atoms is near to π, called strata.Strata can be open, where the first and the last atom of the stratum lie at the boundary of the configuration, or they can be closed forming a closed cycle.In contrast to open strata, closed strata do not contribute to the edge perimeter.Therefore, for a correct estimate, we aim at excluding the existence of closed strata.To this end, we observe that, due to the cycle structure of closed strata, there need to exist angles deviating from π and thus contributing to the three-body energy E 3 .Given specific quantitative assumptions on the potentials similar to the ones in [33], the contribution of E 3 is large enough to allow us to erase a bond from the stratum to turn it into an open stratum.This procedure is made precise in Lemma 3.7 and referred to as stratification.Once all strata are open, the graph satisfies specific properties (see Lemmas 3.3 and 3.6) which reduce our crystallization problem to a simple argument related to an edge isoperimetric inequality on the square lattice.(Compare to [33] for a problem on Z 2 , and see also [6,27] for some related classical issues in Discrete Mathematics.) In contrast to uniqueness of Wulff shapes for continuum crystalline isoperimetric problems, minimizers for a finite number of particles n are in general not unique.For different lattices in 2D, it has been shown that there are arbitrarily large n with ground-state configurations deviating from the hexagonal or square macroscopic Wulff shape by a number of n 3/4 -particles [9,10,33,39].Later, this analysis as been extended to the cubic lattice in higher dimensions [32,34].The proof of such maximal asymptotic deviation, also known as maximal fluctuation estimate, relies on careful rearrangement techniques for atoms at the boundary and edge-isoperimetric inequalities.In our setting of the square lattice, we can immediately reobtain this so-called n 3/4 -law as a mere byproduct of our crystallization proof, see Theorem 2.2.Our argument is similar to [33] with the interesting difference however that our strategy can be applied even if configurations are not subset of Z 2 .We also mention the complementary approach [8], yet restricted to subsets of periodic lattices, where maximal fluctuation estimates are derived via a quantitative version of the edge isoperimetric inequality, based on the quantitative version of the anisotropic isoperimetric inequality proved in [16].
One goal of our work is to revisit finite crystallization results and to suggest a substantially different proof strategy which does not use the induction method over bond graph layers and comes along without arguments from the theory of planar graphs and discrete differential geometry.Besides providing, to our view, a simpler and more direct proof of known results, our main motivation is that our techniques seem promising to tackle more challenging crystallization problems.For example, we expect that our approach can contribute to understand finite crystallization in three dimensions or crystallization for double-bubble problems [11,12,19] (configuration with two types of atoms).
Let us highlight that our proof strategy is tailor-made for the problem of finite crystallization.Concerning the thermodynamic limit, i.e., as the number of particles tends to infinity, other techniques are used and allow to prove results under less restrictive assumptions on the potentials.We refer to [3,14,15,40] for results in the plane and to some few available rigorous results [17,18] in three dimensions.
The article is organized as follows.In Sect. 2 we introduce our setting and state the main results.Section 3 is devoted to the concept of stratification and in Sect. 4 we prove our main results.We close the introduction with basic notation.The Euclidian distance between a point x ∈ R d and a set A ⊂ R d is denoted by dist(A, x).By #A we denote the cardinality of a set A. By B r (x) we indicate the open ball with center x ∈ R d and radius r > 0, and simply write B r if x = 0. We define the ceil function by t := min{z ∈ Z : z ≥ t} for t ∈ R.

Setting and Main Results
We consider particle systems in two dimensions, and model their interaction by classical potentials in the frame of Molecular Mechanics [1,30].Indicating the configuration of particles by C n = {x 1 , . . ., x n } ⊂ R 2 , we define its energy by Here, θ i, j,k denotes the angle formed by the vectors x j −x i and x k −x i (counted clockwisely), and the second sum runs over triples (i, j, k) with |x i − x j | ≤ r 0 and |x i − x k | ≤ r 0 , where r 0 is given in (ii 2 ) below.The factor 1 2 accounts for double counting of bonds and angles.In the following, for simplicity we denote the angle formed by the vectors x − y and z − y by θ x,y,z .We fix 0 < ε < ε 0 for ε 0 < π 6 specified in Lemma 3.2.The two-body potential We briefly comment on the assumptions.Condition (i 2 ) on a unique minimum (here normalized to 1) is natural, e.g., it is valid for Lennard-Jones-type potentials.Assumption (ii 2 ) states that v 2 has compact support.In particular, it ensures that for configurations C n ⊂ Z 2 only atoms at distance 1 interact.These atoms are usually referred to as nearest neighbors in the literature.Eventually, (iii 2 ) prevents clustering of points.In fact, along with (ii 2 ) it shows Definition 3.1(i) in the proof of Lemma 3.2 below.Condition (i 3 ) ensures that the potential v 3 does not depend on how (clockwise or counter-clockwise) bond angles are measured, and (ii 3 ) guarantees that for C n ⊂ Z 2 there is no contribution stemming from the three-body interaction.Slope conditions similar to (iii 3 ) have been used in [20,21,33,35] in order to obtain crystallization on the square or hexagonal lattice.Let us mention that in the other works the condition is needed at all minimum points of v 3 , whereas here only at π.As a consequence, the potential is necessarily non-smooth at π.We also point out that in this work the focus lies on a new proof strategy and all appearing specific numerical constants are chosen for computational simplitcity rather than optimality.The two potentials are illustrated in Fig. 1.We now state the main theorems of the paper.We emphasize that the main theorems have been shown previously in the literature, see [21,33].As outlined in the introduction, the main novelty lies in the proof technique.
Some configurations of minimal energy are depicted in Fig. 2.
Theorem 2.2 (n 3/4 -law) There exists c > 0 such that for all n ∈ N it holds that each ground state C n , up to a rigid motion, satisfies Let us note that the scaling is sharp: the construction in [21,Sect. 3.2] shows that there exists a sequence (n k ) k∈N with n k → +∞ and corresponding ground states C n k such that, up to applying any rigid motion to C n k , it holds that for some 0 < c ≤ c, where c is the constant given in Theorem 2.2.Our proof allows to give an explicit estimate on the constant c, but we do not know if c = c.

Stratification
After a short preliminary on graph theory, this section is devoted to the main technique of this paper: modification of bond graphs, called stratification.

Bond Graph
We denote by G = (V , E) a graph, where V ⊂ R 2 indicates the set of vertices and E ⊂ {{x, y}: x, y ∈ V and x = y} is the set of edges.For x ∈ V , we denote the neighborhood where is the bond energy and the excess energy.For V ⊂ V , we also define the localized elastic energy by where We will identify each C n ⊂ R 2 with its natural bond graph G nat = (V , E nat ), where V = C n and the natural edges are given by for r 0 > 0 as given in (ii 2 ).This definition is motivated by the relation to (2.1), namely In Sect.3.2 below, we will successively modify E nat to a smaller set of edges E ⊂ E nat .
Analogous properties have been derived in [33, Propostion 2.1] and [40, Lemma 2.2].However, as our assumptions on the potentials are slightly different, we include a sketch of the proof for the reader's convenience in Appendix A. For the remainder of this paper, we assume that ε 0 > 0 is chosen small enough such that Lemma 3.2 holds true and that v 2 , v 3 satisfy (i 2 )-(iii 2 ) and (i 3 )-(iv 3 ) for some 0 < ε < ε 0 .Moreover, we suppose that , where r 0 is given in (ii 2 ).This ensures that the bond graph is planar.Indeed, given a quadrilateral with all sides larger or equal than 1 − ε 0 , one diagonal has at least length √ 2(1 − ε 0 ).
(If N = 2, (ii) and (iii) are empty.)Note that paths are ordered subsets of V but they are not oriented, i.e., (x 1 , . . ., x N ) and (x N , . . ., x 1 ) should be considered as the same straight path.
When taking intersections and unions, we will sometimes regard straight paths as subsets of V with a slight abuse of notation.The set of straight paths is denoted by We drop G and write if no confusion arises.If γ ∈ and x 1 = x N , we say that γ is closed and otherwise that γ is open.In the following, we add some strata for degenerate points which will be convenient for Lemma 3.3.Specifically, we define Note that in the second definition, one could equally use the angle open, and degenerate strata are illustrated in Fig. 3.In particular, s(x) for x ∈ V 0 has to be understood as a multiset containing the stratum (x) twice (strictly speaking S(G) is therefore the multiset of all strata).Adding the degenerate stratum (x) with one element twice for V 0 and once for V 1 ∪ V π 2 has no geometrical interpretation but is merely for convenience: for graphs whose straight paths are all open, it allows us to relate the overall number of strata to F bond and ensures that each atom is contained in exactly two strata.More precisely, denoting by l(s) := #s the length of s ∈ S, we have the following.

Lemma 3.3 (Properties of graphs only containing open paths)
Let G = (V , E) be an εregular graph.Assume that all γ ∈ are open.Then, the following holds: Proof We prove the two statements in separate steps.
(i) It suffices to show that each x ∈ V belongs to exactly two s 2 ) lies in exactly two elements of S and in no degenerate stratum.Indeed, as G is ε-regular, we can find two different straight paths that contain x as the only common point and whose union is not a straight path.Here, we used that #N (x, E) ≥ 2 and x / ∈ V π 2 .Since all γ ∈ are open, this guarantees that there exist two different maximal straight paths containing x (left figure of Fig. 5 below is excluded).The ε-regularity of G also implies that there are at most two maximal straight paths through x.
Secondly, each x ∈ V 1 ∪ V π 2 lies in exactly one element of S and x ∈ V 0 is not contained in any element of S .More precisely, each x ∈ V 1 is bonded to exactly one other atom and therefore forms a path.This path is contained in one maximal straight path.If x ∈ V π 2 , it forms a straight path together with N (x, E), according to definition (3.4).Again, this straight path is contained in one maximal straight path.The definition of s(x) for x ∈ V 0 ∪ V 1 ∪ V π 2 now implies that each x ∈ V belongs to exactly two s ∈ S. (This is the very reason for adding the degenerate strata in (3.5).)Hence, (i) follows.
(ii) We prove the statement by induction over m = #E.It is clearly true for m = 0 since, by definition of (3.5), x ∈ V ⇒ x ∈ V 0 and thus Let now #E = m ≥ 1 and let s = (x 1 , . . ., x N ) ∈ S be arbitrary.Consider Ê := E\{x 1 , x 2 } and the corresponding graph Ĝ = (V , Ê).Then, # Ê = m − 1 and thus, by the induction hypothesis, where S( Ĝ) is the set of strata of Ĝ, defined in (3.5).Note that This concludes the proof.
We proceed with two definitions and a lemma on graphs with small angle excess.If γ = (x 1 , x 2 ) ∈ , we set θ ex (γ ) = 0.A stratum s ∈ S and its orthogonal strata are illustrated in Fig. 4. For degenerate strata s = (x) ∈ S (recall the definition below (3.4) and see Fig. 3), we explicitly have where γ ∈ S is the unique maximal straight path containing x, cf.proof of Lemma 3.3(i).The next lemma shows some elementary properties of graphs with small angle excess.Lemma 3.6 (Small angle excess for regular graphs) Let G = (V , E) be an ε-regular graph.The following implications hold true: − ε, then s 1 ∩ s 2 = ∅ for all s 1 , s 2 ∈ S ⊥ (s) and for all s ∈ S.
Proof We first introduce some notation that will be used throughout the proof.Let p = {x 1 , . . ., x N } be such that the edges e i = {x i , x i+1 }, i = 1, . . ., N − 1, form a closed simple polygon.We denote by θ(e i , e i+1 ) the interior angle formed by the edges e i and e i+1 , i = 1, . . ., N − 1, with the convention e 1 = e N .By the interior angle sum of polygons it holds For the reader's convenience, the proof of the three different statements is aided by Fig. 5.
We now come to the stratification of bond graphs.The following lemma allows to reduce the problem of crystallization to a purely geometric problem of minimizing the number of strata in graphs containing only open strata with small angle excess.This is the only step in the proof where (iii 3 ) is needed.
Proof We construct G o = (V , E o ) by iteratively erasing edges.We start by setting G 0 = (V , E) and we suppose that G k = (V , E k ) is already given.We construct G k+1 = (V , E k+1 ) by suitably modifying the set of edges E k .If (i) is satisfied, we may stop.Thus, we assume that there exists γ (3.7) Additionally, due to (iii 3 ), with L := 4(π/6 − ε) −1 > 0 we have Therefore, due to (3.7) and (3.8), we have that where F ex (γ k ) is defined in (3.1).Since G = (V , E) is finite, the procedure terminates for some K ∈ N and we set G o := (V , E K ).By construction, G o satisfies (i).It remains to show (ii).Note that, due to the minimal selection of γ k ∈ , once γ k is selected this way, we will not select any γ ⊂ γ k in any successive step j > k.Thus, using (3.9) and the previous observation, we have with strict inequality whenever K ≥ 1.In particular, if equality holds in (3.10), we have that G o = G.This necessarily gives F ex (G) = 0 which implies that |x − y| = 1 for all x ∈ V , y ∈ N (x, E), and θ ∈ {π/2, π, 3π/2} for all bond angles by (i 2 ) and (ii 3 ).This concludes the proof.

Proof of the Main Results
This section is devoted to the proofs of Theorems 2.1-2.2.

Crystallization
We will show that the minimum of F is given by 2 2 √ n , and that it is attained by subsets of Z 2 .In view of (3.3), this shows Theorem 2.1.Recall the definition of G nat in (3.2).We first state the following upper bound.
Proof The statement is obtained by direct construction of configurations C n with C n ⊂ Z 2 satisfying the energy bound.We refer to [33,Section 4] for details, see also Fig. 2.
The core of the proof now consists in proving a lower bound.
Proof of Theorem 2.1 Let C n be a minimizer of (2.1).Then G nat is ε-regular by Lemma 3. The main part of the proof consists in verifying Hence, there exists s 0 ∈ S such that Recall Definition 3.5 and define l v := max s∈S ⊥ (s 0 ) l(s) and s v ∈ argmax s∈S ⊥ (s 0 ) l(s).We claim that In fact, by (4.1) and Lemma 3.6(ii) we have that #S ⊥ (s 0 ) = l(s 0 ), #S ⊥ (s v ) = l v and, by Lemma 3.6(iii), if s ∈ S ⊥ (s 0 ), then s / ∈ S ⊥ (s v ) (and vice versa).This yields (4.5).We set span(s 0 ) = s ∈S ⊥ (s 0 ) s ⊆ V and we consider two cases: Now, since span(s 0 ) = V , we have in particular that l(s 0 ) • l v ≥ n and therefore, noting that t < t + 1 we obtain by Lemma 3.3(ii) and Young's inequality Since 2#S ∈ 2N, the previous estimate yields the claim (4.2) in case (a).Before proceeding with case (b), we would like to mention that the proof of inequality (4.6) is the analogous step to the continuum isoperimetric inequality proved in Theorem B.1.

Proof in case (b):
We claim that in this case we have that In fact, by definition, there exists x ∈ V \span(s 0 ).Due to Lemma 3.6(iii), for s, s ∈ S ⊥ (s v ) we have that s ∩ s = ∅, and thus there exists at most one s ∈ S ⊥ (s v ) such that s ∩ {x} = ∅.
We also note that s ∩ {x} = ∅ for all s ∈ S ⊥ (s 0 ).Since for all x ∈ V there exist two strata s, s such that x ∈ s, s (see proof of Lemma 3.3(i)), there exists at least one stratum s / ∈ S ⊥ (s 0 ) ∪ S ⊥ (s v ).Therefore (4.7) follows.
We denote by S a := S\(S ⊥ (s 0 ) ∪ S ⊥ (s v )) and observe that by (4.3) and (4.5) it holds that Now, by Lemma 3.6(ii) and the choice of s 0 , see (4.4), and s v respectively, we have #S ⊥ (s 0 ) = l(s 0 ), #S ⊥ (s v ) = l v , l(s) ≤ l(s 0 ) for all s ∈ S, and l(s) ≤ l v for all s ∈ S ⊥ (s 0 ).Due to Lemma 3.3(i) and and thus This together with Lemma 3.3(ii), (4.5), and t < t + 1 implies Again, since 2(l v + l(s 0 ) + S a ) ∈ 2N and, #S a ≥ 1 by (4.5)-(4.7), the claim (4.2) follows also in case (b).This finishes the proof.Finally, we make the following observation: the argument along with Lemma 3.7(ii) also shows that #S a ≥ 2 would induce that G is not a ground state.From this and (4.7) we deduce for the number of strata of a ground state G with span(s 0 ) V .
Estimate (4.6) is related to proving an isoperimetric inequality with respect to the l 1perimeter via slicing.We present a corresponding argument in the continuum setting in Appendix B.

The n 3/4 -Law
We close with a fluctuation estimate for minimizers.

Proof of Theorem 2.2
Clearly, it is enough to prove the statement for n ≥ n 0 for some n 0 ∈ N. We use the notation of the previous proof.In particular, we choose s 0 and l v as done before (4.5).As each x ∈ V belongs to exactly two strata, by (4.3) we find that l(s 0 ) = max s∈S l(s) ≤ 2 √ n .We start by noting that Indeed, if span(s 0 ) = V , we have l(s 0 ) • l v ≥ n.Otherwise, in view of (4.8), the span missed exactly one stratum, consisting of at most 2 √ n points.Now, by (4.4), (4.5), and (4.9) we compute for n sufficiently large Then, a short computation yields √ n − cn 1/4 ≤ l(s 0 ) ≤ √ n + cn 1/4 (4.10) for some universal c > 0 large enough.Using again (4.9) and l v ≤ l(s 0 ), we also find for a larger c > 0. In view of (4.8), we get that, up to a translation and up to one stratum, C n is contained in the rectangular subset of Z 2 defined by Thus, recalling (4.12), to conclude it now suffices to prove that By optimizing with respect to M we get M = (L d (E)) 1−1/d , and thus we conclude Proof of Lemma B. 2 We start by splitting the l 1 -perimeter into where ν E = ((ν E ) 1 , . . ., (ν E ) d−1 ) ∈ R d−1 .Introducing the function the coarea formula, see [31, (18.25)], implies where in the last step we used the fact that (1 − (ν E ) 2 d ) −1/2 ν E ∈ R d−1 is a unit normal to E t .On the other hand, using the notation (∂ * E) x := ∂ * E ∩ {(x , t) : t ∈ R} for x ∈ R d−1 , by slicing properties of BV -functions, we obtain where we used that for H 1

Fig. 2
Fig. 2 Configurations of minimal energy for different cardinality

Fig. 3
Fig. 3 Illustration of some strata in the bond graph of G. Degenerate strata are indicated by short segments crossing the atoms
.2)Once (4.2) is proven, we conclude as follows.First, (2.2) holds due to Lemma 3.7(ii) and(3.3).To characterize the equality case, we get from Lemma 3.7 that G = G o , that all bond lengths are 1, and all bond angles lie in {π/2, π, 3/2π}.This shows that each connected component (in the sense of graph theory) of G lies in a rotated and shifted version of Z 2 .If there existed more than one connected component, one could obtain a modified configuration with an additional bond.This contradicts minimality, and we therefore obtain V ⊂ Z 2 after a rigid motion.We now show (4.2).In the following, we write S in place of S(G o ) for simplicity.By Lemmas 4.1, 3.3(ii), and 3.7 we have that2#S = F bond (G o ) ≤ 2 2