Effect of periodic arrays of defects on lattice energy minimizers

Mathematical results for identifying the lattice ground states of interacting systems have recently attracted a lot of attention. Even though the ‘Crystallization Conjecture’ [16] – the proof of existence and uniqueness of a periodic minimizer for systems with free particles – is still open in full generality, many interesting results have been derived in various settings for showing the global minimality of certain periodic structures including the uniform chain $\mathbb{Z}$, the triangular lattice $\mathsf{A}_{2}$, the square lattice $\mathbb{Z}^{2}$, the face-centred cubic lattice $\mathsf{D}_{3}$ (see Fig. 1), as well as the other best packings $\mathsf{E}_{8}$ and the Leech lattice $\Lambda_{24}$ (see [12, 23] and references therein). Moreover, the same kind of investigation has been made for multi-component systems (e.g. in [29, 30, 35, 10, 36]) where the presence of charged particles yield to rich energetically optimal structures. These problems of optimal point configurations are known to be at the interface of Mathematical Physics, Chemistry, Cryptography, Geometry, Signal processing, Approximation, Arithmetic, etc. The point of view adopted in this work is the one of Material Science where the points are thought as particles or atoms.

In this paper, our general goal is to show mathematically how the presence of periodic arrays of charges (called here ‘defects’ in contrast with the initial crystal ‘atoms’) in a perfect crystal affects the 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 question to know the conditions on the added periodic distribution of defects and on the interaction energy in order to have conservation of the initial ground state structure. Furthermore, only very few rigorous results are available on minimization of charged structures among lattices (see e.g. our recent works [10, 9]).

We therefore assume the periodicity of our systems, and once we restrict this kind of problem to the class of (simple) lattices and radially symmetric interaction potentials, an interesting non-trivial problem is to find the minimizers of a given energy per point among these simple periodic sets of points, with or without a fixed density. In this paper, we keep the same kind of notations we have used in our previous works (see e.g. [14, 8, 10]). More precisely, for any $d\geq 2$ we called $\mathcal{L}_{d}$ the class of $d$-dimensional lattices, i.e. discrete co-compact subgroups or $\mathbb{R}^{d}$,

$$\mathcal{L}_{d}:=\left\{L=\bigoplus_{i=1}^{d}\mathbb{Z}u_{i}:\textnormal{$\{u_{1},...,u_{d}\}$ is a basis of $\mathbb{R}^{d}$}\right\},$$

and, for any $V>0$, $\mathcal{L}_{d}(V)\subset\mathcal{L}_{d}$ denotes the set of lattices with volume $|\det(u_{1},...,u_{d})|=V$, i.e. such that its unit cell $Q_{L}$ defined by

$$Q_{L}:=\left\{x=\sum_{i=1}^{d}\lambda_{i}u_{i}:\forall i\in\{1,...,d\},\lambda_{i}\in[0,1)\right\},$$

(1.1)

has volume $|Q_{L}|=V$. We will also say that $L\in\mathcal{L}_{d}(V)$ has density $V^{-1}$. The class $\mathcal{F}_{d}$ of radially symmetric functions we consider is, calling $\mathcal{M}_{d}$ the space of Radon measures on $\mathbb{R}_{+}$,

$$\mathcal{F}_{d}:=\left\{f:\mathbb{R}_{+}\to\mathbb{R}:f(r)=\int_{0}^{\infty}e^{-rt}d\mu_{f}(t),\mu_{f}\in\mathcal{M}_{d},|f(r)|=O(r^{-p_{f}})\textnormal{ as $r\to\infty$, $p_{f}>d/2$}\right\}.$$

When $\mu_{f}$ is non-negative, $f$ is a completely monotone function, which is equivalent by Hausdorff-Bernstein-Widder Theorem [3] with the property that for all $r>0$ and all $k\in\mathbb{N}$, $(-1)^{k}f^{(k)}(r)\geq 0$. We will write this class of completely monotone functions as

$$\mathcal{F}^{cm}_{d}:=\left\{f\in\mathcal{F}_{d}:\mu_{f}\geq 0\right\}.$$

For any $f\in\mathcal{F}_{d}$, we thus defined the $f$-energy $E_{f}[L]$ of a lattice $L$, which is actually the interaction energy between the origin $O$ of $\mathbb{R}^{d}$ and all the other points of $L$, by

$$E_{f}[L]:=\sum_{p\in L\backslash\{0\}}f(|p|^{2}).$$

(1.2)

Notice that this sum is absolutely convergent as a simple consequence of the definition of $\mathcal{F}_{d}$. We could 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 of parametrized potential such as $r^{-s}$ (see [17]).

One can interpret the problem of minimizing $E_{f}$ in $\mathcal{L}_{d}$ (or in $\mathcal{L}_{d}(V)$ for fixed $V>0$) as a geometry optimization problem for solid crystals where the potential energy landscape of a system with an infinite number of particles is studied in the restricted class of lattice structures. Even though the interactions in a solid crystal are very complex (quantum effects, angle-dependent energies, etc.), it is known that the Born-Oppenheimer adiabatic approximation used to describe the 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 we are deriving in this paper are invariant under rotations, all the results will be tacitly considered up to rotations.

Furthermore, studying such interacting systems with this periodicity constraint is a good method to keep or exclude possible candidates for a crystallization problem (i.e. with free particles). We are in particular interested in a type of lattice $L_{d}$ that is the unique minimizer of $E_{f}$ in $\mathcal{L}_{d}(V)$ for any fixed $V>0$ and any completely monotone potential $f\in\mathcal{F}_{d}^{cm}$. Following Cohn and Kumar [21] notion (originally defined among all periodic configurations), we call this property the universal optimality among lattices of $L_{d}$ (or universal optimality in $\mathcal{L}_{d}(1)$).

Only few methods are available to carry out the minimization of $E_{f}$. Historically, the first one consists to parametrize all the lattices of $\mathcal{L}_{d}(1)$ in an Euclidean fundamental domain $\mathcal{D}_{d}\subset\mathbb{R}^{\frac{d(d+1)}{2}-1}$ (see e.g. [44, Sec. 1.4]) and to study the variations of the energy in $\mathcal{D}_{d}$. It has been done in dimension 2 for showing the optimality of the triangular lattice $\mathsf{A}_{2}$ at fixed density for the Epstein zeta function [42, 28, 19, 27] and the lattice theta function [38] respectively defined for $s>d$ and $\alpha>0$ by

$$\zeta_{L}(s):=\sum_{p\in L\backslash\{0\}}\frac{1}{|p|^{s}},\qquad\textnormal{and}\qquad\theta_{L}(\alpha):=\sum_{p\in L}e^{-\pi\alpha|p|^{2}}.$$

(1.3)

In particular, a simple consequence of Montgomery’s result [38] for the lattice theta function is the universal optimality among lattices of $\mathsf{A}_{2}$ (see e.g. [4, Prop. 3.1]). Other consequences of the universal optimality of $\mathsf{A}_{2}$ among lattices have been derived for other potentials (including the Lennard-Jones one) [15, 4, 14, 7] as well as masses interactions [11]. Furthermore, new interesting and general consequences of universal optimality will be derived in this paper, including a sufficient condition 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 maximality of $\mathsf{A}_{2}$ in $\mathcal{L}_{2}(1)$ – and conjectured the same results in dimensions $d\in\{8,24\}$ for the lattices $\mathsf{E}_{8}$ and $\Lambda_{24}$ – for the alternating and centered lattice theta function respectively defined, for all $\alpha>0$, by

$$\theta_{L}^{\pm}(\alpha):=\sum_{p\in L}\varphi_{\pm}(p)e^{-\pi\alpha|p|^{2}},\quad\textnormal{and}\quad\theta_{L}^{c}(\alpha):=\sum_{p\in L}e^{-\pi\alpha|p+c_{L}|^{2}},$$

(1.4)

where $L=\bigoplus_{i=1}^{d}\mathbb{Z}u_{i}$, $\{u_{i}\}_{i}$ being a Minkowski (reduced) basis of $L$ (see e.g. [44, Sect. 1.4.2]), $\varphi_{\pm}(p):=\sum_{i=1}^{d}m_{i}$ for $p=\sum_{i=1}^{d}m_{i}u_{i}$, $m_{i}\in\mathbb{Z}$ for all $i$, and $c_{L}=\frac{1}{2}\sum_{i}u_{i}$ is the center of its unit cell $Q_{L}$. In particular, the alternate lattice theta function $\theta_{L}^{\pm}(\alpha)$ can be viewed as the Gaussian interaction energy of a lattice $L$ with an alternating distribution of charges $\pm 1$, which can be itself seen as the energy once we have removed $2$ times the union of sublattices $\cup_{i=1}^{d}(L+u_{i})$ from the original lattice $L$. This result shows another example of universal optimality – we will call it universal maximality – among lattices, i.e. the maximality of $\mathsf{A}_{2}$ in $\mathcal{L}_{2}(1)$ for the energies $E_{f}^{\pm}$ and $E_{f}^{c}$ defined by

$$E_{f}^{\pm}[L]:=\sum_{p\in L\backslash\{0\}}\varphi_{\pm}(p)f(|p|^{2}),\quad\textnormal{or}\quad E_{f}^{c}[L]:=\sum_{p\in L}f(|p+c_{L}|^{2}),$$

(1.5)

where $f\in\mathcal{F}_{d}^{cm}$. This kind of problem was actually our first motivation for investigating the effects of periodic arrays of defects on lattice energy minimizers, since removing two times the sublattices $2L+u_{1}$ and $2L+u_{2}$ totally inverse the type of optimality among lattices. Furthermore, this maximality result will also be used in Theorem 2.4, applied – in the general case of a universal maximizer $L_{d}^{\pm}$ for $E_{f}^{\pm}$ in any dimension where this property could be shown – for other potentials $\mathcal{F}_{d}\backslash\mathcal{F}_{d}^{cm}$ 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 programming bound that was successfully used for showing the best packing results in dimensions 8 and 24 for $\mathsf{E}_{8}$ and $\Lambda_{24}$ in [47, 22], as well as their universal optimality among all periodic configurations in [23]. As in the two-dimensional case, many consequences of these optimality results have been 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 a lattice $L\subset\mathcal{L}_{d}$ and $K\subset\mathbb{N}\backslash\{1\}$, a certain real number $a_{k}\neq 0$ of integer sublattices $kL$, $k\in K$, in the original lattice, and where the lattices $kL$ might be shifted by a finite number of lattice points $L_{k}:=\{p_{i,k}\}_{i\in I_{k}}\subset L$ for some finite set $I_{k}$. Writing

$$\kappa:=\{K,A_{K},P_{K}\},\quad K\subset\mathbb{N}\backslash\{1\},\quad A_{K}=\{a_{k}\}_{k\in K}\subset\mathbb{R}^{*},\quad P_{K}=\bigcup_{k\in K}L_{k},\quad L_{k}=\{p_{i,k}\}_{i\in I_{k}}\subset L,$$

(1.6)

the new energy $E^{\kappa}_{f}$ we consider, defined for $f\in\mathcal{F}_{d}$ and $\kappa$ as in (1.6) and such that the following sum on $K$ is absolutely convergent, is given by

$$E^{\kappa}_{f}[L]:=E_{f}[L]-\sum_{k\in K}\sum_{i\in I_{k}}a_{k}E_{f}[p_{i,k}+kL].$$

(1.7)

In particular, in the non-shifted case, i.e. $P_{K}=\emptyset$, then

$$E^{\kappa}_{f}[L]=E_{f_{\kappa}}[L],\quad\textnormal{where}\quad f_{\kappa}(r):=f(r)-\sum_{k\in K}a_{k}f(k^{2}r).$$

(1.8)

Since we are interested in the effects of defects on lattice energy ground states, we therefore want to derive conditions on $\kappa$ and $f$ such that $E_{f}$ and $E_{f}^{\kappa}$ have the same minimizers in $\mathcal{L}_{d}$ or $\mathcal{L}_{d}(V)$ for fixed $V>0$. In particular, we also want to know if the universal minimality among lattices of a lattice $L_{d}$ is conserved while removing or substituting integer sublattices. This a natural step for investigating the robustness of the optimality results stated in the previous section of this paper when the interaction potential is completely monotone or, for instance, of Lennard-Jones type. Furthermore, it is also the opportunity to derive new applications and generalizations of the methods recently developed in [4, 14, 9] for more ‘exotic’ ionic-like structures.

Replacing integer sublattices as described above can be interpreted and classified in two relevant cases in Material Science:

1. 1.

   If $a_{k}=1$, then removing only once the sublattice $kL$ from $L$ creates a periodic array of vacancies (also called periodic Schottky defects [45, Sect. 3.4.3]);

2. 2.

   If $a_{k}\neq 1$, then ‘removing’ $a_{k}$ times the sublattice $kL$ from $L$ creates a periodic array of substitutional defects (also called impurities), where the original lattice points (initially with charges $+1$) are replaced by points with ‘charges’ (or ‘weights’) $1-a_{k}\neq 0$.

In Figure 2, we have constructed three examples of two-dimensional lattices with periodic arrays of defects which certainly do not exist in the real world. In contrast, Figure 3 shows two important examples of crystal structures arising in nature: the Kagome lattice and the rock-salt structure. These examples are discussed further in Section 3.

While the substitutional defects case has different interpretations and applications in terms of optimal multi-component (ionic) crystals (see e.g. Section 3.2), the vacancy case is also of interest when we look for accelerating the computational time for checking numerically the minimality of a structure. Indeed, if the minimizer does not change once several periodic arrays of points are removed from all lattices, then a computer will be faster to check this minimality. This is of practical relevance in particular in low dimensions since the computational time of such lattice energies, which grows exponentially with the dimension, are extremely long in dimension $d\geq 8$ – even with the presence of periodic arrays of vacancies – and shows how important are rigorous minimality results in these cases.

Furthermore, from a Physics point of view, it is well-known (see e.g. [45]) that point defects play an important role in crystal properties. As explained in [1]: ‘Crystals are like people, it is the defects in them which tend to make them interesting’. For instance, they reduce the electric and 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, there is no perfect crystal in nature and it is then interesting and physically relevant to study optimality results for periodic systems with defects, in particular for models at positive temperature where the 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 called self-interstitial – i.e. the presence of extra atoms out of lattice sites – but they are known to be negligible (at least if they are of the same type than the solid’s atoms) compared to the vacancies when disorder appears, excepted for Silicon.

Plan of the paper. Our main results are presented in Section 2 whereas their proofs are postponed to Section 4. Many applications of our results are discussed in Section 3, including explicit 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 and Kumar in [21].

Before stating our results, notice that all of them are stated in terms of global optimality but could be rephrased for showing local optimality properties. This is important, in particular in dimensions $d=3$ where only local minimality results are available for $E_{f}$ (see e.g. [6]) and can be generalized for energies of type $E_{f}^{\kappa}$, ensuring the local stability of certain crystal structures.

We now show that the universal optimalities among lattices in dimension $d\in\{2,8,24\}$ proved in [38, 23] are not conserved in the non-shifted case once we only removed a single integer sublattice a positive number $a_{k}>0$ of times, whereas they are conserved when $a_{k}<0$.

We now present a sufficient condition on $P_{K}$ such that the triangular lattice is universally optimal in $\mathcal{L}_{2}(1)$ for $E_{f}^{\kappa}$. This result is based on our recent work [9] where we have proven the maximality of $\mathsf{A}_{2}$ in $\mathcal{L}_{2}(1)$ for the centred lattice theta functions, i.e. $L\mapsto\theta_{L+c_{L}}(\alpha)$, where $c_{L}$ is the center of the unit cell $Q_{L}$ (see also Remark 2.12).

We now give a general criterion that ensures the conservation of an universal optimizer’s minimality for $E_{f}^{\kappa}$.

The fourth point on Theorem 2.9 generalizes our two-dimensional result [4, Thm. 1.1] to any dimension and with possible periodic arrays of defects. It is an important result since only few minimality results for $E_{f}$ are available for non-completely monotone potentials $f\in\mathcal{F}_{d}\backslash\mathcal{F}_{d}^{cm}$, and this also the first result of this kind for charged lattices (i.e. when the particles are not of the same kind). Condition (2.3) has been used in dimension $d=2$ in [4, 7] for proving the optimality of a triangular lattice at fixed density for non-convex sums of inverse power laws, differences of Yukawa potentials, Lennard-Jones potentials and Morse potentials and we expect the same property to hold in higher dimension. In Theorem 2.17, we will give an example of such application in any dimension $d$ by applying the fourth point of Theorem 2.9 to Lennard-Jones type potentials. We now add a very important remark concerning the adaptation of the fourth point of Theorem 2.9 in the general periodic case, i.e. for crystallographic point packings (see [2, Def. 2.5]).

Using exactly the same arguments as the fourth point of Theorem 2.9, we show the following result which gives a sufficient condition on an interaction potential $f$ for a universal maximizer $L_{d}^{\pm}$ of $\theta_{L}^{\pm}(\alpha)$ to be optimal for $E_{f}^{\pm}$, where

$$\theta_{L}^{\pm}(\alpha):=\sum_{p\in L}\varphi_{\pm}(p)e^{-\pi\alpha|p|^{2}},\quad\textnormal{and}\quad E_{f}^{\pm}[L]:=\sum_{p\in L\backslash\{0\}}\varphi_{\pm}(p)f(|p|^{2}),$$

(2.4)

with $L=\bigoplus_{i=1}^{d}\mathbb{Z}u_{i}$, $\{u_{1},...,u_{d}\}$ being its Minkowski basis, and $\varphi_{\pm}(p)=\sum_{i=1}^{d}m_{i}$ for $p=\sum_{i=1}^{d}m_{i}u_{i}$, $m_{i}\in\mathbb{Z}$ for all $i$. Remark that $E_{f}^{\pm}=E_{f}^{\kappa}$ when $\kappa=\{2,\{2,....2\},\{u_{1},...,u_{d}\}\}$, $L=\bigoplus_{i=1}^{d}\mathbb{Z}u_{i}$. In particular, it holds for the triangular lattice $\mathsf{A}_{2}$ as a simple application of our main result in [9].

The rest of our results are all given in the non-shifted case $P_{K}=\emptyset$. It is indeed a rather difficult task to minimize the sum of shifted and/or non-shifted energies of type $E_{f}$. Very few results are available and the recent work by Luo and Wei [36] has shown the extreme difficulty to obtain any general result for completely monotone function $f$. Shifting the lattices by a non-lattice point which is not the center $c_{L}$ appears to be deeply more tricky in terms of energy optimization.

We remark that, since $\mathcal{F}_{d}^{cm}$ is not stable by difference, it is not totally surprising that Theorem 2.2 holds. Furthermore, identifying the largest space of all functions $f$ such that $E_{f}$ is uniquely minimized by $L_{d}$ in $\mathcal{L}_{d}(1)$ seems to be very challenging (see [14]). Therefore a natural question in order to identify a large class of potentials $f$ such that the minimality of an universal optimizer $L_{d}$ holds for $E_{f}^{\kappa}$ is the following: what are the completely monotone potentials $f\in\mathcal{F}_{d}^{cm}$ satisfying (2.2), i.e. such that $f_{\kappa}\in\mathcal{F}_{d}^{cm}$? The following corollary of Theorem 2.9 gives an example of such potentials, where we define, for $s>0$ and any $A_{K}=\{a_{k}\}_{k\in K}$, $K\subset\mathbb{N}\backslash\{1\}$,

$$\mathsf{L}(A_{K},s):=\sum_{k\in K}\frac{a_{k}}{k^{s}}.$$

(2.5)

Notice that the notation of (2.5) is inspired by the one of Dirichlet L-series that are generalizing the Riemann zeta function (see e.g. [20, Chap. 10]). For us, the arithmetic function appearing in a Dirichlet series is simply replaced by $A_{K}$ and can be finite.

In this subsection, we restrict our study to combinations of inverse power laws, since they are the building blocks of many interaction potentials used in molecular simulations (see e.g. [33]). Their homogeneity simplifies a lot the energy computations and allows us to give a complete picture of the periodic arrays of defects effects with respect to the values of $\mathsf{L}$ defined by (2.5).

In the following result, we show that the values of $\mathsf{L}(A_{K},2s)$ plays a fundamental role in the minimization of $E_{f}^{\kappa}$ when $f$ is an inverse power law.

Many applications of point 4. of Theorem 2.9 can then be shown for non-convex sums of inverse power laws, differences of Yukawa potentials or Morse potentials by following the lines of [4]. In this paper, we have chosen to focus on Lennard-Jones type potentials since it is possible to have a complete description of the effect of non-shifted periodic arrays of vacancies using the homogeneity of the Epstein zeta functions. It is also known that Lennard-Jones type potentials play an important role in molecular simulation (see e.g. [4, Sect. 6.3] and [33, Sect. 5.1.2]).

In our last results, we define the Lennard-Jones type potential by

$$f(r)=\frac{c_{2}}{r^{x_{2}}}-\frac{c_{1}}{r^{x_{1}}}\quad\textnormal{where}\quad(c_{1},c_{2})\in(0,\infty),\quad x_{2}>x_{1}>d/2,$$

(2.6)

which is a prototypical example of function where $\mu_{f}$ is not nonnegative everywhere, and a difference of completely monotone functions. We discuss the optimality of a universally optimal lattice $L_{d}$ for $E_{f}^{\kappa}$ with respect to the values of $\mathsf{L}(A_{K},2x_{i})$, $i\in\{1,2\}$ as well as the shape of the global minimizer of $E_{f}^{\kappa}$, i.e. its equivalence class in $\mathcal{L}_{d}$ modulo rotation and dilation (as previously defined in [14]).

From all our results, we conclude that is possible to remove or substitute several infinite periodic sets of points from all the lattices (i.e. an integer sublattices) and to conserve the already existing minimality properties, but only in a certain class of potentials or sublattices. Physically, it means that adding point defects to a crystal can be without any effect on its ground state if we assume the interaction between atoms to be well-approximate by a pairwise potential (Born model [46]) and the sublattices to satisfy some simple properties. We give several examples in Section 3 and our result 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 energies discussed in this paper in the class of (potentially shifted) lattices $L\backslash 2L$ where $L\in\mathcal{L}_{2}(1)$. This is, as far as we know, the first results of this kind for the Kagome lattice. We also believe that the results and techniques derived in this paper can be applied to other ionic crystals and other general periodic 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 to simplify both numerical investigations – leading to a shorter computational time – and mathematical estimates for these energies. We voluntarily did not explore further this fact since it is only relevant in low dimensions because the computational time of such lattice sums is exponentially growing and gives unreachable durations in dimension $d\geq 4$ for computing many values of the energies, especially in dimensions $d\in\{8,24\}$ 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 ($\mathbb{Z}^{3}$, $\mathsf{D}_{3}$ and $\mathsf{D}_{3}^{*}$) for the Epstein zeta function, the lattice theta function and the Lennard-Jones type energies. We believe that our result will find other very interesting applications in dimension 3 once global optimality properties will be shown for the lattice theta functions and the Epstein zeta functions (Sarnak-Strömbergsson 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 universal optimality among lattices. An interesting problem would be to find a necessary condition for this robustness. Furthermore, we can also ask the following question: is it enough to study this kind of minimization problem in a (small) ball centred at the origin? In other words: can we remove all the points that are far enough from $O$ and conserving the minimality results? Numerical investigations and Figure 5 tend to confirm this fact, and a rigorous proof of such property would deeply simplify the analysis of such lattice energies.

Being the vertices of a trihexagonal tiling, this structure – wich is actually not a lattice as we defined it in this paper – that we will write $\mathsf{K}:=\mathsf{A}_{2}\backslash 2\mathsf{A}_{2}$ is the difference of two triangular lattices of scale ratio $2$ (see Fig. 4). Some minerals – which display novel physical properties connected with geometrically frustrated magnetism – like jarosites and herbertsmithite contain layers having this structure (see [37] and references therein). We can therefore apply our results of Section 2 with $\kappa=\{2,1,\emptyset\}$ or $\kappa=\{2,1,u_{1}+u_{2}\}$. The following optimality results for $E_{f}$ in the class of lattices of the form $L\backslash 2L$ (or $L\backslash(2L+u_{1}+u_{2})$ in the shifted case) are simple consequences of our results combined with the universal optimality of $\mathsf{A}_{2}$ among lattices proved by Montgomery in [38]:

1. 1.

   Universal optimality of $\mathsf{K}$. Applying Theorem 2.4 to $\kappa=\{2,1,u_{1}+u_{2}\}$, it follows that for all $f\in\mathcal{F}_{2}^{cm}$, the shifted Kagome lattice $\mathsf{K}+(1/2,-\sqrt{3}/2)$ (see Fig. 4) is the unique minimizer of $E_{f}$ among lattices of the form $L\backslash(2L+u_{1}+u_{2})$, where $L=\mathbb{Z}u_{1}\oplus\mathbb{Z}u_{2}\in\mathcal{L}_{2}(1)$.

2. 2.

   Minimality of $\mathsf{K}$ at all densities for certain completely monotone potentials. A direct consequence of Theorem 2.9 is the following. For any completely monotone function $f\in\mathcal{F}_{2}^{cm}$ such that $d\mu_{f}(t)=\rho_{f}(t)dt$ and $\rho_{f}$ is an increasing function, the Kagome lattice $\mathsf{K}$ is the unique minimizer of $E_{f}$ among all the two-dimensional sparse lattices $L\backslash 2L$ where $L\in\mathcal{L}_{2}(1)$. This is the case for instance for $f=f_{\sigma,s}$ defined in Example 2.14, including the inverse power laws and the Yukawa potential.

3. 3.

   Optimality at high density for Lennard-Jones interactions. Applying Theorem 2.17, we obtain its optimality at high density: if $f(r)=c_{2}r^{-x_{2}}-c_{1}r^{-x_{1}}$, $x_{2}>x_{1}>1$ is a Lennard-Jones potential, then the unique minimizer of $E_{f}$ at high density among all the two-dimensional sparse lattices $L\backslash 2L$, where $L$ has fixed density, has the shape of $\mathsf{K}$.

4. 4.

   Global optimality for Lennard-Jones interactions with small exponents. Furthermore, using Theorem 2.17 and [4, Thm. 1.2.2] (see also Remark 2.19), we obtain the following interesting result in the Lennard-Jones potential case: if $\pi^{-x_{2}}\Gamma(x_{2})x_{2}<\pi^{-x_{1}}\Gamma(x_{1})x_{1}$, then the unique global minimizer of $E_{f}$ among all the possible sparse lattices $L\backslash 2L$ has the shape of $\mathsf{K}$.

These are the first minimality results for $\mathsf{K}$ in a class of periodic configurations. We recall that a non-optimality result has also been derived by Grivopoulos in [31] for Lennard-Jones potential in the case of free particles, and different attempts have been made for obtaining numerically or experimentally a Kagome structure as an energy ground state (see e.g. [32, 41, 26]).