Stability of classical planar dipole matter on regular and aperiodic lattices

Abstract

The stability of matter is a historical problem that tackles the linearity of the bulk energy with the total number of particles M. The classical and quantum variants have been proved using mostly Coulomb interaction between electrons and nuclei, either fixed or submitted to thermal fluctuation. The classical dipole−dipole interaction is addressed here as the sole energy on regular tilings. We prove that the system on any regular (periodic) grid is always stable. The aperiodic or quasicrystal instance is conjectured and numerically illustrated for the particular cases of the Penrose P2 and the recently discovered hat monotiles.

Graphical abstract

Keywords

dipole−dipole interaction / periodic lattices / aperiodic lattices / stability of matter

1 Introduction

“We show that not only individual atoms but matter in bulk would collapse into a condensed high-density phase. The assembly of any two macroscopic objects would release energy comparable to that of an atomic bomb.” These words by Dyson [1] certify the implications of the proof by Lenard and himself in 1967 [2, 3] that certified the stability of ordinary matter. Matter has to be stable, meaning that the ground state is bounded from below linearly with the particle number. Implicitly in that statement there was, although subtly, the Pauli exclusion principle for fermions. Dyson and Lenard’s proof was eminently mathematically-based, and hard to follow. Lieb and Thirring provided years later a shorter, physically-motivated proof [4, 5]. In plain words, since the energy is an extensible quantity, the sum of two chunks of matter E_1=A\cdot N_1^a and E_2=A\cdot N_2^a has to be less or equal to E_T=A \cdot (N_1+N_2). Hence, in order for the system not to blow up, a\le 1. This is more or less the intuitive mathematical statement for stability. One of course has to bear in mind that there are many interactions at play here, either classical or quantal, but in all cases one basically deals with electrostatic forces.

What these proofs had in common was the consideration of matter formed by positive and negative components, that is, involving Coulomb forces. Further progress considered additional ones, in the classical and quantum realms [6].

Previously, in 1966, Fisher and Ruelle [7] discussed the issue of stability of many particle systems, although without the strength of posterior works. In a brief comment regarding the dipole−dipole (or tensor) force, they mentioned, “As it stands, however, the self-energy is divergent but we may clearly obtain a stable system if the dipole moment m_{\alpha } is distributed with some density {\mathfrak{m}}_{\alpha }(\mathit{r}). The smoothness conditions on {\mathfrak{m}}_{\alpha }(\mathit{r}) are, however, more stringent than on the charge density as is natural since a distribution {\rho }_{\alpha }(\mathit{r}) proportional to the divergence of \mathfrak{m}(\mathit{r}) will have the same electrostatic field.”

We shall tackle a variant of this problem in the present contribution. Fisher and Ruelle’s concern regarding the distribution of dipole moments is certainly founded in the case of classical interacting dipoles in effective two-dimensional manifolds, and in the thermodynamic limit. Dipolar matter does not “crystallize” unless a predefined structure is considered or some highly localized density {\mathfrak{m}}_{\alpha }(\mathit{r}) is considered.

The physics of aperiodic lattices has emerged in the last decades due to the discovery of quasicrystals endowed with impossible symmetries from the crystallographic point of view [8–12]. Physically, they exhibit a rich variety of properties beyond those observed in periodic crystals [13–15]. A prototype example is the Penrose tiling, although quantum approaches are much more demanding that classical studies and only a few number of sites are considered (see, for instance, Ref. [16]). To be specific, they are compared in the following:

Periodic lattices

　● Periodic lattices exhibit a regular, repeating pattern of atoms or molecules in space, occurring at regular intervals in one, two, or three dimensions.

　● They possess translational symmetry, meaning the lattice appears the same when shifted by certain distances.

　● These are periodic lattices that fill space without gaps or overlaps by repeating a unit cell. There are 14 types of Bravais lattices in three dimensions.

　● Most crystalline materials, such as metals and minerals, have periodic lattices.

　● Produce sharp diffraction peaks corresponding to the regular spacing of atoms in their diffraction patterns.

Aperiodic lattices

　● Aperiodic lattices lack a regular, repeating pattern. The arrangement of atoms or molecules does not repeat at regular intervals.

　● These lattices do not possess translational symmetry.

　● The aforementioned quasicrystals and some modulated structures are examples of aperiodic lattices.

　● Produce diffraction patterns with sharp peaks that do not correspond to a simple lattice structure, indicating a more complex order.

　● Can exhibit superior mechanical properties compared to periodic lattices, due to enhanced mechanical rigidity due to their unique structural arrangements, isotropic elastic properties, and superior energy absorption capabilities.

The main differences are: (i) order vs. disorder: periodic lattices have a high degree of order, while aperiodic lattices have a more disordered structure; (ii) symmetry: periodic lattices have translational symmetry, whereas aperiodic lattices do not, and (iii) diffraction: periodic lattices produce simpler and more regular diffraction patterns compared to aperiodic lattices. One has to bear in mind that these properties hold regardless of the interaction considered, for they are exclusively based on topological properties.

Recently, Smith et al. [17, 18] discovered the first example of a single, simply connected tile that covers the plane aperiodically. Obviously, physical interest in the system immediately emerged [19, 20].

Considerable progress has occurred in recent years both experimentally and theoretically with regards to the description of dipolar systems with large dipole moments [21–23]. In the low-temperature regime, it is plausible to consider the formation of classical crystals of dipolar particles. Systems of dipoles whose orientation is perpendicular to the plane of motion have been considered in the past, either classically [24–27] or quantum mechanically [28–33].

Depending on several quantum effects, dipolar interactions can describe unexpected properties in helium [34], as well as accounting for particular effects observed in magnetic colloids [35, 36]. It is well-known that in most realistic cases, the dipole−dipole interaction is weak compared to other interactions or thermal fluctuations. Its strength is usually measured by the parameter \lambda = \frac{{\mu }_0{\mu }^2}{4\pi k_{B}T{a}^3}, with a being the lattice constant and \mu the magnetic dipole moment. Due to the typical length scales in ferrofluids at room temperatures, the value of \lambda never exceeds 5. Thus, we have assumed throughout the present contribution that we can work in the regime of strong dipole moments, which in turn are responsible for the structural stability and ordering of matter.

Luttinger and Tisza [39] treated in their seminal paper the formation of crystals composed solely by dipoles. In their work, they carried out an elegant method for finding the entire energy per dipole spectra in the thermodynamic limit for several three-dimensional structures. This work paved the way for numerical and analytical studies of other systems that were amenable to a classical treatment, either in two or three dimensions.

Several magnetic materials consisting of dipoles owe their properties to the specific nature of the dipole-dipole interaction. Noteworthy, we shall work in the regime of strong dipole moments where a classical treatment is possible.

The present work is divided as follows. In Section 2, from the functional form of the dipole tensor form we derive a simple bound for the minimum energy, irrespective of the dipoles’ orientation. This bound shall depend linearly on the total number of dipoles M. The numerical results for the ground state of two previously unstudied kisrhombille lattices will be performed. In Section 3, we shall study the energetics of finite aperiodic sets of vertices from the Penrose P2 tiling and the first monotile or Ein-Stein ever found, the hat monotile. Based on those results, we shall conjecture the stability of aperiodic dipolar matter in 2D. Finally, some conclusions will be drawn.

2 Periodic dipolar matter

We consider a two-dimensional system formed by dipoles which are assumed to be identical and have a given dipole moment magnitude of \mu. The dipoles have all initially different random orientations while fixed at their sites. The direction of each dipole moment is given in terms of a polar angle, \theta. The dipole moment itself, \mathit{m} is then described in 2D by a vector with two components of the form \mathit{m}=\mu (\cos \theta \sin \theta {)}^{\mathrm{T}}. The interaction energy between any two dipoles, {\mathit{m}}_u and {\mathit{m}}_v localized, respectively, at positions {\mathit{r}}_u and {\mathit{r}}_v is given by

(1)E_{u,v}=D({\displaystyle \frac{{\mathit{m}}_u\cdot {\mathit{m}}_v}{r_{uv}^3}} -{\displaystyle \frac{3\left({\mathit{m}}_u\cdot {\mathit{r}}_{uv}\right)\left({\mathit{m}}_v\cdot {\mathit{r}}_{uv}\right)}{r_{uv}^5}}),

where {\mathit{r}}_{uv} is the vector between the two dipoles u and v, r_{uv}=|{\mathit{r}}_{uv}| is their separation distance. The constant D is either \frac{{\mu }_0}{4\pi } (magnetic dipoles) or \frac{1}{4\pi {ϵ}_0} (electric dipoles). Unless otherwise stated, all energies shall be given in units of D{\mu }^2/a^3, where \mu is the dipole magnitude and a is the lattice constant.

The classical extremal energy states of a magnetic dipole system (either minimum or maximum) is one of equilibrium in which no torque should act on any given dipole. It is then expected that the system of interacting magnetic dipoles has its total energy bounded from below and from above. We shall focus on the case of minimum energy.

Let us derive the lowest possible energy value that a system of dipoles can possess, regardless of their orientations.

Lemma: Given a system of N identical dipoles \mathit{\mu } in 2D placed at positions {\mathit{r}}_i =(x_i,y_i), for i=1,\cdots ,N, its minimal energy E_{min} (in units of D{\mu }^2/a^3) fulfills the following bound

(2)

where r_{ij} is the distance between positions {\mathit{r}}_i and {\mathit{r}}_j. The corresponding proof can be found in the Appendix.

Result (2) is rather intuitive for it considers the minimum energy when all dipoles are pairwise aligned and parallel. The bound (2) is a particular instance of a general mathematical object, namely, the Rietz kernel. Now, proving the linearity of the 1/r_{ij}^3 with respect to the total number of particles M is tantamount as showing that the minimum energy configuration will also possess a lower energy bound of that kind, tightening it.

Theorem: All planar periodic matter composed by classical interacting dipoles at their vertices is stable.

Proof: A lattice is a collection of points of the form \mathrm{\Lambda } =A{\mathbb{Z}}^d for a fixed full-rank matrix A. Any Riesz kernel 1 /r^s with s>d is summable on every lattice in {\mathbb{R}}^d, meaning

(3)

The exact value of the sum depends on the lattice. The function s\mapsto (\text{value of the sum} ) is similar to Riemann’s \zeta-function in 1-dimensional case. For d=2 it is known that for the square lattice,

(4){\displaystyle \sum _{0\ne x\in \mathrm{\Lambda }}{\displaystyle \frac{1}{|x{|}^s}}=4\zeta \left({\displaystyle \frac{s}{2}}\right)\beta \left({\displaystyle \frac{s}{2}}\right),}

where \zeta is the classical Riemann’s \zeta-function, and \beta is the Dirichlet beta function.

Because lattices are translation-invariant, a double sum over a square n\times n tile can be estimated above by extending the finite sums to the entire lattice:

(5)\begin{array}{rl}& \sum _i\sum _{j:j\ne i}\frac{1}{r_{ij}^s}\le \sum _i\sum _{0\ne x\in \mathrm{\Lambda }}\frac{1}{|x{|}^s}\\ & =4M\zeta \left(\frac{s}{2}\right)\beta \left(\frac{s}{2}\right)=:2C_{{\mathbb{Z}}^2}(s)M,s>2.\end{array}

We take half of this sum to have summation over only. For a general periodic lattice \mathrm{\Lambda }, the same argument gives

(6)

Alternatively, we can proceed as follows. The energy of two interacting dipoles is a quadratic form in the moments and has the lowest eigenvalue -2D/r^3 where r is the distance. Hence the energy is bounded below by

(7)-2D{\displaystyle \sum _{x,y\in M}{\displaystyle \frac{1}{r_{xy}^3}}\ge -2D{\displaystyle \sum _{x\in M}{\displaystyle \sum _{y\in L}{\displaystyle \frac{1}{r_{xy}^3}}.}}}

Now the lattice can be written as m{\mathit{e}}_1+n{\mathit{e}}_2 where {\mathit{e}}_1 ,{\mathit{e}}_2 are the vectors defining the lattice. They do not have to be unit vectors. Now the sum

(8)

An elementary way to see this is to consider the quadratic form |m{\mathit{e}}_1+ n{\mathit{e}}_2|=m^2|{\mathit{e}}_1{|}^2+n^2|{\mathit{e}}_2{|}^2+2mn{\mathit{e}}_1\cdot {\mathit{e}}_2 \ge {\ell }^2(m^2 +n^2) where {\ell }^2 is the lowest eigenvalue of this quadratic form. This eigenvalue is not zero since otherwise {\mathit{e}}_1 and {\mathit{e}}_2 have to line up.

Hence, we get as an upper bound on the sum

(9){\displaystyle \frac{1}{{\ell }^3/2}}{\displaystyle \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\displaystyle \frac{1}{(n^2+m^2{)}^{3/2}}},}}

which is finite. ■

The previous argument, based on inequality (2), is only valid for two-dimensional systems (for 1/ r^3 diverges logarithmically with M). Now, finding the constant A such that

(10)

as M\to \mathrm{\infty }, that saturates the lower bound in E> E_{min}> -C_{\mathrm{\Lambda }}(s=2) M is not a simple task. It requires the computation of lattice sums, due to the slow converge of this kind of infinite sums. Thus, for a different set of periodic lattice points, we shall retrieve a unique constant A.

In the particular case of the kisrhombille lattice, that is the substrate on which the hat monotile is grown, a Levenberg−Marquardt non-linear regression [37] for using S_M=a_1 M+a_2{M}^{1/2}+a_3{M}^{1/4}+a_4{M}^{1/8} returns the value a_1= 18.585617\pm 0.0003646 (see online repository [38]). In other words, the minimum energy of the kisrhombille lattice (in units of D{\mu }^2/a^3) is bounded linearly from below as E_{\mathrm{m}\mathrm{i}\mathrm{n}}>-2\times 18.585M.

Now, the total energy of a system of dipoles in terms of (1) can also be cast in the form of the Hamiltonian

(11)\begin{array}{rl}& H=\frac{1}{2}{\mathcal{C}}_E\sum _{i\ne j}{\mathit{m}}_i^{\mathrm{T}}{\mathit{J}}_{ij}{\mathit{m}}_j,\\ & {\mathit{J}}_{ij}=\frac{1}{\Vert {\mathit{r}}_{ij}{\Vert }^3}\left(\mathit{I}-3\frac{{\mathit{r}}_{ij}\times {\mathit{r}}_{ij}}{\Vert {\mathit{r}}_{ij}{\Vert }^2}\right).\end{array}

The quadratic form (11) is particularly suitable for obtaining the spectrum of a system of identical dipoles using the Luttinger−Tisza method [39] under the assumption that the minimum energy configuration exhibits translational symmetry. If T(i) denotes the points generated from i with discrete translations belonging to the T symmetry group the mentioned symmetry corresponds to {\mathit{m}}_i={\mathit{m}}_{i^{\prime }} for all i^{\prime }\in T( i). The system can be split into identical cells and thus limit the summation to one single cell. Therefore, the energy per dipole can be expressed as

(12)E={\displaystyle \frac{1}{2n}}{\displaystyle \sum _{i,j=1}^n{\mathit{m}}_i^{\mathrm{T}}{\mathit{A}}_{ij}{\mathit{m}}_j,}

where n is the number of dipoles per cell, and {\mathit{A}}_{ij} are (symmetric) matrices defined by

(13){\mathit{A}}_{ij}={\displaystyle \sum _{j^{\prime }\in T(j),j^{\prime }\ne i}{\mathit{J}}_{i{j}^{\prime }}.}

The energy per dipole finally reads as

(14)E={\displaystyle \frac{1}{2n}}{\hat{\mathit{m}}}^{\mathrm{T}}\hat{\mathit{A}}\hat{\mathit{m}}.

The method involves the solution of an eigenvalue problem of the nd dimensional matrix \hat{\mathit{A}} (d is the dimension of the dipole), with {\lambda }_k being the eigenvalues and {\hat{\mathit{x}}}_k the corresponding (orthogonal) eigenvectors of the system, with \Vert {\hat{\mathit{x}}}_k\Vert =\sqrt{n}. The final expression for the energy reads as

(15)E={\displaystyle \frac{1}{2}}{\displaystyle \sum _{k=1}^{nd}{\lambda }_k{a}_k^2,}

where a_k denotes the components of \hat{\mathit{m}} in the base \{{\hat{\mathit{x}}}_k\}.

Two conditions must be satisfied for i=1,\cdots ,n, namely,

(16)\Vert {\displaystyle \sum _{k=1}^{nd}{a}_k{\mathit{x}}_k^i}\Vert =\mu ,{\displaystyle \sum _{k=1}^{nd}{a}_k^2={\mu }^2.}

Within the framework of the Luttinger−Tisza method, these two constraints are known as the strong and the weak conditions, respectively. We can thus obtain the minimum energy per dipole as E_{min}=1/2{\lambda }_{min}{\mu }^2, where {\lambda }_{min} denotes the smallest eigenvalue of \hat{\mathit{A}}.

There exists an alternative to the previous Luttinger−Tisza method, derived by us [40–42]. For a certain set of positions \{{\mathit{r}}_{\mathbf{1}},{\mathit{r}}_{\mathbf{2}},\cdots ,{\mathit{r}}_{\mathit{M}}\} corresponding to M identical dipoles (all of them with the same magnitude \mu), one resorts to a minimization procedure for its total dipole−dipole interaction energy. We chose the well-known simulated annealing [43] (SA) method. This Monte Carlo method is able to escape from local minima in the space of M variables corresponding to 2D. For a finite sample of N dipoles it is then easy to infer the minimum energy configuration. The previous overall optimal configuration - sample growing scheme, constitutes a easy-to-implement alternative to Luttinger’s because (i) no periodic boundary conditions are imposed, nor (ii) the computation of Ewald sums is required for obtaining the minimum energy per dipole in the limit M\to \mathrm{\infty }.

The application of the previous numerical schemes on two brand new periodic lattices is carried our for the hexagonal kisrhombille (HK) and the same one without the central vertex (HKC). The application of this Levenberg−Marquardt non-linear regression [37] provides an extremely precise value for the bulk value, and sets a correlation matrix between parameters that approaches an almost perfet agreement with the proposed functional form. See the Appendix for details.

The results are the following:

　● HK, E_{\mathrm{m}\mathrm{i}\mathrm{n}}/M=-14.19738\pm 0.0001,

　● HKC, E_{\mathrm{m}\mathrm{i}\mathrm{n}}/M=-16.26061 \pm 0.00016.

These two lattices display a non-trivial result that does not occur in pure Coulomb systems: the lattice with the unit cell with one less dipole is considerably more bounded. These two cases will be relevant for the next section.

3 Aperiodic dipolar matter

The case of aperiodic tilings has drawn the attention of physicists and mathematicians since the discovery of quasicrystals. Incidentally, the recent discovery of an aperiodic monotile or “Ein-Stein” (a shape that admits tilings of the plane) occurs on a lattice that is periodical. Thus, one can find an aperiodic tiling even with a regular set of points. This result is completely new. All previous known aperiodic tilings needed at least two elements (darts and kites for Penrose P2), and possessed some sort of geometry embedded (five-fold for P2). However the hat tiling does not display such rotational symmetry. However, the substrate upon which is constructed in indeed a periodic one, namely, the HK one (and the HKC, for that matter).

In Fig.1 we show a detail of the (1,\sqrt{3}) hat tiling. This nomenclature means that 1 and \sqrt{3} are the lengths of the sides of the hat, although we shall employ 2(1,\sqrt{3} ) because we take the side of the hexagon in HK and HKC to be unity. Note that each hat disregards two vertices from HK. Thus, although we cannot employ the previous numerical tools from periodic lattices, and given the tensor form of the dipole force, we may venture to ascertain that two less vertices might well increase (i.e., less bounded) the total ground state energy as compared to both HK and HKC. Indeed, after numerical minimizing the total energy for M=537 dipoles, E_{\mathrm{m}\mathrm{i}\mathrm{n}}/M=-10.69. Taking into account finite-size effects, we heuristically conclude that the true value shall not be smaller than -13, still far from HK and HKC (recall that the bound for HK is - 2\times 18.585).

The Penrose P2 tiling is shown in Fig.2. We have considered a finite set of 1000 vertices, with E_{\mathrm{m}\mathrm{i}\mathrm{n}}/M=-4.59. The rhombus that forms the tiling (dart+kite) has unit edge length. Remarkably enough, even though having adjacent vertex at smaller distances, the energy per dipole seems much higher (less bounded) that for the hat monotile or HK/HKC. One might be tempted to change the inter-spacing of the overall lattice to investigate the changes. Of course that would imply considerable amount of computational work for each case. However, and luckily for us, that is not required because all energies, even with difference sizes, shall scale by the same geometrical factor. Distorting the lattices is not an option and would make the overall analysis less straightforward and considerably more involved.

The concomitant equilibrium configuration of dipoles clearly displays a patchwork of five-fold flower petals and stars. As opposed to the hat monotile, in the case of the Penrose P2 one we have no frame of reference to compare with. Physically, it is most likely that bound (2) also applies to aperiodic lattices by virtue of successive coarse-grained approaches. However, a mathematical proof is lacking.

For the sake of completeness, we have numerically performed the Fourier transform of both samples corresponding to the previous aperiodic lattices. The results are depicted in Fig.3. The spread of intensity in the Fourier transform indicates the presence of various frequency components in the original image, reflecting the complex and aperiodic nature of the lattice structure (Penrose). Due to the overall angular dependency of the dipole tensor, the energies per particle in equilibrium are considerably different in both cases.

In the light of the previous results, we formulate the following:

Conjecture: Classical matter interacting solely by dipole-dipole forces in any aperiodic lattice \mathrm{\Lambda } is unstable for dimensions D>2. Additionally, for D=2 the bound

(17)

holds for averaged pairwise distances {\rho }_{ij}.

4 Conclusions

The issue of two-dimensional matter formed entirely by classic dipoles has been considered. The stability of these systems is proved for any periodic lattice by means of a lower bound that increases linearly with the size of the system in the thermodynamic limit. We have also considered two aperiodic lattices, namely, the Penrose P2 and the recently discovered hat monotile. On the grounds of the unique properties of the latter one, we conjecture, based on numerical computations, that all planar aperiodic dipolar matter is also stable.

5 Declarations

The authors declare that they have no competing interests and there are no conflicts.

6 Appendix A: Proof of the lemma

In this appendix, we provide the general bound that allows us to eventually set the stability of the dipolar matter in two-dimensions. Let us first rewrite the expression for the total energy from to \frac{1}{2}\sum _{i\ne j}, where a double-counting appears whilst we expand all terms in arising from scalar products in polar coordinates. Rearranging the terms, we have 1/r_{ij}^3 times

(18)\begin{array}{rl}& \cos {\theta }_i[\cos {\theta }_j-3\frac{x_i-x_j}{r_{ij}}(\cos {\theta }_j\frac{x_i-x_j}{r_{ij}}+\sin {\theta }_j\frac{y_i-y_j}{r_{ij}})]\\ & +\sin {\theta }_i[\sin {\theta }_j-3\frac{y_i-y_j}{r_{ij}}(\cos {\theta }_j\frac{x_i-x_j}{r_{ij}}+\sin {\theta }_j\frac{y_i-y_j}{r_{ij}})].\end{array}

It is apparent from the previous expression that we have made explicit the separation between dipoles i and j. Let us define C_{ij} \equiv (\cos {\theta }_j \frac{x_i-x_j}{r_{ij}} +\sin {\theta }_j\frac{y_i-y_j}{r_{ij}}), and the other two quantities A_{ij}\equiv \cos {\theta }_j-3\frac{x_i-x_j}{r_{ij}}{C}_{ij} and B_{ij} \equiv \sin {\theta }_j-3\frac{y_i-y_j}{r_{ij}}{C}_{ij}. Notice that A_{ij}^2+ B_{ij}^2= 1+3C_{ij}^2. With these definitions, the energy looks like

(19){\displaystyle \frac{1}{2}}{\displaystyle \sum _{i\ne j}{\displaystyle \frac{1}{r_{ij}^3}}(\cos {\theta }_i{A}_{ij}+\sin {\theta }_i{B}_{ij}).}

Let us recall the following relation: max{}_{\alpha }(\cos \alpha X+\sin \alpha Y)=\sqrt{X^2+Y^2}. We shall now take the absolute value for the total energy | E|, in such a way that we optimize the expression with respect to {\theta }_i irrespective of the other j terms. By doing so, we obtain

(20)\begin{array}{rl}& |E|\le \frac{1}{2}\sum _{i\ne j}\sqrt{A_{ij}^2+B_{ij}^2}\\ & =\frac{1}{2}\sum _{i\ne j}\sqrt{1+3[\cos {\theta }_j\frac{x_i-x_j}{r_{ij}}+\sin {\theta }_j\frac{y_i-y_j}{r_{ij}}{]}^2}.\end{array}

Carrying out the optimization with respect to {\theta }_j, and recalling that max{}_{{\theta }_j}{C}_{ij}=\sqrt{(\frac{x_i-x_j}{r_{ij}}{)}^2+(\frac{y_i-y_j}{r_{ij}}{)}^2}=1, we add a new layer in the inequality. Finally, we obtain

(21)

which concludes the proof of the lemma.

7 Appendix B: Numerical results for the ground state energy for the kisrhombille lattice with and without the central vertex

The kisrhombille tiling with all the vertices populated by dipoles constitutes the base for the hat ein-stein or monotile. This lattice is rather particular for the equilibrium configuration depends on the number of dipoles N (a simmilar instance occurs in the kagomé lattice). As shown in (a), two dipoles in the unit cell have the same slight {\theta }_c deviation (in degrees). For N=75,{\theta }_c=5.83906; N=51686,{\theta }_c=4.22; N=207773,{\theta }_c=4.14; N=466459,{\theta }_c=4.08. Thus, we have assumed {\theta }_c\approx 4 for N\to \mathrm{\infty }. Likewise, for the kisrhombille tiling with the central vertex removed [(b)], we have N=15351,{\theta }_c=7.642; N=61903,{\theta }_c=7.986; N=173173,{\theta }_c=8.1; N=388759,{\theta }_c=8.23. Therefore, we take {\theta }_c^{\mathrm{\prime }}\approx 8.3 for N\to \mathrm{\infty }. The final calculation retrieves an energy per dipole [in unit of C{\mu }^2/a^3)] in the kisrhombille lattice of −14.1977382 \pm 0.0001421. The energy per dipole in the kisrhombille lattice without the central vertex is of −16.2611647 \pm 0.0002419. The corresponding data can be found in an online repository [38].

8 Appendix C: Generation of the Penrose P2 and the hat monotile lattices

This section is more formal in content and is aimed at providing more rigor in how to generate the points present in the two aperiodic structures that we have studied. Be that as it may, we nevertheless provide a set of generated points for the interested reader.

The Penrose P2 lattice or any other one requires an efficient means of generating the tiling or, in a more physical way, their set of vertices. In crystallography the shaping of crystalline and quasicrystalline states of matter is a subject of importance. An approach to the modeling of crystal formation, based on successive encirclements in a periodic tiling of space into polyhedra [44] was generalized in Refs. [45, 46] to arbitrary adjacency graphs, and the concept of the form of layer-by-layer growth was precisely defined.

The equivalent layer-by-layer growth of vertex graph of the Penrose tiling was performed in Ref. [47], and is the one we shall follow. It corrects the original Baake et al. procedure [48–50]. This method consists of an approach where the set of Penrose tiling vertices is given in the form of the set {\mathrm{\Lambda }}_W=\{{\pi }_1(x):x\in L,{\pi }_2(x)\in W\}, where L is an integer lattice in {\mathbb{Z}}^4 with coordinates (h,j ,k,l), {\pi }_1 is the projection of the lattice L onto the complex place \mathbb{C}, and {\pi }_2 is another projection of L onto A=\mathbb{C}\times C_5 (C_5 is a fifth-order cyclic group). Given \zeta = e^{\frac{2\pi i}{5}} the solution to z^5=1, the projections {\pi }_1 and {\pi }_2 are given by

(22)\begin{array}{rl}& {\pi }_1((h,j,k,l))=h+j\zeta +k{\zeta }^2+l{\zeta }^3\\ & {\pi }_2((h,j,k,l))=(h+j{\zeta }^2+k{\zeta }^4+l\zeta ,(h+j+k+l)\mathrm{mod}5).\end{array}

The window W\subset A is an aggregation of five sets W=\bigcup _{i=0}^4( {\mathrm{\Omega }}_i ,i), with {\mathrm{\Omega }}_0 =\{0 \},{\mathrm{\Omega }}_1=P, {\mathrm{\Omega }}_2=- \varphi P,{\mathrm{\Omega }}_3=\varphi P ,{\mathrm{\Omega }}_4= -P. P is the pentagon defined by \{1, \zeta ,{\zeta }^2,{\zeta }^3,{\zeta }^4\} and \varphi is the Golden Ratio. For the sake of clarity, a considerable set of vertices is provided inside an online repository [38].

The hat monotile is described by the parameters 2(1,\sqrt{3} ) (recall that side of the hexagon is unity in our case). The generation of lattices (via the concomitant tilings) follows more or less the same patterns as the Penrose P2 case, but more intricate since now the five-fold symmetry no longer exists. We have used the procedure as in Ref. [51] and added the corresponding set of vertices to the online repository. Note that since each vertex is shared between 2, 3 or 4 hats, vertices appear repeated.

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