Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies

Bétermin, Laurent; Faulhuber, Markus

doi:10.48550/arxiv.2007.15977

Cited by 7 publications

(18 citation statements)

References 50 publications

(93 reference statements)

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“…Whereas different rigorous techniques exist for studying lattice energies in dimension d = 2 (see e.g. [3,6,9,50]), where dim L 2 = 3 and dim L 2 (V ) = 2 for all V > 0, the situation in dimension d = 3 is much more complicated. Indeed the so-called fundamental domain containing only one copy of each lattice (up to dilation, isometry and symmetry) is 5-dimensional for L 3 (1) and has a complicated shape (see e.g.…”

Section: Introduction Setting and Main Resultsmentioning

confidence: 99%

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

Bétermin¹,

Šamaj²,

Travěnec³

2021

Preprint

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The Riesz potential fs(r) = r −s is known to be an important building block of many interactions, including Lennard-Jones type potentials f LJ n,m (r) := ar −n − br −m , n > m that are widely used in Molecular Simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices of Riesz and Lennard-Jones energies. We discuss the minimality of the Body-Centred-Cubic lattice (BCC), Face-Centred-Cubic lattice (FCC), Simple Hexagonal lattices (SH) and Hexagonal Close-Packing structure (HCP), globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for s < 0 and the HCP lattice is computed to have higher energy than the FCC (for s > 3/2) and BCC (for s < 3/2) lattices. In the Lennard-Jones case, the ground state among lattices is confirmed to be a FCC lattice whereas a HCP phase occurs once added to the investigated structures. Furthermore, phase transitions of type "FCC-SH" and "FCC-HCP-SH" (when the HCP lattice is added) as the inverse density V increases are observed for a large spectrum of exponents (n, m). In the SH phase, the variation of the ratio ∆ between the inter-layer distance d and the lattice parameter a is studied as V increases.

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Section: Introduction Setting and Main Resultsmentioning

confidence: 99%

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

Bétermin¹,

Šamaj²,

Travěnec³

2021

Preprint

Self Cite

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“…One way of interpreting the results in this article is that we study two families of two-dimensional lattice theta functions. They are particular restrictions of Riemann theta functions and can be seen as a canonical extension of the restricted Jacobi theta θ 2 and θ 4 , providing an alternative to the lattice theta-functions studied in [12]. The functions we study complement the functions studied by Montgomery [61], just as θ 2 and θ 4 accompany the Jacobi θ 3 function.…”

Section: 4mentioning

confidence: 93%

“…However, the equality in (3.1) obtained by the Poisson summation might no longer hold, it may even not be clear what the dual structure of Γ could be. In this case, one deals with two separate problems (compare with [12]). Usually, the problem of universal optimality involves a limiting procedure to define the energy of a class of radial potentials, called completely monotone, of squared distance (see [22,24] for the details).…”

Section: Preliminaries and Notationmentioning

confidence: 99%

A variational principle for Gaussian lattice sums

Bétermin¹,

Faulhuber²,

Steinerberger³

2021

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We consider a two-dimensional analogue of Jacobi theta functions and prove that, among all lattices Λ ⊂ R 2 with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be interpreted as the dual of a 1988 result of Montgomery who proved that the hexagonal lattice minimizes the maximal values. Our inequality resolves a conjecture of Strohmer and Beaver about the operator norm of a certain type of frame in L 2 (R). It has implications for minimal energies of ionic crystals studied by Born, the geometry of completely monotone functions and a connection to the elusive Landau constant.

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“…The next definition concerns the notion of universal optimality originally introduced by Cohn and Kumar in [20] and showed in many contexts [7,8,21,51]. Contrary to [20], we do not stay in the set of configurations with fixed density but we simply say that an optimality property is universal if it holds for E f for any completely monotone function f among lattices in a certain subset.…”

Section: Potentials and Lattice Energiesmentioning

confidence: 99%

“…Gaussians [6,8,13,21,25,51], Lennard-Jones potentials [2,3,4,63] and more general functions [5,14] or energies [11,19,46,47,58]. Most of them investigate the minimization problem at fixed density and look for optimality results for the usual best (densest) lattice packing that are the triangular lattice, the FCC lattice, the Gosset lattice E 8 and the Leech lattice Λ 24 .…”

Section: Introductionmentioning

confidence: 99%

On energy ground states among crystal lattice structures with prescribed bonds

Bétermin

2021

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We consider pairwise interaction energies and we investigate their minimizers among lattices with prescribed minimal vectors (length and coordination number), i.e. the one corresponding to the crystal's bonds. In particular, we show the universal minimality -i.e. the optimality for all completely monotone interaction potentials -of strongly eutactic lattices among these structures. This gives new optimality results for the square, triangular, simple cubic (SC), face-centred-cubic (FCC) and body-centred-cubic (BCC) lattices in dimensions 2 and 3 when points are interacting through completely monotone potentials. We also show the universal maximality of the triangular and FCC lattices among all lattices with prescribed bonds. Furthermore, we apply our results to Lennard-Jones type potentials, showing the minimality of any universal minimizer (resp. maximizer) for small (resp. large) bond lengths, where the ranges of optimality are easily computable. Finally, a numerical investigation is presented where a phase transition of type "square-rhombic-triangular" (resp. "SC-rhombic-BCC-rhombic-FCC") in dimension d = 2 (resp. d = 3) among lattices with more than 4 (resp. 6) bonds is observed.

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Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies

Abstract: We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H. Montgomery. Minimal theta functions.

Cited by 7 publications

References 50 publications

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies

A variational principle for Gaussian lattice sums

On energy ground states among crystal lattice structures with prescribed bonds

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