Maximal theta functions universal optimality of the hexagonal lattice for Madelung-like lattice energies

Bétermin, Laurent; Faulhuber, Markus

doi:10.1007/s11854-022-0254-z

Cited by 4 publications

(3 citation statements)

References 44 publications

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“…Notice that similar problems have been investigated by the first author in Refs. 65–67 concerning charged systems and related lattice optimality for frame bounds in time‐frequency analysis.…”

Section: Introduction Setting and Main Resultsmentioning

confidence: 99%

“…Whereas different rigorous techniques exist for studying lattice energies in dimension

d=2

(see, e.g., Refs. 22, 54, 59, 66, 67), where

\dim \mathcal {L}_2=3

and

\dim \mathcal {L}_2(V)=2

for all

V&gt;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 five‐dimensional for

\mathcal {L}_3(1)

and has a complicated shape (see, e.g., Terras 69 ).…”

Section: Methodsmentioning

confidence: 99%

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

2022

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The Riesz potential fs(r)=r−s$f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard‐Jones–type potentials fn,mLJ(r):=ar−n−br−m$f_{n,m}^{\rm {LJ}}(r):=a r^{-n}-b r^{-m}$, n>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‐centered‐cubic (BCC) lattice, face‐centered‐cubic (FCC) lattice, simple hexagonal (SH) lattices, and hexagonal close‐packing (HCP) structure, 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$s<0$ and the HCP lattice is computed to have higher energy than the FCC (for s>3/2$s>3/2$) and BCC (for s<3/2$s<3/2$) lattices. In the Lennard‐Jones case with exponents 3

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Section: Introduction Setting and Main Resultsmentioning

confidence: 99%

“…Whereas different rigorous techniques exist for studying lattice energies in dimension

d=2

(see, e.g., Refs. 22, 54, 59, 66, 67), where

\dim \mathcal {L}_2=3

and

\dim \mathcal {L}_2(V)=2

for all

V&gt;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 five‐dimensional for

\mathcal {L}_3(1)

and has a complicated shape (see, e.g., Terras 69 ).…”

Section: Methodsmentioning

confidence: 99%

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

2022

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“…Many physical, chemical and number theoritic problems can be formulated to the following minimization problem on lattices: (1.1) See e.g. [2,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,28,29,38,39,36,34,40,42,43,44,45,46,47,4]. The summation ranges over all the lattice points except for the origin 0 and the function f denotes the potential of the system.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On minima of difference of theta functions and application to hexagonal crystallization

Luo

Wei

2022

Math. Ann.

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Let L = 1 Im(z) Z ⊕ zZ where z ∈ H = {z = x + iy or (x, y) ∈ C : y > 0} be the two dimensional lattices with unit density. Assuming that α ≥ 1, we prove that min L P∈L,|L|=1|P| 2 e −πα|P| 2 is achieved at hexagonal lattice. More generally we prove that for α ≥ 1 minis achieved at hexagonal lattice for b ≤ 1 2π and does not exist for b > 1 2π . As a consequence, we provide two classes of non-monotone potentials which lead to hexagonal crystallization among lattices. Our results partially answer some questions raised in [1,7,10,14] and extend the main results in [34] on minima of difference of two theta functions.

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On symmetry adapted bases in trigonometric optimization

Metzlaff

2025

Journal of Symbolic Computation

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Maximal theta functions universal optimality of the hexagonal lattice for Madelung-like lattice energies

Cited by 4 publications

References 44 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

On minima of difference of theta functions and application to hexagonal crystallization

On symmetry adapted bases in trigonometric optimization

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