# Crystallization to the Square Lattice for a Two-Body Potential

@article{Btermin2019CrystallizationTT, title={Crystallization to the Square Lattice for a Two-Body Potential}, author={Laurent B{\'e}termin and Lucia De Luca and Mircea Petrache}, journal={arXiv: Analysis of PDEs}, year={2019} }

We consider two-dimensional zero-temperature systems of $N$ particles to which we associate an energy of the form $$ \mathcal{E}[V](X):=\sum_{1\le i<j\le N}V(|X(i)-X(j)|), $$ where $X(j)\in\mathbb R^2$ represents the position of the particle $j$ and $V(r)\in\mathbb R$ is the {pairwise interaction} energy potential of two particles placed at distance $r$. We show that under suitable assumptions on the single-well potential $V$, the ground state energy per particle converges to an explicit…

## 13 Citations

On the optimality of the rock-salt structure among lattices with charge distributions

- Mathematics
- 2020

The goal of this work is to investigate the optimality of the $d$-dimensional rock-salt structure, i.e., the cubic lattice $V^{1/d}\mathbb{Z}^d$ of volume $V$ with an alternation of charges $\pm 1$…

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

- MathematicsCommunications in Mathematical Physics
- 2021

We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of…

Finite Crystallization and Wulff shape emergence for ionic compounds in the square lattice

- Materials ScienceNonlinearity
- 2020

We present two-dimensional crystallization results in the square lattice for finite particle systems consisting of two different atomic types. We identify energy minimizers of configurational…

Optimality of the triangular lattice for Lennard-Jones type lattice energies: a computer-assisted method

- Mathematics
- 2021

It is well-known that any Lennard-Jones type potential energy must a have periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method…

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

- Mathematics
- 2021

The Riesz potential fs(r) = r is known to be an important building block of many interactions, including Lennard-Jones type potentials f n,m (r) := ar − br, n > m that are widely used in Molecular…

On energy ground states among crystal lattice structures with prescribed bonds

- Computer ScienceJournal of Physics A: Mathematical and Theoretical
- 2021

The universal minimality—i.e. the optimality for all completely monotone interaction potentials— of strongly eutactic lattices among these structures gives new optimality results for the square, triangular, simple cubic (sc), face-centred-cubic (fcc) and body-centre-cUBic (bcc) lattices in dimensions 2 and 3 when points are interacting through completely Monotone potentials.

Vectorial crystallization problems and collective behavior.

- Materials ScienceJournal of mathematical biology
- 2021

A class of vectorial crystallization problems, with applications to crystallization of anisotropic molecules and collective behavior such as birds flocking and fish schooling, is proposed and analyzed, focusing on two-dimensional systems of "oriented" particles.

Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture

- MathematicsProceedings of the American Mathematical Society
- 2020

The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the
E
8
E_8
lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the…

Note on Crystallization for Alternating Particle Chains

- Mathematics, Materials ScienceJournal of statistical physics
- 2020

The crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential, is proved.

Effect of Periodic Arrays of Defects on Lattice Energy Minimizers

- Materials ScienceAnnales Henri Poincare
- 2021

In the case of inverse power laws and Lennard-Jones-type potentials, necessary and sufficient conditions are given on non-shifted periodic vacancies or substitutional defects for the conservation of minimality results at fixed density.

## References

SHOWING 1-10 OF 68 REFERENCES

Optimality of the Triangular Lattice for a Particle System with Wasserstein Interaction

- Mathematics
- 2012

We prove strong crystallization results in two dimensions for an energy that arises in the theory of block copolymers. The energy is defined on sets of points and their weights, or equivalently on…

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

- MathematicsJ. Nonlinear Sci.
- 2018

It is shown that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems by appealing to a discrete Gauss–Bonnet theorem which relates the sum/integral of the curvature to topological invariants.

NEXT ORDER ASYMPTOTICS AND RENORMALIZED ENERGY FOR RIESZ INTERACTIONS

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2015

We study systems of $n$ points in the Euclidean space of dimension $d\geqslant 1$ interacting via a Riesz kernel $|x|^{-s}$ and confined by an external potential, in the regime where $d-2\leqslant…

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

- Mathematics
- 2014

We study the Hamiltonian of a two-dimensional log-gas with a confining potential V satisfying the weak growth assumption—V is of the same order as $$2\log \Vert x\Vert $$2log‖x‖ near…

Finite Crystallization and Wulff shape emergence for ionic compounds in the square lattice

- Materials ScienceNonlinearity
- 2020

We present two-dimensional crystallization results in the square lattice for finite particle systems consisting of two different atomic types. We identify energy minimizers of configurational…

$$\varGamma $$Γ-Convergence of the Heitmann–Radin Sticky Disc Energy to the Crystalline Perimeter

- MathematicsJ. Nonlinear Sci.
- 2019

A compactness result is proved which shows the emergence of polycrystalline structures when the limit configuration is a single crystal, and it is shown that the anisotropic perimeter is the Finsler metric determined by the orientation of the single crystal.

Optimal and non-optimal lattices for non-completely monotone interaction potentials

- MathematicsAnalysis and Mathematical Physics
- 2019

We investigate the minimization of the energy per point $$E_f$$Ef among d-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function…

A Proof of Crystallization in Two Dimensions

- Computer Science
- 2006

This work shows rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant E*: where E* ∈ ℝ is the minimum of a simple function on [0,∞).

Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

- Mathematics
- 2019

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they…

Nonhexagonal Lattices From a Two Species Interacting System

- Materials ScienceSIAM J. Math. Anal.
- 2020

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic…