# Crystallization to the square lattice for a two body potential

created by deluca on 23 Oct 2019
modified on 26 Oct 2019

[BibTeX]

Submitted Paper

Inserted: 23 oct 2019
Last Updated: 26 oct 2019

Year: 2019

Abstract:

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 constant $\overline{\mathcal E}_{\mathrm{sq}}[V]$ which is the same as the energy per particle in the square lattice infinite configuration. We thus have $N{\overline{\mathcal E}_{\mathrm{sq}}[V]}\le \min_{X:\{1,\ldots,N\}\to\mathbb R^2}\mathcal E[V](X)\le N{\overline{\mathcal E}_{\mathrm{sq}}[V]}+O(N^{\frac 1 2}).$ Moreover $\overline{\mathcal E}_{\mathrm{sq}}[V]$ is also re-expressed as the minimizer of a four point energy.

In particular, this happen{s} if the potential $V$ is such that $V(r)=+\infty$ for $r<1$, $V(r)=-1$ for $r\in [1,\sqrt{2}]$, $V(r)=0$ if $r>\sqrt{2}$, in which case ${\overline{\mathcal E}_{\mathrm{sq}}[V]}=-4$.

To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.