*Published Paper*

**Inserted:** 23 oct 2019

**Last Updated:** 13 sep 2021

**Journal:** Arch. Rational Mech. Anal.

**Volume:** 240

**Number:** 2

**Year:** 2021

**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.

**Keywords:**
crystallization, square lattice, pairwise interactions, ground state

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