Crystallization to the square lattice for a two-body potential

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 $\bar{\mathcal E}_{\mathrm{sq}}[V]$ which is the same as the energy per particle in the square lattice infinite configuration. We thus have $$ N{\bar{\mathcal E}_{\mathrm{sq}}[V]}\le \min_{X:\{1,\ldots,N\}\to\mathbb R^2}\mathcal E[V](X)\le N{\bar{\mathcal E}_{\mathrm{sq}}[V]}+O(N^{\frac 1 2}). $$ Moreover $\bar{\mathcal E}_{\mathrm{sq}}[V]$ 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 ${\bar{\mathcal E}_{\mathrm{sq}}[V]}=-4$. To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.
Comment: 58 pages, 8 figures. Clarified proofs throughout the paper, added appendix with list of notation. Submitted version of the paper