Crystallization to the Square Lattice for a Two-Body Potential

Abstract

We consider two-dimensional zero-temperature systems of N particles to which we associate an energy of the form $$\begin{aligned} \mathcal {E}[V](X):=\sum _{1\leqq i<j\leqq N}V(|X(i)-X(j)|), \end{aligned}$$ 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 $$\begin{aligned} N{\overline{\mathcal E}_{\mathrm {sq}}[V]}\leqq \min _{X:\{1,\ldots ,N\}\rightarrow \mathbb R^2}\mathcal E[V](X)\leqq N{\overline{\mathcal E}_{\mathrm {sq}}[V]}+O(N^{\frac{1}{2}}). \end{aligned}$$ Moreover $$\overline{\mathcal E}_{\mathrm {sq}}[V]$$ is also re-expressed as the minimizer of a four point energy. In particular, this happens 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.