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 E[V](X):=?1?i<j?NV(|X(i)-X(j)|),where X(j) ? R represents the position of the particle j and V(r) ? 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 E¯ [V] , which is the same as the energy per particle in the square lattice infinite configuration. We thus have NE¯sq[V]?minX:{1,...,N}->R2E[V](X)?NE¯sq[V]+O(N12).Moreover E¯ [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) = + ? forr< 1 , V(r) = - 1 for r?[1,2], V(r) = 0 if r>2, in which case E¯ [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.