Face-centered cubic crystallization of atomistic configurations

We address the question of whether three-dimensional crystals are minimizers of classical many-body energies. This problem is of conceptual relevance as it presents a significant milestone towards understanding, on the atomistic level, phenomena such as melting or plastic behavior. We characterize a set of rotation- and translation-invariant two- and three-body potentials V2, V3 such that the energy minimum of $$\frac{1}{\#Y}E(Y) = \frac{1}{\# Y} \left(2\sum_{\{y,y'\} \subset Y}V_2(y, y') + 6\sum_{\{y,y',y''\} \subset Y} V_3(y,y',y'')\right)$$ over all \({Y \subset \mathbb{R}^3}\), #Y = n, converges to the energy per particle in the face-centered cubic (fcc) lattice as n tends to infinity. The proof involves a careful analysis of the symmetry properties of the fcc lattice.