A Proof of Crystallization in Two Dimensions

Abstract

Many materials have a crystalline phase at low temperatures. The simplest example where this fundamental phenomenon can be studied are pair interaction energies of the type where y(x) ∈ℝ² is the position of particle x and V(r) ∈ ℝ is the pair-interaction energy of two particles which are placed at distance r. Due to the Mermin-Wagner theorem it can't be expected that at finite temperature this system exhibits long-range ordering. We focus on the zero temperature case and show rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant E _*: where E _* ∈ ℝ is the minimum of a simple function on [0,∞). Furthermore, if suitable Dirichlet- or periodic boundary conditions are used, then the minimizers form a triangular lattice. To the best knowledge of the author this is the first result in the literature where periodicity of ground states is established for a physically relevant model which is invariant under the Euclidean symmetry group consisting of rotations and translations.