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A proof of crystallization in two dimensions. (English) Zbl 1113.82016
Summary: 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 \[ E\left(\left|\{y\}\right|\right) = \sum_{1 \leq x < x' \leq N} V\left(| | y(x)-y(x')\right), \] where \(y (x) \in \mathbb R^2\) is the position of particle \(x\) and \(V(r) \in \mathbb R\) 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 \(E_*\): \[ \lim_{n \to \infty} \frac1N \min_{y:\{1\ldots N\}\to\mathbb R^2} E\left(\{y\}\right) = E_* \] where \(E_* \in \mathbb R\) is the minimum of a simple function on \([0,\infty)\). 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.

82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
74N05 Crystals in solids
74G65 Energy minimization in equilibrium problems in solid mechanics
82D25 Statistical mechanical studies of crystals
Full Text: DOI
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