Two-Dimensional Theta Functions and Crystallization among Bravais Lattices

In this paper, we study minimization problems among Bravais lattices for finite energy per point. We first prove that if a function is completely monotonic, then the triangular lattice minimizes its energy per particle among Bravais lattices for any given density. Second, we give an example of convex decreasing positive interacting potential for which the triangular lattice is not a minimizer for a class of densities. We use Montgomery method presented in [L. Betermin and P. Zhang, Commun. Contemp. Math., 17 (2015), 1450049] to prove the minimality of the triangular lattice among Bravais lattices at high density for a general class of potentials. Finally, we deduce the global minimality among all Bravais lattices, i.e., without a density constraint, of a triangular lattice for some Lennard-Jones-type potentials and attractive-repulsive Yukawa potentials.