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The Thermodynamic Stability of Two Dimensional Crystals with an Extended Coupling Scheme

We calculate mean square deviations for crystals in one and two dimensions. For the two dimensional lattices, we consider several distinct geometries (i.e. square, triangular, and honeycomb), and we find the same essential phenomena for each lattice structure. We investigate the stability of long-range crystalline order for a variety of coupling schemes, including short-range exponentially decaying inter-atomic potentials and long-range interactions with a power law dependence r^{-alpha}. For the latter in the 1D case, we find a critical value alpha_c(1D) = 1.615 +/- 0.005 for the power law decay exponent below which crystalline order is intact, and above which thermal fluctuations destroy long-range order when T > 0. The corresponding critical value for two dimensional lattices with displacements confined to the plane is alpha_c(2D) = 3.15 +/ 0.025. If motion perpendicular to the crystal plane is permitted, thermally induced distortions diverge rapidly (i.e. linearly) in dual layer systems with local stiffness provided by an extended coupling scheme, even if the interaction is long ranged, decaying as a power law in the separation between lattice sites.