Crystalline Order in Two Dimensions

If N classical particles in two dimensions interacting through a pair potential Φ(→r) are in equilibrium in a parallelogram box, it is proved that every →k≠0 Fourier component of the density must vanish in the thermodynamic limit, provided that Φ(→r)-λr²|∇²Φ(→r)| is integrable at r=∞ and positive and nonintegrable at $r=0$, both for λ=0 and for some positive λ. This result excludes conventional crystalline long-range order in two dimensions for power-law potentials of the Lennard-Jones type, but is inconclusive for hard-core potentials. The corresponding analysis for the quantum case is outlined. Similar results hold in one dimension.