Crystalline Order in Two Dimensions

If $N$ classical particles in two dimensions interacting through a pair potential $\ensuremath{\Phi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are in equilibrium in a parallelogram box, it is proved that every $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{\ne}0$ Fourier component of the density must vanish in the thermodynamic limit, provided that $\ensuremath{\Phi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})\ensuremath{-}\ensuremath{\lambda}{r}^{2}|{\ensuremath{\nabla}}^{2}\ensuremath{\Phi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})|$ is integrable at $r=\ensuremath{\infty}$ and positive and nonintegrable at $r=0$, both for $\ensuremath{\lambda}=0$ and for some positive $\ensuremath{\lambda}$. 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.