Abstract
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.
Keywords
PhysicsOrder (exchange)Dimension (graph theory)LambdaNabla symbolMathematical physicsQuantum mechanicsCondensed matter physicsCombinatoricsMathematics
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Publication Info
- Year
- 1968
- Type
- article
- Volume
- 176
- Issue
- 1
- Pages
- 250-254
- Citations
- 1286
- Access
- Closed
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Cite This
N. David Mermin
(1968).
Crystalline Order in Two Dimensions.
Physical Review
, 176
(1)
, 250-254.
https://doi.org/10.1103/physrev.176.250
Identifiers
- DOI
- 10.1103/physrev.176.250