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Prior to the discovery of graphene, the Mermin-Wagner theorem was used to argue that purely two-dimensional materials would not be stable as two-dimensional order would show logarithmic divergences at long range and fluctuations would spontaneously melt the crystal. Yet, graphene exists and it has very good mechanical properties. Are the predictions of the Mermin-Wagner theorem inapplicable in this case or are there still observable effects of the theorem at large scale?

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This Physics Stack Exchange thread answers this question very well.

Apparently, the length scale of such "logarithmic divergence" is actually merely in the order of atomic distances even if the entire graphene sheet is as large as the solar system! So practically, the results of the theorem is irrelevant to the real-world graphene, even though mathematically the theorem is perfectly applicable to graphene. (Note: Sometimes the MW theorem is misunderstood to be prohibiting ANY continuous symmetry breaking in 2D short-ranged systems, even though this is not true: "hexatic" order in 2D hard sphere systems is a good example for this).

This is so cool because it essentially challenges our belief in the "thermodynamic limit" as a good approximation of macroscopic substances!

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