I have been told that the Mermin-Wagner theorem disallows the existence of the crystal of graphene. However, I don't have enough knowledge to understand the Mermin-Wagner theorem. If possible can someone please explain to me:
It is pretty funny that people continue to cite the Mermin-Wagner theorem in a context where credit should be given to David Mermin for a paper he wrote alone. There, he derived the theorem that directly applies to the problem of crystalline order in 2D. As an example of the ongoing citation confusion, there is a very highly cited paper by M I Katsnelson, freely readable online, where, in the text, Mermin-Wagner theorem is mentioned, but the corresponding bibliographic item is N.D. Mermin Phys. Rev., 176 (1968), p. 250. The reason for such a confusion is probably that the Mermin-Wagner theorem predates by two years the paper by Mermin. Moreover, the two theorems are connected but describe different things. Mermin-Wagner theorem was about the possibility of ferromagnetic or antiferromagnetic order in one and two-dimensional lattice systems, as measured by the spin-spin correlation functions. It was not directly linked to the existence of one- or two-dimensional crystals of atoms. Mermin's theorem of 1968 has the title Crystalline order in two dimensions and it specifically addresses the existence of two-dimensional crystals.
In the following, I'll describe the content of the theorem without going into the technical details of the proof, and I'll summarize the conclusions people already got many years before the discovery of graphene. Some of these conclusions have been rediscovered recently in connection with the strong momentum of graphene research.
What Mermin's (not Mermin-Wagner) theorem is about:
A broken symmetry crystalline solid can be characterized straightforwardly by the presence of a periodic one-particle density, $\rho({\bf r})$, or by its D-dimensional Fourier components, $\rho_{\bf G}$, where ${\bf G}$ is a generic reciprocal lattice vector.
Mermin was able to prove that in less than $3$ dimensions $\rho_{\bf G}$, for all the non-zero reciprocal lattice vectors, must vanish at the thermodynamic limit. The proof is a tour de force of estimates about the asymptotic behavior of the selected quantity. The result implies that if a 2-D crystal is defined by non-vanishing Fourier components $\rho_{\bf G}$, then such a crystal cannot exist in one or two dimensions at the thermodynamic limit. Notice that in statistical mechanics, the thermodynamic limit is a prerequisite for finding a non-analytic behavior in thermodynamics, which is taken as the definition of the existence of a phase transition.
It is worth noticing that the theorem establishes, in a mathematically sound way, what had been previously conjectured by Rudolph Peierls based on a more physical argument. Peierls' intuition was that, in low dimensions, long-wavelength excitations (long-wavelength phonons) destroy the crystalline order by making the mean square displacement of the particles logarithmically diverging with the size of the system.
Apparently, the theorem seems to forbid the existence of systems, like graphene, which can be experimentally characterized in terms of non-zero $\rho_{\bf G}$ (STM experiments). This no-go theorem should apply to graphene, but even before discovering graphene, other indications of real two-dimensional crystals were challenging the applicability of the theorem to the real world. The case of rare gases adsorbed on the surface of graphite was a first example, although some doubts could remain about the role of the underlying graphite lattice. It was much more challenging the case of the crystallization of electrons trapped on the surface of liquid Helium. Also, computational physics experiments showed the possibility that in practice, the theorem could not be valid.
So, what is the way to escape the consequence of the theorem?
Since the early eighties, a consensus was reached about the practical irrelevance for laboratory samples of the asymptotic vanishing of the Fourier coefficients. Regarding Peierls' argument, it is true that long-wavelength phonons make the mean square displacement increase logarithmically. However, quantitative analysis shows that even for a crystal of the size of the solar system, this value would remain a fraction of the interatomic distance (see for instance https://arxiv.org/abs/0807.2938 ). So, in practice, the consequence of the theorem can be avoided.
Interestingly, such an attitude means that in some cases (low-dimensional systems), one of the basic tenets of Statistical Mechanics (the crucial role of thermodynamic limit) has to be weakened. For such systems, the thermodynamic limit is not the best possible approximation for finite macroscopic systems.
