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:

  • The basics of the theorem, what it is talking about?
  • Why it prevents graphene from existing?
  • Why graphene exists if it is prevented by MW theorem?

It is quite funny that people continues to cite the Mermin-Wagner theorem in a context where credit should be given to David Mermin for a paper he wrote alone, in which he derived the theorem which 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 on-line, 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 year the paper by Mermin. Moreover, the two theorems are connected but describe different things. Mermin-Wagner theorem was originally about the possibility of ferromagnetic or anti ferromagnetic order in one and two-dimensional lattice systems, as measured by the spin-spin correlation functions. As such, it was not directly linked to the existence of one- or two-dimensional crystals of atoms. Mermin's theorem of 1968 has a title Crystalline order in two dimensions and it is specifically addressing the problem of 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 got already many years before the discovery of graphene. Some of these conclusions have been rediscovered recently in connection with the strong momentum of research on graphene.

What Mermin's ( not Mermin-Wagner ) theorem is about:

A broken symmetry crystalline solid can be characterized in a straightforward way 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 being able to find a non-analytic behavior in the thermodynamics which is taken as definition of the existence of a phase transition.

It is worth to notice that the theorem establishes, in a mathematically sound way, what had been previously conjectured by Rudolph Peierls on the basis of a more physical argument. Peierls' intuition was that, in low dimensions, long-wavelength excitations (long-wavelength phonons) destroy the crystalline order by making the 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 term of non zero $\rho_{\bf G}$ (STM experiments). This no-go theorem should apply to graphene, but even before the discovery of graphene, other indications of real two-dimensional crystals were challenging the applicability of the theorem to real world. The case of rare gases adsorbed on the surface of graphite was a first example, although some doubts could remail about the role of the underlying graphite lattice. Much more challenging the case of the crystallization of electrons trapped on the surface of liquid Helium. Also computational physics experiments were showing the possibility that in practice the theorem could not be valid.

So, what's the way to escape the consequence of the theorem?

Since the early eighties, consensus was reached about the practical irrelevance for laboratory samples, of the asymptotic vanishing of the Fourier coefficients. With reference to Peierls' argument, it is true that long wavelength phonons make the mean square displacement increasing logarithmically. But a 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. So, in practice the consequence of the them can be avoided.

Interestingly, such an attitude means that in some cases (low-dimensional systems) one of the basic tenets of Statistical Mechanics (the key role of thermodynamic limit) has to be weakened: for such systems, the thermodynamic limit is not the best possible approximation for finite macroscopic systems.

  • $\begingroup$ Hello thank you for answering that well! I have few other questions on the topic, not sure if it's the correct place to ask: Which assumptions were used for Mermin-Wagner theorem? Do you have examples of $\rho_g$ for a lattice like graphene for a beginner? $\endgroup$ – PackSciences May 8 '19 at 7:37
  • $\begingroup$ Maybe one should also add that, in the case of graphene, the sheet is actually not flat, but displays ripples due to thermal fluctuations. The nonlinear coupling between those "transverse" fluctuations and the "longitudinal" ones induces an effective long-range component to the interaction , which may invalidate Mermin's theorem (and its numerous extensions). In any case, the fact that this is a 2d "crystal" inside a 3d space is certainly relevant. $\endgroup$ – Yvan Velenik May 8 '19 at 7:39
  • $\begingroup$ @PackSciences The strongest mathematical result to date is given in this paper; see also this one. (Strongest in the sense of weakest assumptions; in particular, these results apply to systems with hard-core exclusion, in contrast to the earlier proofs.) $\endgroup$ – Yvan Velenik May 8 '19 at 8:04
  • $\begingroup$ Thank you for this great answer. Do you happen to have a reference, or a pointer to further reading, for the "quantitative analysis [that] shows that even for a crystal of the size of the solar system this value would remain a fraction of the interatomic distance"? $\endgroup$ – user8153 May 9 '19 at 17:35
  • $\begingroup$ @user8153 I have to find the exact reference. Unfortunately I do not have access to my folders until next week. $\endgroup$ – GiorgioP May 10 '19 at 5:42

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