According to Peierls and Landau, 2D crystals were thermodynamically unstable. They can't exist!
Of course, this theory was disapproved in 2004 (example: graphene).
What is the general definition of stability of a general system?
What is the thermodynamics' stability?
Best Answer
Well, when we talk of stability of systems, at least for equilibrium systems, we require the free energy to be bounded below and be convex. As the free energy is obtained by a Legendre transformation (which preserves convexity), the energy functional is required to be convex. This essentially allows us to minimize energies to find ground states. Within the realm of regular equilibrium thermodynamics, higher order quantities (depending on fluctuations) like the heat capacity or susceptibility/compressibility are put in by hand to be finite quantities.
Now, coming to your point about stability of ordered states, it is precisely these fluctuations (that in thermodynamics you assume are finite, but in statistical mechanics, are calculable) that diverge for arbitrarily large systems for some dimensions and/or at some critical points corresponding to phase transitions. So, the so called argument against the stability of 2D crystals with short ranged interactions is the Mermin-Wagner theorem, which basically shows that in 2 dimensions, fluctuations about the ordered state (in this case the lattice structure) de-correlate over large distances, thereby destroying any large scale order in the system (again only in the thermodynamic limit of large system sizes).
Coming to the specific case of graphene as a stable 2D crystal, the loophole it exploits to "violate" the Mermin-Wagner theorem is rather subtle. Mind you that, just because it fluctuates as a sheet embedded in a higher dimensional space does not make it a non-2D structure. As an aside, fluid membranes that can self-intersect (this means that all the interactions are only local) are equivalent 2D sheets embedded in a higher dimensional space, and they do not have a flat phase (ordering of normals) and are always crumpled. So just by allowing fluctuations into another dimension is not enough to have ordering in graphene.
What instead happens is that the lattice structure of graphene is fixed, so it corresponds to what is called a tethered elastic membrane (which unlike the fluid membrane can stretch.) It is this in plane stretching that allows for phonon mode to propagate and couple in-plane degrees of freedom to transverse fluctuations out of the plane. This effectively mediates a long range interaction, thereby circumventing Mermin-Wagner. Another way to see this technically, is that the bending rigidity of the sheet gets renormalized such that it diverges at large length scales, effectively making the sheet stiffer on larger scales. So in short, the thermal undulations are essential for the stability of graphene as a 2D crystal.