I'm just starting out in my mathematics career by looking at some simple stuff on broken symmetries in statistical mechanics. Since 3D is 'hard' it would be very nice to look at 2D toy models of crystals. But by (what I assume is down to) the awkwardness of logarithms and two dimensions, this leads to some complications in the form of the Mermin-Wagner theorem, which most people state in words as 'it is impossible to break symmetry in two dimensions'.

However, many people including Mermin himself in Crystalline Order in Two Dimensions, point out that this doesn't apply to the action of rotations, and the gap between mathematical statements and physics starts to get difficult for me. There was an excellent question that gives a precise statement of the Mermin Wagner theorem. Starting from this and Mermin's paper I have two questions:

Suppose we have a simple model where we sit an atom at each point in $\mathbb{Z}^2$, and we restrict their motion to some ball in $\mathbb{R}^2$ centered on each lattice site so we have somtheing like a crystal, and equip them with something like harmonic interactions for simplicity. Then as Mermin points out in his paper, we can show rotational symmetry is broken, but translational symmetry isn't. In this case-

  1. Does the 'Mermin Wagner theorem' not apply in this case because the atoms actually take values in different compact spaces, and not all in the same one $\mathcal{S}$, so we're really looking at a different thing all together with rotations? But then it does apply to translational symmetry, so I guess this isn't why.
  2. Can anyone explain to me exactly how Mermin concludes as he does in this remarks about rotational symmetry breaking, if you happen to have read the paper? I'm not quite sure what the reasoning is, which I guess is exactly why I'm asking on here!

Thank you very much, and sorry the question is a little vague.

  • $\begingroup$ I am not sure I get you point. But the idea of Mermin Wagner theorem is that there cannot be any symmetry breaking of a continuous and non compact symmetry, like translation but not rotation. $\endgroup$ – gatsu Jul 31 '14 at 8:31
  • $\begingroup$ Another comment is that Mermin Wagner has to do with the existence of well behaved probability measures in the thermodynamic limit. In such a limit, the size of the ball in $\mathbb{R}^2$ you are imagining has to increase to infinity so as to keep the particle density finite. On another hand, this effect does not happen with the rotations. $\endgroup$ – gatsu Jul 31 '14 at 9:54
  • $\begingroup$ The theorem I linked to uses compact in the statement, rather than non-compact group, so I'm not sure I follow the first comment. Do you know of a source that gives the statement you're using? All the ones I can find are for things like XY rotators where the spins are internal rather than crystals. For the second, how does this then allow symmetry breaking? Is it simply because looking at neighbours to define a rotational ordering starts behaving well? $\endgroup$ – user56366 Jul 31 '14 at 12:08
  • $\begingroup$ From a technical point of view, it is the fact that under such a rotation, far away atoms "move with arbitrarily large speed". There are no rigorous proof that there is a breaking of rotation invariance in two-dimensional particle systems (one expects existence of "soft crystals", which would break rotation invariance); there are heuristic arguments (I think you can find a discussion in Nelson and Halperin's paper in Phys.Rev.B 19, 2457 (1979). [...] $\endgroup$ – Yvan Velenik Jul 31 '14 at 19:17
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    $\begingroup$ One of the ways you can understand the preservation of rotational invariance in 2D crystals is that once translational symmetry is broken, the soft phonon modes mediate a long range interaction between local rotations at different lattice points. This immediately invalidates the applicability of Mermin-Wagner. $\endgroup$ – surajshankar Sep 29 '14 at 1:31

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