Question

Why is there a 'loophole' in Mermin Wagner for rotations?

Answer 1

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 Z2Z2, and we restrict their motion to some ball in R2R2 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

Answer 2

The Mermin-Wagner Theorem In two dimensions, crystals provide another loophole in a well-known result, known as the Mermin–Wagner theorem.