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There are many presentations of the proof of the Mermin-Wagner theorem in many different contexts (which talk about quantum vs. classical, existence of unique Gibbs measure or non-zero mean magnetization, talk about a Lie group of symmetries or just the XY model). I am aware of the relations between the various statements of the theorem (correlation inequalities e.g.) but:

I am looking for the simplest, most immediate and direct proof one could find for the following statement, which I guess is slightly stronger than the statement about mean magnetization:

Theorem: If $S:\{1,\dots,L\}^2\to\mathbb{S}^{N-1}$ is spin field of the $O(N)$ model ($N\geq2$), whose partition function is defined as $$ Z_L = \prod_{x\in\{1,\dots,L\}^2}\int_{S_x\in\mathbb{S}^{N-1}}\mathrm{d}\mu(S_x)\exp(\beta\sum_{x,y\in\{1,\dots,L\}^2:x\sim y}S_x\cdot S_y) $$ where $\mu$ is the uniform volume measure on $\mathbb{S}^{N-1}$ then $$ \lim_{x\to\infty}\lim_{L\to\infty}\mathbb{E}_L[S_x\cdot S_0] = 0\qquad(\beta > 0)\,. $$

The best is if you could prove (or sketch a proof) in a paragraph or few right here, otherwise if there is a great reference for this it would also be appreciated. Just as an example, googling the term, the first result is Scholarpedia and I think one has to work a bit to get from what they state to the statement above. It would be nice to have something direct.

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The simplest way to prove this statement (actually, it gives you more, namely an upper bound on the 2-point function that decays with a power law) is the argument by McBryan and Spencer.

You can actually prove much more, namely the full rotation invariance of the limiting Gibbs state. In my opinion, the simplest proofs are due to Dobrushin and Shlosman and Pfister. (They are actually applicable to much more general systems.)

Both proofs are explained pedagogically and in detail in Chapter 9 of this book, with some improvements. The McBryan-Spencer proof is given in Section 9.4.2, while (a variant of) Pfister's proof is given in Section 9.2.2.

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  • $\begingroup$ Thanks! I somehow neglected McBryan-Spencer in this regard because I didn’t associate that proof with the usual Mermin-Wagner intuition. Oops! $\endgroup$
    – PPR
    Nov 10, 2020 at 7:50

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