# Talk:Mermin-Wagner Theorem

Review A: this is an exhaustive and detailed exposition of a key result in Statistical Mechanics. The Authors provide a well documented analysis of the physical meaning and interest of the theorem, complete it with a full derivation (including the discussion of the Bogoliubov inequality) and provide comments illustrating more recent applications.

The article is recommended "as is".

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Review B:

Summary.

This article is a short and very nice review of the Mermin-Wagner theorem for quantum Heisenberg (anti)ferromagnets. After a an introduction of the model and of its relevant (continuous) symmetries, the author briefly reviews spin-wave theory and some of its implications, including the fact that no magnetic long range order (of ferromagnetic or antiferromagnetic type) is expected at low temperatures in the quantum Heisenberg model in one or two dimensions. Next, the main theorem is stated and proved. Finally, some extensions of the main theorem are discussed and a proof of Bogoliubov's inequality is presented.

Comments.

I recommend the publication of this paper, after the consideration of the following (minor) comments.

1) In the section ``Heisenberg model, symmetry", in the first line after the second equation, it may be useful to define $I_{2S+1}$.

2) In the section ``Heisenberg model, symmetry", in the second line after the second equation, it may be useful to specify whether the couplings J(x-y) have definite sign or not.

3) In the section ``Spin waves", after Eq.(3), it may be useful to define the quantity a(k) explicitly and to correspondingly explain more clearly the meaning of the sentence ``a(k) arises from the fact that the alternating magnetisation is not a conserved quantity", which I find a bit obscure.

4) In the section ``Proof", a couple of lines before Eq.(6), I think that ``A and B" should be replaced by ``A and C"

5) In the section ``Proof", in the second line of the last equation, the constant should be $c_2$ rather than $c_1$.

6) In the section ``Supplement: Bogoliubov Inequality" and in the References, the citations to Bratteli and Robinson should be corrected (they now read BRATTELLI and Robinson - the double L in Bratteli should be eliminated).

7) In the section ``Supplement: Bogoliubov Inequality", in Eq.(18), I think that the last identity should be $(B^\dagger|A^\dagger)$ rather than $(A^\dagger|B^\dagger)$.

8) Finally, two typos: the word Neel at the end of the section ``Heisenberg model, symmetry" and the reference to Hohenberg at the beginning of the section ``Proof" are not displayed correctly: they should be written in the scholarpedia style (rather than in LaTeX).