# Limitations/breakdown of Mermin-Wagner Theorem

Mermin-Wagner theorem says that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions $$d ≤ 2$$. (this is directly copied from wiki).

I'm just wondering that, if we could add some interaction, like Dzyaloshinskii-Moriya (DM) interaction interaction, or change other conditions (although currently I don't know which condition to change), to make Mermin-Wagner theorem no longer work?

Also I'm wondering that, is there any method to make Mermin-Wagner theorem work in higher dimension, like $$d=3$$?

• What do you mean by DM interaction? It is always better to avoid acronyms in questions. May 19 '20 at 16:24
• It's sort of trivial, but if you allow zero temperature then you can get spontaneously broken symmetry even for finite ("zero dimensional") systems. May 19 '20 at 18:53
• @fenglin Finite temperature is one of the assumptions for the Mermin-Wagner theorem to hold, at zero temperature it no longer applies. So there is no contradiction. A very simple example is the Hamiltonian for two spins $H = - \vec{S}_1 \cdot \vec{S}_2$, which has an SU(2) symmetry that is spontaneously broken in the ground state(s). May 19 '20 at 19:59
• How would that be a counterexample to the theorem? There are no counterexamples to theorems, only to conjectures. You should consider fixing your title. May 19 '20 at 22:22
• @NorbertSchuch : I took the question as finding a counterexample to a "physicists' theorem", which to me is not the same thing. In this particular case, the physicists' version would state something like "a continuous symmetry cannot be spontaneously broken in dimensions 1 and 2". Such a statement is of course not generally true. The corresponding mathematical theorem has a list of nontrivial assumptions attached. Note that this is true of most of the results to which physicists attach the word theorem... May 20 '20 at 7:15

As @NorbertSchuch comments, a theorem cannot have counterexamples. Well, at least this is true for what mathematicians call theorems. I thus take the question as asking for a way to violate the "physicists' version of Mermin-Wagner theorem", which would state something like "a continuous symmetry cannot be spontaneously broken in dimensions $$1$$ and $$2$$ at positive temperature". In this form (which is the form you often see this result stated in the physics literature), there are of course counterexamples and the latter can be found by trying to remove some of the assumptions of the mathematically precise versions of this theorem.

Probably the simplest way to violate the (physicists' version of) Mermin-Wagner theorem is to consider a system with sufficiently long-range interactions. For instance, consider the (classical) XY model on $$\mathbb{Z}^2$$ with Hamiltonian $$H=-\sum_{i\neq j} J_{|j-i|} S_i\cdot S_j,$$ where $$J_r = r^{-\alpha}$$. Then, for any $$\alpha\geq 4$$, the Mermin-Wagner theorem applies (see, for instance, this paper), but for any $$\alpha<4$$, it fails: there is spontaneous magnetization at low temperatures (see, for instance, this paper).

Concerning your second question, I don't think that there is any way of making the Mermin-Wagner theorem works in systems of dimension genuinely larger than $$2$$.

The theorem is quite robust to variations in the interaction - it only requires that it does not violate local continuity. The biggest physical limitation is that one needs A LOT of infra-red fluctuations to destabilize the ordered state, which are cut off by the size of the sample in any real setting. In 2D, the effect of the theorem is only logarithmic in the size. For superconductivity in particular, the theorem is physically ineffective as it stands. The sample would have to be larger than the observable universe for long-wavelength fluctuations to have an effect on the SC Tc in 2D. See our paper Physical limitations of the Hohenberg–Mermin–Wagner theorem, also available on https://arxiv.org/abs/2107.09714.

$$\newcommand{\pd}{\partial}\newcommand{\d}{\mathrm{d}}$$I bring back this old question to supplement the other answer with what happens at $$d\geqslant 3$$. A version of Mermin–Wagner valid for higher dimensions can be formulated using higher-form symmetries.

Let me first remind you what higher-form symmetries are. Since you are only interested in the continuous case, I will formulate everything having continuous symmetries in mind but it can be extended to discrete symmetries. Noether's theorem asserts that to every continuous symmetry there is a conserved current: $$\pd_\mu J^\mu(x) = 0 \tag{1}.$$ Equivalently you can construct a one-form, $$J_{1}(x):= J_\mu(x)\,\d x^\mu$$ and then (1) becomes $$\d \star J_{1}(x) = 0.$$ A $$p$$-form symmetry is a symmetry with a conserved $$(p+1)$$-form current, $$J_{p+1}(x)$$, $$\d \star J_{p+1}(x) = 0.\tag{2}$$ In index notation (2) reads $$\pd_{\mu_0} J^{[\mu_0\mu_1\cdots\mu_{p}]}(x)=0$$, where $$[\cdots]$$ denotes antisymmetrisation with respect to all indices. In this wording, an ordinary symmetry is a $$0$$-form symmetry.

With respect to these indices one can formulate a Mermin–Wagner theorem as follows [GKSW15] :

Higher Mermin–Wagner theorem: A continuous $$p$$-form symmetry cannot be spontaneously broken in dimensions $$d ⩽ p + 2$$.

For example continuous one-form symmetries can break in $$d>3$$, they must be unbroken in $$d=3$$ and $$d=2$$, while they do not exist at all in $$d=1$$ (there are no $$2$$-forms in one dimension). Two-form symmetries must be unbroken in $$d=4$$ and $$d=3$$, and they do not exist in $$d\leqslant 2$$, and so on.

References:

[GKSW15] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02, 172 (2015), doi:10.1007/JHEP02(2015)172, (arXiv: 1412.5148 )