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The Mermin-Coleman-Wagner theorem states that quantum fluctuations of Goldstone modes in d=1+1 dimensions are "strong enough" to destroy the possibility of Spontaneous Symmetry Breaking (SSB) at $T=0$. Now, SSB cannot occur means that the ground state respects the symmetry of the action. Therefore, the quantum fluctuations of Goldstone modes must be responsible for restoring the symmetry of the vacuum. How does this happen in $d=1+1$ dimensions?

From Wikipedia, if I understand correctly, propagator of Goldstone modes are logarithmically divergent and that forbids SSB. But I do not understand, how does this divergence responsible for precluding SSB?

In other words, the necessary criterion to preclude SSB is that the vacuum remains invariant under the full symmetry of the action. This divergence in propagator, must therefore be responsible for taking the system to a vacuum state which respects the symmetry of the action. Is that way of understanding erroneous?

Can anyone help me in understanding this point.

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  • $\begingroup$ The original paper "There are no Goldstone bosons in two dimensions" by Coleman (Mermin and Wagner's parts of the theorem are more about lattice theories than full quantum field theories) is rather short and accessible. Have you read it (it's openly accessible on projecteuclid)? What specifically is unclear to you about the proof? The point about the diverging propagator is simply the fact that there is no 1+1 dimensional massless scalar field (precisely because the propagator diverges), and Coleman also gives further references for that. $\endgroup$ – ACuriousMind Dec 23 '16 at 15:27
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    $\begingroup$ @ACuriousMind- I haven't seen it yet. But is the Wikipedia explanation wrong? Does Coleman explain why there is no SSB in terms of large fluctuations of Goldstone modes? $\endgroup$ – SRS Dec 23 '16 at 15:32
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    $\begingroup$ Just to clarify the terminology here, which also seems to be confused by the article on Wikipedia. The "Mermin-Wagner-Coleman theorem" really are two different results. The Coleman theorem is about absence of SSB in 1+1 dimensional Lorentz-invariant systems at $T=0$. The Mermin-Wagner theorem is about absence of SSB is certain 2+1 dimensional lattice systems at $T>0$. Mermin and Wagner place a rigorous upper bound on the order parameter and prove that it vanishes. Coleman shows that the Goldstone boson stemming from SSB would give rise to diverging long-distance correlations. $\endgroup$ – Tomáš Brauner Feb 20 '17 at 22:25
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If you're okay with some hand-waving, this explanation isn't too hard to follow: http://www.weizmann.ac.il/condmat/oreg/sites/condmat.oreg/files/uploads/2015/tutorial_3.pdf. The idea is that you can write the correlation function as an integral over momentum space, and by analyzing the divergence or convergence of the integral at low momenta, you can determine the long distance behavior of the correlation function for various dimensions. For classical systems, it turns out that the correlation functions always decay to zero at finite temperature in $d = 1$ or $d = 2$, preventing the onset of off-diagonal long-range order and thus spontaneous symmetry breaking (by the cluster decomposition theorem). For quantum systems at finite temperature, the imaginary time direction is compact so the Matsubara frequencies are discrete, and you can't get an IR divergence at low Matsubara frequency. But at zero temperature, the imaginary time dimension is infinitely long and effectively adds a dimension to your field theory, so a 1D quantum system at zero temperature is "like" a 2D classical system, and you again can't have continuous symmetry breaking.

A fully rigorous derivation of all this is quite mathematically technical and intimidating; see for example https://www.math.ucdavis.edu/~bxn/mermin-wagner.pdf.

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