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.