Possibility of Bose-Einstein condensation in low dimensions

I remember having a problem (for practice preliminary exams at UC Berkeley) to prove that Bose-Einstein condensation(BEC) is not possible in two dimensions (as opposed to three dimensions): For massive bosons (in 2D), a short calculation with density of states shows that the number of particles is independent of energy and diverges when chemical potential $\mu\rightarrow 0$.

But, if you consider massless bosons then a calculation shows that the density of states does depend on energy and there is a critical temperature where condensation occurs.

So now I am a little confused, because the Mermin-Wagner theorem states that for systems in dimensions $d\le 2$, long-range fluctuations will be created that destroy any existence of a BEC. So does this theorem not apply for massless bosons? There seems to be no confusion in 1D, since a similar calculation of density of states show that neither massive nor massless bosons become a BEC.

It is definitely the case that a BEC in 2D can exist for particles in certain potential traps, though.

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