# 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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For massless bosons you can check that, performing a similar calculation (but with the density of energy states different from the usual massive case because the dispersion is linear) you will find that the total number of particles converges in 2D but still diverges in 1D. As far as I know, and I'm no expert on BEC, this would indicate that BEC for massless bosons (e.g. photons) is possible in 2D. –  Vinsanity Mar 23 at 14:09