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.
Best Answer
This is a good example of when the theoretical rubber meets the proverbial experimental road.
The two issues you bring up are actually completely independent. The chemical potential problem is purely kinematic, and may be solved by simply introducing a harmonic trapping potential or any other way to modify the density of states.
Mermin-Wagner is more subtle and is a dynamic (or fluctuation) problem. However, one needs to bear in mind the difference between "long ranges" theoretically vs experimentally accessible. In 2D one has algebraic decay of the correlations. Compare with the Coulomb force which we usually call long-ranged, what's the difference? For most accessibly sized gas clouds, this amount of correlation is good enough to treat things as condensed.
Theoretically, one talks of the Kosterlitz-Thouless transition, and the mechanism is that of binding-unbinding of dipoles.