# What is difference between Bose-Einstein condensation in 3D and 2D?

I read in the reference [1] below that

An infinite, non-interacting two-dimensional gas of bosons has no phase transition and never develops spontaneous coherence. However, adding interactions leads to the Berezhinski-Kosterlitz-Thouless(BKT) transition, below which power law correlations, and super fluidity can develop. For a finite, trapped two-dimensional system of non-interacting particles, macroscopic occupation of a single particle state can also occur.

This means, if we introduce interaction (trapping potentials) in a finite 2D system, it can undergo a phase transition that is similar to BE condensate. But I thought even a 3D bosonic system requires a trapping potential so that they can occupy a quantized energy levels and reach the critical phase space density to undergo the macroscopic occupation. In this respect, how is 2D condensate different from 3D condensate ? Also, reference [2] emphasizes similar requirement for BKT in 2D. Can anyone explain what point I am missing here. Any comments are highly appreciated. Thanks

[1] J. Kasprzak et al., Nature 443, 409 (2006)

[2] D. S. Petrov, M. Holzmann and G. V. Shlyapnikov, Phys. Rev. Lett. 84, 2551 (2000)

In principle you can get a condensate in an infinite 3D system. In practice however it is problematic, as the requirement is that you need a certain finite density and therefore an infinite number of atoms. That's why you introduce a trap to make your system finite and get away and obtain BEC with finite number of atoms.

In 2D however you can't have a condensate in an extended system in principle at non-zero temperature (as any other long-range order; see "Mermin-Wagner theorem"), because fluctuations smear out the order entirely for any dimension $$D \leq2$$ (footnote: if you ask what is the significance of D=2, I am tempted to say it is the power $$n=2$$ of the gradient term in the free energy expansion $$F(\phi) \propto |\nabla \phi|^2+...$$).

Nevertheless the divergences in D=2 are very weak (logarithmic) therefore, therefore in finite systems all Mermin-Wagner-like phenomena are exponentially suppressed, and hence can be ignored for most practical purposes (like graphene people who seemingly are never bothered by the fact that their sample are not supposed to exist at finite temperatures).

• Thank you for clearing that up. Regarding the Ref. 1, I have another confusion. In my opinion, the critical temperature should be about equal to or lower than the energy spacing between the ground state and the first excited state. But in Ref.1, they estimate the critical temperature to be around 19K but if we use the trapping potential (interaction for BKT) from disordered potential (1meV potential fluctuation in 10 µm ) they have in the microcavity system, the energy spacing is 0.7K. Is it due to the 2D KT transition but not the 3D BEC transition? Or, is my opinion wrong? Commented Oct 25, 2022 at 4:58
• I think your opinion is mistaken. In the original work of Bose, he considered a gas of bosons, i.e. a system with continuous spectrum and found that the condensate forms at a finite temperature. Level spacing and BEC transition are not related the way you have described.
– John
Commented Oct 25, 2022 at 5:46
• Thank you again for the comment. But isn't BEC a kind of phase transition where you have spontaneous macroscopic occupation of the lowest energy state and for the particle to stay in that state, the thermal energy should be less than the energy spacing between the ground state and the first excited state (formed by the trapping potential) ? Commented Oct 25, 2022 at 7:40