For noninteracting 2D system of bosons, many textbooks have a statement that no BEC exists as the system is capable to accommodate infinite number of bosons when chemical potential $\mu \rightarrow 0 $. But what happens if interaction is taken into account?
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Bose-Einstein condensation per se is a non-interacting effect, solely driven by particle statistics. In $d=2$ dimensions, free space, the energy density is such that particles can still be accommodated in the excited states for $T \neq 0$ thereby not triggering the macroscopic occupation of the ground state.
This does not mean that you cannot superfluidity (SF) in 2D. Superfluidity is not the same thing as BEC. SF just means that you have a critical velocity $v_{\mathrm{c}}$ below which the fluid experiences no dissipation. But $v_{\mathrm{c}}$ depends on the interaction strength, so if you want in 3D the BEC is a "boring SF" with $v_{\mathrm{c}} = 0$.
You can still have a mechanism that allows SF in a 2D fluid. It's called the BKT transition, and is driven by the energy favourability of creating free/bound vortices. The vortex size is determined by the interaction length (healing length of the wavefunction), so it only plays a rôle for an interacting gas. You don't consider BKT in a 3D gas because they are unstable anyway (line vortex instability).
Interesting literature:
this paper, where they connected the BEC and BKT phase transitions by changing the interactions strength of a 2D/3D trapped atomic gas.
this paper, where they look at a BEC on the surface of a sphere, where, depending on the radius of curvature, you can have BEC or BKT.
answered Mar 17, 2020 at 5:35
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$\begingroup$ Thanks for this very helpful answer, not least because of the referenced papers, and also the statement about $v_c$. In the classic non-interacting case in a simple geometry (flat plane or volume, not a spherical surface) the ground state population tends to $N$ (particle number) as $T \rightarrow 0$ in any number of dimensions, so this is not the right consideration to decide whether there is a phase transition. In 3D I guess there is a discontinuity in $dn_0/dT$ at some $T$, and this is the transition temperature. I guess there is no such discontinuity in 2D. Is that right? $\endgroup$ Commented Mar 24, 2023 at 18:12
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$\begingroup$ I think the GS population tending to N as T goes to 0 is true for any (single particle) system, just from the limit of the Boltzmann distribution. I don't know who coined the term but I usually say that there is a difference between "boson proliferation" and "boson condensation", in that the latter is triggered by a phase transition, and is associated with either an order parameters, topological ordering, discontinuities etc. $\endgroup$ Commented Mar 27, 2023 at 17:04
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1$\begingroup$ For BEC in 2D, if you do the integral, the critical temperature is Tc = 0. This agrees with the Mermin-Wagner theorem, that says that symmetries cannot be spontaneously broken at finite temperature for $d\leq 2$. Also, the order parameter for BEC usually is Off Diagonal Long Range Order (ODLRO), also quantified by some formula with the wavefunction. For BKT and other 2D or finite system transitions, one sometimes talk about quasi-off-diagonal-long-range order $\endgroup$ Commented Mar 27, 2023 at 17:07