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On the Wikipedia page on superfluidity one can find the sentence

not all Bose-Einstein condensates can be regarded as superfluids, and not all superfluids are Bose–Einstein condensates.

So I was wondering if someone here can give me an example of a system which is superfluid, but not a Bose-Einstein condensate.

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In my opinion the wikipedia quote in the question is inherently misleading.

The two concepts (superfluidity and condensation) are certainly related, but also very much different, as they describe things at different scales. Superfluidity comes about from the hydrodynamic description, thus having to do with macroscopic properties of the system, and transport in particular. Bose-Einstein condensation, on the contrary, occurs at the level of the microscopic description.

I would say that only in systems of weakly interacting bosonic atoms is the relation straightforward. In particular, in this system, superfluid fraction is (to a good approximation) equal to the fraction of condensed atoms. This is not the case in other systems, where microscopic description is much more involved. Helium is a prime example, as its atoms interact strongly.

To reiterate, the point is that the microscopic description is sometimes so far from the transport properties of the system that the connection between the two becomes very involved.

For example, if one defines superfluidity for transport of momentum, a rigid crystal is a superfluid (c.f. my related question). Now if you consider that to be a condensate of something (e.g. phonons) is another question. People typically think of crystals in terms of atoms on a lattice, and not as condensates of phonons.

Similarly, an easy-plane ferromagnet is a spin superfluid, even though this has arguably to do with magnon condensation. See this review by Sonin for more details.

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Some may exclude superfluid 3He from being a Bose-Einstein condensate because it obeys Fermi-Dirac statistics. However, this viewpoint is also not quite clear cut as the 3He form Cooper pairs which then condense. However, even those pairs do not obey Bose-Einstein statistics but nonetheless condense.

Therefore this question is a little murky and Wikipedia is making a claim that is not as transparent as it seems. Annet's book Superfluids, Superconductors and Condensates is a good reference for this topic.

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