# Are BECs superfluid or vice versa or what?

## Main Question or Discussion Point

I'm a physicist doing a course on condensates, superfluidity and superconductivity and I'm confused as to how these states overlap. Are all BECs necessarily superfluid? Are all superfluids necessarily BECs? The literature is incredibly ambiguous

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Superfluidity is often produced at the same time as BEC. BEC is often produced without superfluidity. Superfluidity is occasionally produced without BEC.

Superfluidity is occasionally produced without BEC.
What!? Can you provide some more details for us, as this sounds interesting.

Superfluidity is characterised by non-classical rotational inertia, i.e. the existence of vortices of a quantum nature. BEC is a macroscopic condensation. In 2D or 1D, it is not possible have an ordered state such as a BEC; but nevertheless, it is possible to have superfluidic properties.

Superfluidity is often produced at the same time as BEC. BEC is often produced without superfluidity. Superfluidity is occasionally produced without BEC.

Cthugha
In 2D or 1D, it is not possible have an ordered state such as a BEC; but nevertheless, it is possible to have superfluidic properties.
It is widely accepted that microcavity polaritons also undergo Bose-Einstein condensation and form a 2D BEC. However this is quite an unusual BEC as it can only have limited spatial size and is always in a state of nonequilibrium.

For completeness, I should mention that there are (at least) three widely used definitions of BEC. The budding physics student should probably not worry too much about the differences, but they do matter.

1. Order via broken U(1) symmetry. Usually signalled by the existence of a non-zero <c>. Downside: can't actually happen in any finite system.
2. Off-diagonal long-range order. The existence of <c*(r) c(0)> as r goes to infinity. Again, not really appropriate for any system where the limit can't actually be taken (most atomic BECs are pretty small).
3. The third is somewhat more technical: the existence of a "macroscopic" eigenvalue to the reduced density matrix <c*(r) c(0)>. Really, it's just saying that there is some eigenstate in the density matrix which is dominant over the others. Usually, the ratio will be something like 1:N where N is the number of particles. One can obviously have more marginal cases (1:100 or less). Neverthless, it actually includes the previous two as special cases.

The first definition is probably the most widely used, usually taught starting from advanced undergraduate level. In that definition, a BEC requires a broken symmetry order, which as a matter of principle is not possible in 2D or 1D because thermal fluctuations would destroy it at any non-zero temperature.

I'm not a polariton expert, but my understanding is that the transition seen is actually of a Kosterlitz-Thouless kind, rather than the usual textbook BEC. I personally do not think these two transitions are the same, essentially because one is topological and one actually involves a more classical symmetry change.