Many texts and most web sources suggest quite strongly, without
explicitly saying so, that the so-called lambda
phase
transition at 2.17 K from normal helium I to superfluid helium II
indicates Bose-Einstein condensation.
One reason given that is that the temperature at which it occurs is
comparable in magnitude to the temperature for Bose-Einstein
condensation in a corresponding system of noninteracting particles.
However, that argument is very weak; the similarity in temperatures
merely suggests that the main energy scales involved are the classical
energy
Still, there is not much doubt that the transition is due to the fact
that helium atoms are bosons. The isotope
While the fact that the helium atoms are bosons is apparently essential to the lambda transition, the conclusion that the transition should therefore be Bose-Einstein condensation is simply not justified. For example, Feynman [18, p. 324] shows that the boson character has a dramatic effect on the excited states. (Distinguishable particles and spinless bosons have the same ground state; however, Feynman shows that the existence of low energy excited states that are not phonons is prohibited by the symmetrization requirement.) And this effect on the excited states is a key part of superfluidity: it requires a finite amount of energy to excite these states and thus mess up the motion of helium.
Another argument that is usually given is that the specific heat varies with temperature near the lambda point just like the one for Bose-Einstein condensation in a system of noninteracting bosons. This is certainly a good point if you pretend not to see the dramatic, glaring, differences. In particular, the Bose-Einstein specific heat is finite at the Bose-Einstein temperature, while the one at the lambda point is infinite.. How much more different can you get? In addition, the specific heat curve of helium below the lambda point has a logarithmic singularity at the lambda point. The specific heat curve of Bose-Einstein condensation for a system with a unique ground state stays analytical until the condensation terminates, since at that point, out of the blue, nature starts enforcing the requirement that the number of particles in the ground state cannot be negative, {D.57}.
Tilley and Tilley [47, p. 37] claim that the
qualitative correspondence between the curves for the number of atoms
in the ground state in Bose-Einstein condensation and the fraction of
superfluid in a two-fluid description of liquid helium “are
sufficient to suggest that
Since the specific heat curves are completely different, Occam’s
razor would suggest that helium has some sort of different phase
transition at the lambda point. However, Tilley and Tilley
[47, pp. 62-66] present data, their figure 2.17,
that suggests that the number of atoms in the ground state does indeed
increase from zero at the lambda point, if various models are to be
believed and one does not demand great accuracy. So, the best
available knowledge seems to be that Bose-Einstein condensation,
whatever that means for liquid helium, does occur at the lambda point.
But the fact that many sources see evidence
of
condensation where none exists is worrisome: obviously, the desire to
believe despite the evidence is strong and widespread, and might
affect the objectivity of the data.
Snoke & Baym point out (in the introduction to Bose-Einstein Condensation, Griffin, A., Snoke, D.W., & Stringari, S., Eds, 1995, Cambridge, p. 4), that the experimental signal of a Bose-Einstein condensate is taken to be a delta function for the occupation number of the particles [particle state?] with zero momentum, associated with long-range phase coherence of the wave function. It is not likely to be unambiguously verified any time soon. The actual evidence for the occurrence of Bose-Einstein condensation is in the agreement of theoretical models and experimental data, including also models for the specific heat anomaly. However, Sokol points out in the same volume, (p. 52): “At present, however, liquid helium is the only system where the existence of an experimentally obtainable Bose condensed phase is almost universally accepted” [emphasis added].
The question whether Bose-Einstein condensation occurs at the lambda point seems to be academic anyway. The following points can be distilled from Schmets and Montfrooij [39]:
The statement that no Bose-Einstein condensation occurs for photons applies to systems in thermal equilibrium. In fact, Snoke & Baym, as mentioned above, use lasers as an example of a condensate that is not superfluid.