N.21 Superfluidity versus BEC

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 ${k_{\rm B}}T$ and the quantum energy scale formed from $\hbar^2$$\raisebox{.5pt}{$/$}$$2m$ and the number of particles per unit volume. There are likely to be other processes that scale with those quantities besides macroscopic amounts of atoms getting dumped into the ground state.

Still, there is not much doubt that the transition is due to the fact that helium atoms are bosons. The isotope $\fourIdx{3}{}{}{}{\rm {He}}$ that is missing a neutron in its nucleus does not show a transition to a super­fluid until 2.5 mK. The three orders of magnitude difference can hardly be due to the minor difference in mass; the isotope does condense into a normal liquid at a comparable temperature as plain helium, 3.2 K versus 4.2 K. Surely, the vast difference in transition temperature to a super­fluid is due to the fact that normal helium atoms are bosons, while the missing spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$ neutron in $\fourIdx{3}{}{}{}{\rm {He}}$ atoms makes them fermions. (The eventual super­fluid transition of $\fourIdx{3}{}{}{}{\rm {He}}$ at 2.5 mK occurs because at extremely low temperatures very small effects allow the atoms to combine into pairs that act as bosons with net spin one.)

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 [17, 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.58}.

Tilley and Tilley [45, 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 $T_\lambda$ marks the onset of Bose-Einstein condens­ation in liquid $\strut^4$He.” Sure, if you think that a curve reaching a maximum of one exponentially has a similarity to one that reaches a maximum of one with infinite curvature. And note that this compares two completely different quantities. It does not compare curves for particles in the ground state for both systems. It is quite generally believed that the condensate fraction in liquid helium, unlike that in true Bose-Einstein condens­ation, does not reach one at zero temperature in the first place, but only about 10% or so, [45, pp. 62-66].

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 [45, 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 [37]:

1. Bose-Einstein condens­ation is a property of the ground state, while super­fluidity is a property of the excited states.
2. Ideal Bose-Einstein condensates are not super­fluid.
3. Below 1 K, essentially 100% of the helium atoms flow without viscosity, even though only about 7% is in the ground state.
4. In fact, there is no reason why a system could not become a super­fluid even if only a very small fraction of the atoms were to form a condensate.

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.