Expt 6) Superfluidity in 4He

In 1938, Pyotr Kapitza and, independently, John F. Allen and Don Misener demonstrated that on cooling below the “lambda temperature”, the viscosity of liquid ⁴He suddenly reduced by orders of magnitude. Kaptiza coined the name “superfluidity” in his famous paper, where he also presciently compared superfluidity to superconductivity.

He used a rather ingenious experimental setup where the liquid flowed between two disks spaced about half a micron apart. By showing that the liquid helium in the superfluid phase flowed easily, while that in the normal phase did not, he estimated that the viscosity drops by about a factor of 1500. Strikingly, there is no data in the paper, but only a diagram of the apparatus he used to measure the viscosity (and to ensure that the flow of the liquid was laminar). Allen and Misener also published similar results, but their upper bound for the viscosity was orders of magnitude higher than Kapitza’s. They used flow through narrow capillaries to estimate the viscosity. While these results were quite definitive, it is worth remembering that experiments prior to these two had already shown a quite large drop in the viscosity on entering the superfluid state but with less spectacular results, presumably due to more turbulent flow. In these experiments, the researchers tried their best to ensure that the flow was largely laminar.

Below is a video showing a few of the dramatic effects in superfluid ⁴He.

Landau Theory and the Ginzburg Criterion

The Landau theory of second order phase transitions has probably been one of the most influential theories in all of condensed matter. It classifies phases by defining an order parameter — something that shows up only below the transition temperature, such as the magnetization in a paramagnetic to ferromagnetic phase transition. Landau theory has framed the way physicists think about equilibrium phases of matter, i.e. in terms of broken symmetries. Much current research is focused on transitions to phases of matter that possess a topological index, and a major research question is how to think about these phases which exist outside the Landau paradigm.

Despite its far-reaching influence, Landau theory actually doesn’t work quantitatively in most cases near a continuous phase transition. By this, I mean that it fails to predict the correct critical exponents. This is because Landau theory implicitly assumes that all the particles interact in some kind of average way and does not adequately take into account the fluctuations near a phase transition. Quite amazingly, Landau theory itself predicts that it is going to fail near a phase transition in many situations!

Let me give an example of its failure before discussing how it predicts its own demise. Landau theory predicts that the specific heat should exhibit a discontinuity like so at a phase transition:

However, if one examines the specific heat anomaly in liquid helium-4, for example, it looks more like a divergence as seen below:

So it clearly doesn’t predict the right critical exponent in that case. The Ginzburg criterion tells us how close to the transition temperature Landau theory will fail. The Ginzburg argument essentially goes like so: since Landau theory neglects fluctuations, we can see how accurate Landau theory is going to be by calculating the ratio of the fluctuations to the order parameter:

where is the error in Landau theory, quantifies the fluctuations and is the order parameter. Basically, if the error is small, i.e. , then Landau theory will work. However, if it approaches , Landau theory begins to fail. One can actually calculate both the order parameter and the fluctuation region (quantified by the two-point correlation function) within Landau theory itself and therefore use Landau theory to calculate whether or not it will fail.

If one does carry out the calculation, one gets that Landau theory will work when:

where is the reduced temperature, is the dimension, is the dimensionless mean-field correlation length at (extrapolated from Landau theory) and is the change in specific heat in units of , which is usually one per degree of freedom. In words, the formula essentially counts the number of degrees of freedom in a volume defined by  . If the number of degrees of freedom is large, then Landau theory, which averages the interactions from many particles, works well.

So that was a little bit of a mouthful, but the important thing is that these quantities can be estimated quite well for many phases of matter. For instance, in liquid helium-4, the particle interactions are very short-ranged because the helium atom is closed-shell (this is what enables helium to remain a liquid all the way down to zero temperatures at ambient conditions in the first place). Therefore, we can assume that , and hence and deviations from Landau theory can be easily observed in experiment close to the transition temperature.

Despite the qualitative similarities between superfluid helium-4 and superconductivity, a topic I have addressed in the past, Landau theory works much better for superconductors. We can also use the Ginzburg criterion in this case to calculate how close to the transition temperature one has to be in order to observe deviations from Landau theory. In fact, the question as to why Ginzburg-Landau theory works so well for BCS superconductors is what awakened me to these issues in the first place. Anyway, we assume that is on the order of the Cooper pair size, which for BCS superconductors is on the order of . There are about particles in this volume and correspondingly, and Landau theory fails so close to the transition temperature that this region is inaccessible to experiment. Landau theory is therefore considered to work well in this case.

For high-Tc superconductors, the Cooper pair size is of order and therefore deviations from Landau theory can be observed in experiment. The last thing to note about these formulas and approximations is that two parameters determine whether Landau theory works in practice: the number of dimensions and the range of interactions.

*Much of this post has been unabashedly pilfered from N. Goldenfeld’s book Lectures on Phase Transitions and the Renormalization Group, which I heartily recommend for further discussion of these topics.

Precision in Many-Body Systems

Measurements of the quantum Hall effect give a precise conductance in units of . Measurements of the frequency of the AC current in a Josephson junction give us a frequency of times the applied voltage. Hydrodynamic circulation in liquid 4He is quantized in units of . These measurements (and similar ones like flux quantization) are remarkable. They yield fundamental constants to a great degree of accuracy in a condensed matter setting– a setting which Murray Gell-Mann once referred to as “squalid state” systems. How is this possible?

At first sight, it is stunning that physics of the solid or liquid state could yield a measurement so precise. When we consider the defects, impurities, surfaces and other imperfections in a macroscopic system, these results become even more astounding.

So where does this precision come from? It turns out that in all cases, one is measuring a quantity that is dependent on the single-valued nature of the (appropriately defined) complex scalar  wavefunction. The aforementioned quantities are measured in integer units, , usually referred to as the winding number. Because the winding number is a topological quantity, in the sense that it arises in a multiply-connected space, these measurements do not particularly care about the small differences that occur in its surroundings.

For instance, the leads used to measure the quantum Hall effect can be placed virtually anywhere on the sample, as long as the wires don’t cross each other. The samples can be any (two-dimensional) geometry, i.e. a square, a circle or some complicated corrugated object. In the Josephson case, the weak links can be constrictions, an insulating oxide layer, a metal, etc. Imprecision of experimental setup is not detrimental, as long as the experimental geometry remains the same.

Another ingredient that is required for this precision is a large number of particles. This can seem counter-intuitive, since one expects quantization on a microscopic rather than at a macroscopic level, but the large number of particles makes these effects possible. For instance, both the Josephson effect and the hydrodynamic circulation in 4He depend on the existence of a macroscopic complex scalar wavefunction or order parameter. In fact, if the superconductor becomes too small, effects like the Josephson effect, flux quantization and persistent currents all start to get washed out. There is a gigantic energy barrier preventing the decay from the current-carrying state to the current non-carrying state due to the large number of particles involved (i.e. the higher winding number state is meta-stable). As one decreases the number of particles, the energy barrier is lowered and the system can start to tunnel from the higher winding number state to the lower winding number state.

In the quantum Hall effect, the samples need to be macroscopically large to prevent the boundaries from interacting with each other. Once the states on the edges are able to do that, they may hybridize and the conductance quantization gets washed out. This has been visualized in the context of 3D topological insulators using angle-resolved photoemission spectroscopy, in this well-known paper. Again, a large sample is needed to observe the effect.

It is interesting to think about where else such a robust quantization may arise in condensed matter physics. I suspect that there exist similar kinds of effects in different settings that have yet to be uncovered.

Aside: If you are skeptical about the multiply-connected nature of the quantum Hall effect, you can read about Laughlin’s gauge argument in his Nobel lecture here. His argument critically depends on a multiply-connected geometry.

Just a little thought on Aharonov-Bohm Destruction of Superconductivity

Many experimenters in the past have exploited the similarities between superconductors and superfluids to come up with new ideas. One of the most important of these analogies is the Hess-Fairbank effect in a superfluid (pdf!) and the corresponding Meissner effect in a superconductor. The Hess-Fairbank effect is often taken as the pre-eminent experimental signature of a superfluid.

For those who are unfamiliar with the Hess-Fairbank effect, let me describe it briefly. It was found that if one rotates a cylindrical container of helium slowly (i.e. with angular velocity less than half the critical velocity, ), and one cools below the transition temperature into the superfluid phase towards , that the liquid in the container remains at rest despite the rotating walls!

We can write the Hamiltonian for the liquid in the rotating container in the rotating frame (not the lab frame!):

Besides an unimportant centrifugal term which I’ve buried in the definition of , there is a strong mathematical analogy between the Hamiltonian above and that which describes a superconductor in the presence of a vector potential (in the lab frame):

One can see that corresponds to . Furthermore, if we assume a constant magnetic field (as in the Meissner effect), we can write for and the correspondence becomes .

Consider now what happens when one rotates the container containing the superfluid quickly (faster than the critical angular velocity). I have blogged previously about this scenario and the elegant experiment showing the proliferation of quantized vortices. In fact, if one rotates the cylinder quickly enough, the superfluidity is destroyed entirely.

Now, let us consider switching to a multiply connected geometry. It is important to note that the kinetic energy terms in the Hamiltonians remain the same if we were to use an annular container for the superfluid and a ring in the superconducting case with an Aharonov-Bohm flux through the center. For the annular container, the superfluidity would still be destroyed if the container were rotated quickly enough. This would lead one to conclude that for a superconducting ring, a strong enough Aharonov-Bohm flux would also destroy the superconductivity. There would not be an actual magnetic field applied to the superconductor (i.e. not the Meissner effect in the usual sense), but the superconductivity would be destroyed through a pseudo-Meissner effect nonetheless.

I have to admit that I have not come across an experimental paper demonstrating this effect, so please share if you know of one, as I’m sure this idea has been around for a while! It is just interesting to think that one could destroy superconductivity without actually ‘touching’ it with any kind of measurable field.

Balibar and his Beef with Science Magazine’s Depiction of a BEC

I’m coming to the end of reading Sebastian Balibar’s physics book (intended for a general audience) entitled The Atom and the Apple. Thematically, the book works by asking a basic question at the beginning of each chapter and seeing the wondrous science that has to be understood to answer such a basic question. The author pulls on a dangling thread and watches entire garments unravel. The book is interspersed with personal anecdotes, which gives the physics some semblance of humanity.

Just to whet your appetite a little, let me recount one of the amusing stories Balibar relays. In 1995, following the discovery of Bose-Einstein Condensation by the Colorado and MIT groups, Science had the following image on the cover (sorry, I couldn’t find a large picture in color anywhere, it’s supposed to be primarily blue if that helps). It shows marching soldiers, a supposed metaphor for BEC.

Balibar takes issue with this representation and goes onto say about the cover:

No, what bothered me was actually that march—those orderly
troops. True, I had bad memories of my own experiences with military marches. Before May 1968, the hierarchy at the École Polytechnique had little patience for the antics of its rebel students, and my deviance had cost me fifteen days in prison and gotten me barred from marching with my fellow students on the Champs Élysées. But regardless of these youthful memories, I saw in that march of atoms a basic error of interpretation as to what the recently discovered “Bose-Einstein” condensation was. The order of the actual condensate seemed radically different to me from the regimentation depicted on the magazine cover.
The artist at Science hadn’t just dreamt up this march all by him or herself, though: the military analogy had been hanging around in the public scientific discourse for a long time. Nevertheless, in becoming famous, this image threatened to distort the understanding of the discovery. I intended to denounce it firmly.

What Balibar had in mind about BEC is much more accurately depicted in this great little video.

The book also has some anecdotes about the history and controversy surrounding the Nobel Prizes awarded for superfluidity in liquid helium-4 among many other interesting historical detours. The discussion on radioactivity is also noteworthy. This charming little book is written with an approach that I feel more popular physics books should take, or Balibar could write a couple more himself.