If a branch of physics were to be judged by the number of Nobel laureates it has produced, then superfluidity would surely rank among the most successful. The field has borne nearly 20 laureates, from the award of the 1913 Nobel Prize for Physics to Heike Kamerlingh Onnes, who discovered superconductivity, to that of the 2003 prize to Alexei Abrikosov, Vitaly Ginzburg and Tony Leggett for their contributions to the theory of superconductors and superfluids. The reason why is simple: these counterintuitive phenomena, whereby below a certain temperature matter flows without resistance, are rare examples of quantum-mechanical behaviour seen at the macroscopic scale.

“There is a saying among condensed-matter physicists that what the Big Bang is to cosmology the ‘supers’ are to atomic physics,” says Philip Anderson of Princeton University in the US, who shared the 1977 Nobel Prize for Physics for his work on the electronic structure of magnetic and disordered materials. “Most people are unaware how much of our conceptual understanding of the world around us comes from this field, such as broken symmetry and the Higgs mechanism.”

Superfluidity was discovered in the liquid phase in 1938, when Pjotr Kapitsa – who shared the 1978 Nobel prize for the work – found that liquid helium-4 suddenly behaves as if it has zero viscosity when cooled below a temperature of about 2 K. With no resistance to flow, a superfluid can do bizarre things such as creep up the sides of a vessel containing the material or pass through holes just a few atoms wide. Superconductivity, a similarly dramatic low-temperature phenomenon in which electrical current flows without resistance, is due to the superfluidity of electron pairs. However, in 2004 Moses Chan of Penn State University in the US and his then graduate student Eun-Seong Kim reported evidence for superfluidity in a much more unlikely setting: the atomic lattice of bulk-solid helium-4.

Such a “supersolid” phase of matter would flow through a classical solid as if it were not there. Like superfluidity in a liquid, this weird behaviour is predicted to be a consequence of Bose–Einstein condensation – a phase transition in which all the particles in a system collapse to the same ground state and can therefore no longer be treated as individual entities moving at random. Such quantum degeneracy is possible because helium-4 atoms are bosons, i.e. particles that have integer multiples of spin angular momentum.

Since 1995, when physicists in the US created the first Bose–Einstein condensate (BEC) in the gaseous phase by cooling bosonic rubidium and sodium atoms to a few hundred nanokelvin – an achievement for which they were awarded the Nobel prize in 2001 – these systems have provided an unprecedented “laboratory” in which to study the mechanisms responsible for superfluidity. If supersolids do indeed exist, it would mean that Bose–Einstein condensation has been observed in the solid as well as in the liquid and gas phases.

Although several groups have since verified Chan’s 2004 claim, recent experiments that reveal the role played by crystal disorder have raised doubts about whether such a supersolid phase has been observed at all. Moreover, theorists disagree over precisely what the mechanism behind supersolidity might be. “The situation is very murky,” admits Chan.

Super signature

The possibility of a supersolid, in which a Bose–Einstein condensate would coexist with the regular atomic lattice of a solid, is not new: it was first predicted by Russian theorists Alexander Andreev and Ilya Liftshitz in 1969. Rather than individual atoms undergoing condensation, they suggested that the supersolid state emerged from the condensation of atomic vacancies. In most solids vacancies are created when an atom at a particular lattice site is liberated, usually by thermal energy. But in the case of helium-4, which only solidifies at extremely low temperatures and high pressures (see figure 2), the atoms are so weakly bound that vacancies may exist even at absolute zero due to quantum “zero-point” energy.

It was the possible existence of such zero-point vacancies that convinced John Goodkind of the University of California in San Diego back in the 1980s that the supersolid state was worth investigating. “The existence of BEC in the solid phase would be as significant as the discovery of superfluidity in liquid helium,” he says. “It would be a new state of matter that is counterintuitive.” Using ultrasound to probe the microscopic properties of helium-4 as it was cooled, Goodkind noticed a sudden increase in the velocity and dissipation of the sound waves near 200 mK, which he interpreted as being due to a thermodynamic phase change – possibly a BEC (1997 J. Low Temp. Physics 109 409). Unfortunately, at that point Goodkind’s funding ran out, but his anomalous result had caught the attention of Kim and Chan.

Excited by the prospect of observing a new phenomenon that would push their experimental ingenuity to the limit, Kim and Chan set about searching for the supersolid state in 1999. They used a torsional oscillator, which consisted of a cylindrical cell filled with high-pressure helium-4 embedded within a porous Vycor glass disk. The cell, which was suspended from a rod, could then be rotated back and fore. By monitoring the oscillation period while the cell was cooled close to absolute zero, the researchers were able to look for signs of nonclassical rotational inertia – a sudden drop in the oscillation period of the cell that would mark the onset of superfluidity in the solid helium-4 inside it. When the sample reached a temperature of 175 mK, this is exactly what they observed (see figure 3).

“At sufficiently low temperature, solid helium-4 does not behave as a solid,” says Chan. “I lost count of how many control experiments we performed with different cells to convince ourselves of the phenomenon.” Kim and Chan published their results in January 2004, concluding that the drop in rotational inertia they had observed was “probably” due to 2% of the helium-4 undergoing Bose–Einstein condensation to the supersolid state (Nature 427 225). Flying in the face of classical physics, this ghostly component of the system remained at rest in the laboratory frame, passing effortlessly in and out of the normal atomic lattice as the cell rotated about it.

Crucially, Kim and Chan saw no such behaviour when shortly afterwards they repeated the experiment using helium-3. Unlike their heavier bosonic cousins, helium-3 atoms are fermions – that is, they have half-integer spins and are thus prevented by the exclusion principle from forming a BEC. It is, however, possible for helium-3 atoms to produce a BEC if they first pair up to form bosons, a process akin to the pairing of electrons in superconductivity, which can take place at much lower temperatures. Indeed, the first observation of superfluidity in helium-3 in 1972 at just 2 mK – a feat for which David Lee, Douglas Oscheroff and Robert Richardson were awarded the 1996 Nobel Prize for Physics – was a clear sign of the link between superfluidity and Bose–Einstein condensation.

Given the potential significance of discovering the supersolid phase, however, it was clear that there was still some work to do before Kim and Chan could drop the word “probable” from their claim. In particular, there was the possibility that the “non-classical inertia” that they recorded was simply due to a layer of liquid helium-4 in the sample that had become trapped in the nanometre-sized pores of the Vycor glass disk in which their helium-4 had to be contained to keep it under sufficient pressure. The pair therefore repeated the experiment using a bulk sample of solid helium-4, observing a drop in rotational inertia that suggested 1% of the sample had become supersolid. “What this showed was that we were observing a macroscopic, not a local, quantum phenomenon,” remarks Chan. The pair’s definitive claim of the discovery of a supersolid was duly published in September 2004 (Science 305 1941).

Perfect confusion

Spurred on by the results from Penn State, other research groups soon began attempts to replicate Kim and Chan’s torsional-oscillator experiment. By the start of 2006 three such groups had confirmed the supersolid result: Keiya Shirahama and co-workers at Keio University in Japan (arXiv.org/abs/cond-mat/0607032); Minoru Kubota and his students at Tokyo University (arXiv.org/abs/cond-mat/0702632); and John Reppy – who searched unsuccessfully for the supersolid state in the late 1970s – and his student Sophie Rittner at Cornell University in the US (Phys. Rev. Lett. 97 165301).

All of these groups had confirmed that below a temperature of about 200 mK some 1% of solid helium-4 flows in a non-classical way. However, in the Cornell experiment Reppy and Rittner also found that by maintaining the temperature of the solid helium-4 close to its melting point for several hours and then slowly cooling it down again, they could reduce the supersolid signal to less than 0.05% and even make it disappear completely. Since such “annealing” is expected to reduce the level of imperfection in the solid, this suggested that the observed supersolid behaviour is not a universal property of bulk-solid helium-4 but the result of defects or imperfections in the crystal structure.

Several further experiments have supported this disorder interpretation. Earlier this year, for instance, Reppy repeated Kim and Chan’s bulk helium-4 experiment but with samples that were heated and re-frozen extremely rapidly so as to introduce disorder, finding that up to 20% of the solid had become superfluid (Phys. Rev. Lett. 98 175302). Meanwhile, one of Chan’s new students, Tony Clark, has found that when an ultra-pure single crystal of helium-4 is placed in a torsional oscillator, the supersolid fraction is just 0.3%. “I found all these new results very confusing,” says Chan. “What is puzzling is that the solid helium that was confined in Vycor glass in our initial experiment, and in porous gold in another [2005 J. Low Temp. Phys. 138 159], should have been of even worse quality than that in Reppy’s latest results – yet we found a supersolid fraction of just 2%.”

The case for supersolids appeared to be further weakened by independent “DC flow” experiments performed by John Beamish and co-workers at the University of Alberta in Canada shortly after Kim and Chan’s 2004 result. The team placed solid helium-4 in an array of capillaries and searched for direct evidence of supersolidity by creating a pressure difference in the sample and seeing whether any mass flowed as a result (2006 Phys. Rev. Lett. 96 105304 and 95 035301). “The behaviour of solid helium-4 is quite different to that of superfluids,” says Beamish. “Our results show that right down to temperatures of 30 mK, solid helium-4 does not flow.”

Theorists to the rescue

Many theorists are not surprised that the outcome of torsional-oscillator experiments should depend strongly on the conditions under which the solid helium-4 sample is prepared. In part this is due to calculations performed by Nikolay Prokofev and Boris Svistunov at the University of Massachusetts and several others, including David Ceperley at the University of Illinois, showing that vacancies cannot exist at absolute zero. As a result, superfluidity in a helium-4 crystal may not be due to the Bose-Einstein condensation of vacancies. Indeed, last year Chan himself cast doubt on this explanation when he found that the supersolid fraction did not decrease as a function of pressure, as it should have done if the observed drop in rotational inertia was due to the formation of a BEC (Phys. Rev. Lett. 97 115302).

But an even more compelling reason to doubt Andreev and Liftshitz’s 1969 prediction is the recent progress made in understanding how certain types of crystal defect might produce a supersolid-like signature. “Based on first-principle numerical calculations, we guarantee the existence of at least two supersolid phases of helium-4,” says Svistunov. One of these, he claims, occurs in superfluid grain boundaries, the layers about three atoms wide that separate regions of different crystal orientation (Phys. Rev. Lett. 98 135301). The other is a superfluid glass phase, in which the helium-4 atoms form a spatially disordered but metastable “superglass” state (Phys. Rev. Lett. 96 105301).

Sebastien Balibar at the Laboratory for Statistical Physics at the Ecole Normale Supérieure in Paris has recently found evidence for superfluidity in helium-4 crystals with grain boundaries. Although a network of such entities would produce non-classical rotational inertia that would show up in a torsional oscillator, he and his colleagues instead used a barometer-like device to look for direct signs of supersolidity (see figure 6). Similar to the flow experiments of Beamish and coworkers, their idea was to contain solid helium-4 inside a glass tube and use a camera to visualize the flow of mass in response to a height difference between the inside and outside of the tube (Science 313 1098).

“Good-quality crystals do not exhibit flow, while those with grain boundaries and hence a definite amount of disorder do,” says Balibar. However, he concedes that it is difficult for a network of superfluid grain boundaries to account for the large supersolid fractions observed in the torsional-oscillator experiments. “It could be that grain boundaries connect liquid or glassy regions in disordered crystals to produce a large supersolid signature,” he says. Since such grain boundaries are unlikely to be aligned in narrow channels in a material, this may explain why Beamish and co-workers did not see such flow in their capillary experiments.

In February this year, Victor Grigor’ev and colleagues at the Academy of Sciences of Ukraine reported evidence for Prokofev and Svistunov’s glass phase in solid helium-4 (arXiv.org/abs/cond-mat/0702133). By precisely measuring the pressure of their sample as a function of temperature, T, they found a departure from the expected classical T⁴ dependence to a T² dependence at temperatures below 300 mK. The team claims this is what one would expect if a glassy phase had formed, and suggests that such a phase might explain the anomalous results seen so far in solid helium-4.

Chan hopes that studies like these, which look for direct thermodynamic signatures of supersolidity, will help settle the issue of what is responsible for the supersolid signature seen in his experiment. “Other than Goodkind’s original ultrasound experiment, the most clear-cut evidence for a supersolid so far still comes from torsional-oscillator measurements,” he says. “In my opinion, Balibar’s results are probably due to liquid helium-4 films flowing along “cracks” or grain boundaries and are not relevant to the supersolid phenomenon seen in our experiments.”

Chan’s group is now performing thermodynamic studies of its own. At the March meeting of the American Physical Society in Denver this year one of his new students, Xi Lin, presented measurements of the heat capacity of solid helium-4 as a function of temperature, showing that it has the expected classical form plus an additional peak at the same temperature (about 80 mK) where the torsional-oscillator signals show up. “We need to do one more control experiment to nail this before we can say whether the peak is related to the supersolid,” says Chan.

Super questions

So is Chan in line for the next Nobel prize in superfluidity? “Based on current experimental evidence, we do not know whether he is observing the supersolid state and therefore we do not know if it exists,” says Goodkind. “Only torsional-oscillator experiments explicitly show a signal – other studies of direct flow plus neutron and X-ray scattering experiments show no evidence for supersolidity and attempts to look for a thermodynamic signature have so far not been successful.” Goodkind is hopeful, however, that the variety of current experimental searches for supersolids – which include recent studies performed by Beamish that are similar to his own 1997 experiment – will be able to identify the cause of the anomalous behaviour seen by Chan and by him.

Tony Leggett at the University of Illinois, who in 1970 showed that non-classical rotational inertia was a marker of the supersolid state, thinks that the recent Rittner–Reppy results with disordered crystals are very telling. “To me these data suggest that what people have seen to date is anomalous kinetics, rather than non-classical rotational inertia,” he says. But he adds that to find out for sure we need a “Hess–Fairbank” experiment, in which solid helium-4 is subjected to DC rather than AC rotation as in a torsional oscillator.

Philip Anderson basically agrees. “None of the experiments are seeing superfluidity proper, but they do see evidence for a quantum fluid,” he says. Anderson calls this quantum fluid a vortex fluid because, he says, the evidence to date supports the idea that vorticity – a key parameter in describing superfluidity – is quantized but not that the vortices are frozen in.

Meanwhile, Chan – who prefers not to dwell on the possibility of a trip to Stockholm – is concentrating on the role of crystal defects, by systematically introducing disorder to solid helium-4 samples and measuring the supersolid response. “It is fun to be involved in an experiment that has got many theorists scratching their heads and encouraged experimental colleagues to join in the search,” he says. “Sometimes progress seems slow, and it can be frustrating to find yourself up a blind alley. But it is more fun to be the detective than to read the detective novel.”