# Solid Helium

In recent months, I’ve mentioned super-solids a couple of times, which is a bit unusual for something we haven't been sure actually exists. However, a recent paper seems to offer some quite strong confirmation that super-solids are real. That means it is time to delve into the weird and wonderful world of low-temperature helium.

Helium is, without a doubt, the Universe’s weirdest material, beating out molecular hydrogen by a rather long nose. The key to helium’s strangeness is that it is normally a boson: a helium-4 atom consists of two protons, two neutrons, and two electrons, which sums to an even number, making a composite boson.

A sensitive torsional oscillator filled with solid helium appeared to indicate supersolid behavior in a 2004 experiment by Eunseong Kim and Moses Chan. (a) In a torsional-oscillator experiment, the period of the torsional oscillator is related to the moment of inertia of the sample cell and its contents and to the spring constant of the torsion rod. If the contents decouple from the motion, the oscillation period shifts.

## Helium is confusing

What does all that mean? It means that when cold enough, a group of helium atoms can enter the same quantum state. Even though they are spread out over a whole vessel, they all know something about the condition of their distant neighbors. This enables the helium atoms to flow without resistance, a state called a superfluidity. It's good company among other weird and wonderful properties of helium.

There is another type of helium that only has a single neutron (so two protons, one neutron, and two electrons), which means that it is not a composite boson. Instead, it is a fermion. When cold, these helium atoms cannot enter the same quantum state, so they don’t become superfluid. But, cool them enough, and two helium atoms can pair up to create a composite boson. At that very low temperature, superfluidity also emerges in helium-3.

1. Helium can exist as a solid. However, not only does the temperature need to be low enough, the pressure must also be high enough. This is due to repulsive interatomic forces, which (if I recall correctly) are electrostatic and quantum (the latter in the form of ground state energy that for helium is unusally high).
2. Feb 11, 2021 One of the unusual properties of helium is that it cannot exist as a solid or liquid at normal pressures, even at extremely low temperatures. At a pressure of roughly 360 pounds per square inch (2.5 megapascals), the transition between liquid and solid, or the melting point, is -458°F (0.95 Kelvin). The boiling point is -452°F (4.22 Kelvin).

Neither helium-3 nor helium-4 can become solids at atmospheric pressure. Instead, they become solids at 20-40 atmospheres. As a solid, at the right temperature, there are predictions that helium-4 can enter the super-solid state, while helium-3, which is not a boson, will not. The problem is that the super-solid is also very hard to detect. It hides among other changes to the elastic properties of solid helium.

## What is a super-solid?

A super-solid makes its presence known by flowing without resistance. However, what does it mean to say that a solid flows?

When helium (either type) becomes a solid, it crystalizes. That means that all the atoms hold themselves in a fixed arrangement with each other—to give one example, atoms can line up so that they're at the corners of a cube. As solids form, however, some positions that should have atoms do not. Others are out of position. When pressure is applied, atoms can move into these vacant positions, creating new vacant positions. As the atoms shuffle along, the solid flows.

To flow, the atoms have to have sufficient energy to leave their current location before they can move to new locations. As the temperature goes down, atoms have less energy and can no longer move. That means that the rate of flow should decrease with temperature.

If a material is in a super-solid state, however, then atoms can move from hole to hole because the quantum properties of the superfluid state tell the atoms where the holes are (so to speak) and allow them to move. These quantum effects get stronger with reduced temperature, so the rate of flow increases with decreasing temperature.

Increasing flow with reduced temperature has been observed in Helium-4. Unfortunately, it was not quite the smoking gun that the researchers were looking for, because the elastic properties of the solid also play a role. As the temperature decreases, there is competition between reduced movement of the atoms because they have less energy and increased movement because the solid as a whole is more able to transmit any applied force to a sensor. Maybe the increased flow was actually a change in elastic properties?

## Helium-3 to the rescue

To make the case for super-solidity, researchers turned to a form of helium that does not turn into a super-solid: helium-3.

The researchers repeated their experiments with solid helium-3 and observed that the rate of flow decreased with decreasing temperature, exactly as expected for a normal solid. And, because the elastic properties of helium-3 are nearly identical to that of helium-4, the researchers were able to eliminate that as an effect.

Indeed, the researchers were able to distinguish the elastic motion of the solid and the flow of the solid via atomic motion between vacancies. They showed that flow proceeds quite differently for helium-4 compared to helium-3.

Then came the surprise. At the lowest temperatures, the flow rate of helium-3 stopped decreasing. It didn’t exhibit super-solid properties, but it also stopped behaving like a normal solid.

If you recall from above, helium-3 can become a superfluid at very low temperatures, because the individual atoms can pair up to create bosons. It should also be possible for this to occur for solid helium-3. The researchers were not at a low-enough temperature to expect a helium-3 super-solid. But the temperature was low enough that maybe, just maybe, some pairing was occurring, which was allowing some super-solid properties to start to become apparent.

That last conclusion is a bit speculative in my opinion. However, the contrast between the behavior of helium-3 and helium-4 is quite stark. That alone makes the case for the existence of the super-solid state much stronger.

Physical Review Letters, 2018, DOI: 10.1103/PhysRevLett.121.225304 (About DOIs)

Experiments indicate that, as in a superfluid, mass can flow through solid helium-4 without viscous resistance. Recent calculations shed light on how this may happen thanks to defects in the crystal lattice.

Is it really possible that mass flows without viscous dissipation through a quantum solid? In other words, could a solid be “supersolid”—that is, a superfluid and a solid at the same time? Some experiments suggest so. No one really understands this paradoxical phenomenon yet [1], but a group of six theorists has given us some insight on how this may happen. Writing in Physical Review Letters, L. Pollet and M. Troyer (ETH), M. Boninsegni (University of Alberta), N. Prokof’ev and B. Svistunov (University of Massachusetts), and A. Kuklov (City University of New York) explain how some types of defects in solid helium-4 support mass superflow [2]. It is now up to the experimental groups to check if they are right.

The story began four years ago, when Eunsong Kim and Moses Chan (Pennsylvania State University) filled a little box with solid helium-4. The box was suspended by a rod with which it made a so-called “torsional oscillator” [3,4] that rotates one way, then the other, as in Video 1[5]. The oscillator had a resonance at a frequency of about 1 kHz, which depended on the oscillating mass. Below a temperature of order 100 mK they observed that the resonance frequency increased as if some of the mass inside the box had decoupled from the moving walls, and from this they proposed that solid helium was perhaps a “supersolid.”

Although paradoxical, in the 1960s A. F. Andreev and L. M. Lifshitz and other theorists thought that this phenomenon was possible in quantum crystals like helium-4 [6]. Their idea was the following: even in the zero-temperature limit, most of the atoms would be localized at the nodes of a periodic lattice but some of the nodes would be vacant. These vacancies could tunnel very quickly from site to site and be delocalized, which would lower the vacancy energy so much that even the ground state of the crystal would contain a nonzero density of these so-called “zero-point vacancies.” As a consequence, the crystal would be “incommensurate,” which means that the number of atoms would be different from the number of sites of the crystal lattice.

In helium-4, where atoms are Bose particles, the vacancies would also be Bose particles. At low temperature, these vacancies could then form a “Bose-Einstein condensate,” that is, a macroscopic matter wave. A vacancy moving one way being equivalent to an atom moving the other way, superfluid mass transport would be possible through this surprising crystal. But at the same time, the crystal would show solid properties due to its localized atoms: it would react elastically to a shear stress; it would be both elastic and superfluid!

This idea is interesting but it could not apply to helium: solid helium-4 is commensurate because the energy of vacancies is not zero but rather on the order of 10 K, so that at 100 mK, the probability of finding a vacancy is of order $e-100$, which is totally negligible. Moreover, most physicists in this field agree that a commensurate crystal cannot be supersolid [7]. So what is happening in these helium crystals? The experiments by Kim and Chan have been repeated by several groups working at Cornell, Keio University (Tokyo), and others. But successive experiments have shown that, if it really exists, supersolidity is related to disorder in the solid samples [8,9]. This disorder could be dislocations, grain boundaries, or ill-crystallized regions, which could be liquid or even glassy.

As a result of these experimental developments, predictions were made in 2007 by the aforementioned theorists [10,11]. As shown in Fig. 1, they found that the region inside screw dislocations is superfluid [10] and predicted that helium inside the grain boundaries between crystal grains in polycrystals should also be superfluid [11]. This confirms that such defects may allow superfluidity, but why does it occur? Thanks to the new work by Pollet et al.[2] the underlying physical mechanisms are becoming clear. They find that inside a dislocation or a grain boundary, the local stress is anisotropic and sufficient to bring the vacancy energy to zero, so that the defect is invaded by vacancies that are mobile and superfluid. In a perfect crystal, an invasion by vacancies would lead to melting but inside a dislocation it does not. As a result, Pollet et al. show that solid helium could contain a network of defects, and if these defects are connected to each other, mass could flow from one side of the crystal to the other without friction.

With Pollet’s new results, have we really understood what happens in torsional oscillators? We have clearly progressed, but I don’t think the story is finished. One reason for my doubts is that dislocations are very small objects, about 1 nm in diameter. In order to build a measurable superfluid fraction, one would need an enormous density of dislocations. Some researchers such as Philip Anderson think that there could be a kind of proximity effect, as in superconductors [12]. In this scenario, dislocations are sources of vacancies that may invade the crystal and make it superfluid, but this does not seem to be observed in numerical simulations.

Another difficulty comes from measurements of the shear modulus of helium-4 crystals. Day and Beamish [13] have found that below 100 mK, when the crystals seem to be supersolid, they become stiffer, not softer as one could have expected since mass is supposed to move inside. Furthermore, the way that the elastic shear modulus increases below 100 mK is very similar to the manner in which the frequency of the torsional oscillator increases. The stiffening could be the result of dislocations becoming pinned by isotopic impurities (for example, helium-3 atoms even at very small concentrations). But why would the pinning of dislocations be necessary for supersolidity to appear? As far as I can tell, at the moment nobody knows.

After nearly five years of hard work in several laboratories, helium crystals still appear to be very strange and it is our understanding of the solid state of matter that needs to be reconsidered in depth. I believe that Pollet et al. are on the right track in trying to understand defects in crystals, but the game is far from over. After many years of work on quantum crystals, the understanding of quantum defects has only just started. Of particular interest would be experiments on single defects or on solid samples with well-characterized disorder.

## References

### Solid Helium Phase Diagram

1. Sébastien Balibar and Frédéric Caupin, J. Phys. Condens. Matter20, 173201 (2008)
2. L. Pollet, M. Boninsegni, A. B. Kuklov, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Phys. Rev. Lett.101, 097202 (2008)
3. E. Kim and M. H. W. Chan, Nature427, 225 (2004)
4. E. Kim and M. H. W. Chan, Science305, 1941 (2004)
5. A. F. Andreev and L. M. Lifshitz, Sov. Phys. JETP29, 1107 (1969)
6. For a review of these theoretical aspects, see N. Prokof’ev, Adv. Phys.56, 381 (2007)
7. A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett.97, 165301 (2006); 98, 175302 (2007); A. S. Rittner and J. Reppy (to be published)
8. S. Sasaki, R. Ishiguro, F. Caupin, H. J. Maris, S. Balibar, Science313, 1098 (2006); S. Sasaki, F. Caupin, and S. Balibar, Phys. Rev. Lett.99, 205302 (2007)
9. M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Phys. Rev. Lett.99, 035301 (2007)
10. L. Pollet, M. Boninsegni, A. B. Kuklov, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Phys. Rev. Lett.98, 135301 (2007)
11. P. W. Anderson, personal communication (2008)
12. James Day and John Beamish, Nature450, 853 (2007)

#### Local Stress and Superfluid Properties of Solid $\mathrm{He}_{4}$

L. Pollet, M. Boninsegni, A. B. Kuklov, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer

### Solid Hydrogen

Published August 25, 2008

## Related Articles

Superfluidity

### Excitations in the Ultimate Quantum Fluid

Researchers have measured superfluid helium’s full dispersion spectrum, explaining discrepancies in previous studies and constraining theories of superfluidity. Read More »

Condensed Matter Physics

### Can Helium Be A Solid

Sound waves reveal the unique properties of the unitary Fermi gas, a model system for describing certain superconductors and forms of nuclear matter. Read More »

Superfluidity

### Solid Helium 3

Trapping a molecule inside a liquid helium nanodrop allows clean measurements of the molecule’s vibrations. Read More »