Superfluid helium-3 in confined quarters

Liquid helium-3 and helium-4 are remarkable substances. They are quantum liquids, meaning that their behavior is governed by the laws of quantum mechanics. Because of their small atomic mass, each isotope exists in a liquid state down to the temperature of absolute zero. And at sufficiently low temperature, each becomes a superfluid. However, the two isotopes have very different properties because ³He is a fermion and ⁴He is a boson. As a result of their different statistics, superfluidity in ³He appears at a temperature one-thousandth of that at which superfluid ⁴He forms. A second difference is that ³He has multiple thermodynamic phases.

The discovery of superfluid ³He was recognized with two Nobel Prizes in Physics, one for experimental work and one for theory. The 1996 prize was awarded to David Lee, Douglas Osheroff, and Robert Richardson for their pioneering ³He NMR measurements, carried out in 1971 at Cornell University. ¹ (See Physics Today, December 1996, page 17 .) As the Cornell experiments were being conducted, Anthony Leggett, then at the University of Sussex, showed that superfluid ³He is the realization of the Bardeen-Cooper-Schrieffer (BCS) pairing theory of superconductivity in a quantum liquid; he thus made an important connection that is still relevant to research on superconductivity. ² Leggett’s theory also accounted for the two observed phases of pure superfluid ³He, and for it, he received a share of the 2003 Nobel Prize in Physics. (See Physics Today, December 2003, page 21 .)

In this article we highlight the discovery of new superfluid phases in ³He infused into amorphous, extremely light, high-porosity solids known as aerogels. Before 1995 it was not clear that the superfluid could actually survive in such a messy environment. (See the article by Moses Chan, Norbert Mulders, and John Reppy, Physics Today, August 1996, page 30 .) And if it could survive, what would be the properties of the disordered superfluid?

Research by the three of us addressed those questions on several fronts. Parpia used a torsional oscillator (described below) to look for the superfluid mass fraction, ³ and Halperin performed NMR measurements ⁴ that were key to the identification of the new phase of ³He as a superfluid of bound pairs with a nuclear spin of ℏ. Sauls and colleagues immediately began to develop a theory of superfluid ³He confined in random media. ⁵ We found that not only does the superfluid exist but also its phase transitions are well defined in temperature, pressure, and magnetic field. More recently, various researchers have demonstrated that by engineering anisotropy in the mean free path of the confined superfluid, they can control which superfluid phases are stable and can even force the appearance of new phases. But before discussing those phases, we begin with a brief description of the properties of unconfined superfluid ³He.

Pure helium-3

For a wide range of temperature below the Fermi temperature of about 1 K, ³He is well described by Lev Landau’s Fermi-liquid theory, which is formulated in terms of low-energy, fermionic excitations called quasiparticles. The thermodynamic and transport properties of the normal (not superfluid) Fermi liquid are determined by the quasiparticles and their interactions, which can lead to composite bosonic excitations. One such composite is a so-called paramagnon, a longed-lived spin excitation that is responsible for the binding of two spin-½ quasiparticles into a Cooper pair and thus for the formation of a superfluid. ⁶

Figure 1.

The experimental and theoretical developments that followed the initial discovery of superfluid ³He have influenced many aspects of condensed-matter physics. ⁷ In particular, they have been central to developments in the field of unconventional superconductivity, which describes materials in which symmetries of the normal metallic state are broken in the superconducting phase. For ³He, spin- and orbital-rotation symmetries, mirror symmetry, and time-reversal symmetry are all broken, as is the so-called U(1) gauge symmetry, the characteristic broken symmetry of all superconductors and superfluids. Many ground states are possible in which states with orbital angular momentum equal to ℏ (called p-wave) are spin-triplets with ℏ spin angular momentum. Figure 1 shows the phase diagram for two of the p-wave, spin-triplet states—the A and B phases that exist in pure ³He absent a magnetic field. Figures 2a and 2b show the energy gaps that separate the Fermi surface from the next-highest energy state in those two phases.

Figure 2.

The A phase is a superposition with equal amplitudes for the oppositely spin-polarized triplet states |↑↑⟩ and |↓↓⟩, a combination called equal-spin pairing (ESP). The orbital wavefunction of the A phase breaks both time-reversal and mirror symmetries; for that reason, the A phase is chiral. The B phase is a superposition of all three triplet spin states: |↑↑⟩, |↓↓⟩, and |↑↓ + ↓↑⟩. The spin states and orbital states are organized so that the B phase has zero total angular momentum. It is an isotropic state with an energy gap that is isotropic in momentum space. For most of the superfluid part of the zero-field phase diagram, the B phase is the stable one. The stability of the A phase at high pressure is a consequence of strong quasiparticle coupling to long-lived spin fluctuations.

Because the B phase incorporates |↑↓ + ↓↑⟩ pairs, its magnetic susceptibility is less than that of the A phase. The transition from A to B is first order and accompanied by a susceptibility jump. For B-phase ³He in a magnetic field, the Zeeman energy can be comparable to the binding energy of a |↑↓ + ↓↑⟩ pair, and the B phase is suppressed in favor of the A phase. For large magnetic fields, a third phase of the pure superfluid can exist. Called A₁, it is a chiral state like the A phase, but instead of including both the |↑↑⟩ and |↓↓⟩ states, it includes only the one that remains stable at high field. In 2015 Vladimir Dmitriev and colleagues reported a fourth p-wave phase. ⁸ That is the P (for polar) phase, also an ESP state like the A phase. The P phase has not been observed in pure ³He. Rather, it was stabilized in a highly anisotropic alumina aerogel whose Al₂O₃ strands were nematically aligned—that is, oriented predominantly in parallel—and separated on average by about 50 nm. Liquid ³He confined in that structure forms a phase with a single p orbital along the strands. The theoretical structure of the energy gap for the P phase in pure ³He is shown in figure 2c.

Silica aerogel

Dry silica aerogels are low-density solids whose porosities may be as high as 99.5%. In aerogels used for superfluid ³He investigations, the porosity value is typically 98%. As shown in figure 3a, the aerogel structure consists of a dilute network of thin silicon dioxide strands whose diameter is typically 3–5 nm. Aerogels have found a wide range of applications that include Cherenkov counters used in particle physics; lightweight, transparent thermal insulation for space vehicles; and heterogeneous catalysis.

Figure 3.

A silica aerogel can be fabricated with a base-catalyzed synthesis that produces nanometer-sized SiO₂ particles from tetramethylorthosilicate dissolved in alcohol. In a process called diffusion-limited cluster aggregation (DLCA), the particles diffuse in the fluid and aggregate to generate strands and clusters that bond to form a gel. That wet gel is dried supercritically in a high-pressure autoclave. Supercritical drying means that the wet gel never crosses the liquid–gas phase line; the technique ensures that the microstructure doesn’t collapse from capillary forces at the liquid–gas interface.

The resulting material is hydrophobic and stable in air. A 98% porosity aerogel is fractal on length scales less than about 50 nm; on much longer length scales, it can be prepared to be very homogeneous and isotropic, as verified by small-angle x-ray scattering measurements and optical birefringence. According to our simulations, the strand-like structures of the gel have a spatial correlation length ξ of 30–50 nm; you can think of ξ as a measure of the largest voids, as shown in figure 3b.

A length scale even larger than ξ is associated with the open structure of an aerogel: the straight-line ballistic mean free path that terminates at aerogel surfaces. A DLCA simulation for 98% porous structure gives a mean-free-path length λ of about 180 nm, a value that agrees well with the measured mean free path for ³He quasiparticles in the normal Fermi liquid. That agreement provided impetus for Sauls and colleagues to use scattering theory ⁵ with model parameters ξ and λ to describe the effects of an aerogel on the superfluid phases of ³He.

Although aerogels can be homogeneous and isotropic on large scales, they need not be. As mentioned above, the P phase of superfluid ³He was discovered in an anisotropic, nematically ordered aerogel; figure 3c shows an example.

Disorder in superfluid helium-3

Impurities and structural defects are unavoidable in almost all forms of condensed matter. For superconducting materials, including superfluids, relatively modest concentrations of chemical impurities or structural imperfections can mask intrinsic behavior or even destroy an unconventional superconducting state that has broken rotation or reflection symmetry.

In contrast, liquid ³He at low temperature is the purest known form of matter. Even ⁴He will not dissolve measurably in ³He if the temperature is below 10 mK. The intrinsic properties of the normal and superfluid phases are rich, well studied, and quantitatively understood in terms of BCS pairing in a material free of defects and impurities.

Superfluid ³He, presumed always to be pristine, had widely been considered unique among superconductors. That changed with the discovery of superfluidity of liquid ³He in a silica aerogel, which opened wide-ranging opportunities to study the effects of disorder on an unconventional BCS condensate whose intrinsic properties were quantitatively understood.

Superfluid ³He in a silica aerogel was discovered with the use of a torsional oscillator—a torsion rod attached perpendicular to a disk containing the ³He–aerogel sample. In the so-called two-fluid model, the superfluid state of ³He is a superposition of normal-fluid excitations that are viscously clamped to the porous structure and an inviscid, decoupled superfluid of Cooper pairs that does not contribute to the moment of inertia of the mechanical oscillator. At the onset of superfluidity, the moment of inertia and the resonant period of the oscillator decrease in a way that is directly related to the superfluid fraction. However, in contrast to pure superfluid ³He, the superfluid fraction of the confined liquid is much less than unity as the temperature approaches absolute zero (see figure 4a).

Figure 4.

As the superfluid fraction was being measured at Cornell, researchers at Northwestern University were obtaining a frequency shift Δω of the superfluid’s NMR spectrum from that of the normal fluid; figure 4b shows their result. For pure ³He, Δω was theoretically understood to be proportional to the square of the amplitude of the bound-pair wavefunction. For each specific p-wave state, according to theory, the shift depends in a characteristic way on the amplitude of the NMR excitation. ² Thus a measurement of shift versus RF amplitude provides a powerful means for state identification.

The onset of the NMR frequency shift is at the critical temperature T_c of the superfluid transition. Proportional to the square of the amplitude of the pair wavefunction, Δω grows with temperature below that second-order transition. Like the superfluid fraction, the frequency shift is smaller in an aerogel than in the pure superfluid. The reduction of both is due to ³He quasiparticle scattering from the aerogel structure. The scattering not only breaks Cooper pairs, it also destroys the energy gap of the pure superfluid, even in the B phase.

Gapless superfluids in an isotropic aerogel

Pure ³He in the B phase is a fully gapped BCS superfluid, “fully” meaning that the gap exists for momenta in all directions. When the temperature is much less than the gap energy divided by Boltzmann’s constant—about 2 mK—essentially no normal fluid exists. The heat capacity, thermal conductivity, and normal fluid fraction are exponentially suppressed with temperature, as would be expected for an ideal BCS superconductor. However, the superfluid B phase in an isotropic aerogel is different: As the temperature approaches zero, the thermal conductivity and heat capacity go to zero linearly.

Those results are consistent with the theoretical prediction that superfluid ³He in a high-porosity aerogel is “gapless”—that is, the superfluid has a nonzero density of states at the Fermi energy—over the entire pressure range such that a normal-fluid component coexists with the superfluid condensate at arbitrarily low temperature. ⁹ In fact, the calculated density of states as a function of energy shows only slight distinction among the various superfluid phases.

Acoustic-impedance measurements reveal that when no magnetic field is present, the equilibrium superfluid phase for isotropic aerogels is the gapless B phase, as shown in figure 1. The absence of the A phase in isotropic aerogel is a consequence of suppression of the strong-coupling effect, evident in pure ³He, due to pair breaking by quasiparticle scattering. The observed stability of the B phase confirms the important prediction that isotropic quasiparticle scattering favors a more isotropic phase. ⁵ However, as with the pure superfluid, the A phase is more stable than the B phase in a magnetic field because the B phase has higher Zeeman energy. When the magnetic field is small, the transition temperature from the B phase to the A phase depends on the square of the magnetic field.

New phases in anisotropic aerogels

Perhaps the most remarkable effects of an aerogel on superfluid ³He result from controlling quasiparticle scattering with engineered anisotropic aerogels. Uniaxial stretching or compression of the silica aerogel framework introduces in the quasiparticle mean free path a globally uniform anisotropy that can be conveniently quantified via optical birefringence.

Stretching is achieved by using extra catalyst during aerogel synthesis; the result is a uniform radial shrinkage of the aerogel away from the walls of a glass tube during the drying process. ¹⁰ Figure 5 shows the pressure–temperature phase diagram for superfluid ³He in a stretched aerogel with a strain of 0.14. Equal-spin pairing phases, stabilized by anisotropic quasiparticle scattering, replace the B phase throughout the entire diagram. ^{5 , 11} NMR measurements with magnetic fields oriented parallel and perpendicular to the strain show that the ESP-1 phase is a chiral state with angular momentum aligned along the strain axis. At a temperature below the normal-to-ESP-1 transition is a second transition to a new ESP state, labeled ESP-2 in the figure. Theory predicts that phase to be a polar-distorted, biaxial chiral phase whose orbital structure is distorted away from the strain axis. ¹¹ NMR measurements demonstrate that its angular momentum is perpendicular to the strain axis.

Figure 5.

Anisotropy in an aerogel can also be created by compressing isotropic samples in the glass tube after drying. The unexpected result is that the B phase becomes more stable and preempts ESP A phases. As figure 6 shows, at a sufficiently high magnetic field the A phase reappears; thus there exists a new polycritical point describing conditions where the normal, A, and B phases can coexist in a nonzero magnetic field. The square of the field H_c at that point is proportional to the strain in the aerogel; a representative value is H_c = 0.1 T, which was obtained for a strain of −0.20 (negative for compression) and a pressure of 26.3 bar. ¹² Further work has shown that the B phase in the compressed aerogel is no longer isotropic. The significant polar distortion of its orbital structure in the direction of the strain is a subject of ongoing investigation.

Figure 6.

Topology and the polar phase

Evidence for the existence of a polar phase of superfluid ³He came independently from experiments using NMR and torsional-oscillator techniques. ^{8 , 13} The investigations were conducted with nematically oriented alumina aerogels—sometimes called nafen or obninsk—manifesting extreme anisotropy: The ratio of the ballistic mean free path of quasiparticles moving along the nematic axis can be eight times that of the mean free path for perpendicular motion. The P phase has a single p-wave orbital state, illustrated in figure 2c for pure ³He; the state’s polar axis is aligned along the nematic axis of the aerogel.

The observation of the polar state is of great significance. It does not occur in pure superfluid ³He, but theoretical work suggested it would be stable in anisotropic aerogels ¹⁴ and channels. ¹⁵ Moreover, the polar state was recognized to be a favorable candidate to support the elusive half-quantum vortex that Grigory Volovik and Vladimir Mineev predicted for p-wave, spin-triplet superfluids more than 40 years ago. ¹⁶ Indeed, the half-quantum vortex was experimentally detected ¹⁷ shortly after discovery of the P phase in ³He confined in rotating nafen and cooled to as low as 300 µK.

But what is a half-quantum vortex? It is a topological defect that combines mass and spin vortices. For conventional superconductors, the change in the phase of the wavefunction around a closed path enclosing a quantized vortex is 2π; for the half-quantum vortex in superfluid ³He, half of that phase change is associated with the circulation of superfluid mass current and half is associated with the circulation of spin current. ¹⁶

The discoveries of new phases of ³He confined in high-porosity disordered solids have demonstrated the effects of disorder on superfluids with complex symmetry breaking. Together with theoretical developments in topological condensed matter, the discoveries have opened up new research directions in the study of topological phases of ³He confined on the nanoscale. Those include high-quality-factor nanofluidic mechanical oscillators, nanoscale NMR cells, ultrasonic cavities for quantum transport, and the dynamics in quantum fluids. Such investigations are helping to advance the frontier of hybrid quantum systems, a field that marries quantum systems with other nanoscale materials and nanofabrication techniques. To this day, one of the simplest of those quantum systems, superfluid ³He, continues to astound.