Observation of a new superfluid phase for 3He embedded in nematically ordered aerogel

In bulk superfluid 3He at zero magnetic field, two phases emerge with the B-phase stable everywhere except at high pressures and temperatures, where the A-phase is favoured. Aerogels with nanostructure smaller than the superfluid coherence length are the only means to introduce disorder into the superfluid. Here we use a torsion pendulum to study 3He confined in an extremely anisotropic, nematically ordered aerogel consisting of ∼10 nm-thick alumina strands, spaced by ∼100 nm, and aligned parallel to the pendulum axis. Kinks in the development of the superfluid fraction (at various pressures) as the temperature is varied correspond to phase transitions. Two such transitions are seen in the superfluid state, and we identify the superfluid phase closest to Tc at low pressure as the polar state, a phase that is not seen in bulk 3He.

where and are 3x3 matrices with components and respectively, * is the 3 He quasiparticles renormalized mass, and is the Landau parameter.
, is the Yosida function, which is related to the energy density distribution along the Fermi sphere, : ,

Supplementary Equation 4
with the information for the gap structure contained in ⁄ .
The presence of impurity scattering in aerogels leads to a reduction of the superfluid gap, as well as the superfluid temperature compared to the bulk [ 1,3]. (superfluid density is proportional to the square of the gap). The ratio that best fits the experimental data is plotted as filled red circles in Supplementary Figure 1 for each of the experimental pressures. These were chosen to be temperature independent, but varied with pressure. As seen in the figure, the factors by which the gap is suppressed that best match the data vary roughly linearly with superfluid transition temperature suppression in a similar fashion as previously observed for the fluid in isotropic silica aerogels.

Supplementary Note 2: Slow-mode sound resonances
When the torsion pendulum frequency crosses the resonant frequency of a standing sound wave mode, these sound resonances appear as dips in the amplitude of the pendulum for the same value of drive, thus they are seen as peaks in the measured dissipation ( ). Our procedure for determining the resonant frequency of the pendulum (digital phase locked loop) relies on a Lorentzian resonance response. The presence of sound modes in the "head" of the torsional pendulum results in a distorted complex response which appears as a non-monotonic calculated resonance frequency, and hence the inferred superfluid fraction. The resonances are sufficiently far spaced that we can none-the-less reliably produce a phase diagram.
Starting from the two fluid model for superfluids, there are many ways in which sound can propagate though the liquid medium. In bulk 3 He one finds first sound (normal and superfluid components move in-phase), second (normal and superfluid components move out-ofphase) and fourth sound (normal component is clamped, but the superfluid component oscillates) [4]. In aerogel, the normal fluid is well clamped to the aerogel strands, while the superfluid is free to move, as in the case of fourth sound. However, the aerogel is not perfectly rigid, but can 7 flex as well. Thus sound modes for 3 He in aerogel are composite modes in which the aerogel, the normal and the superfluid components all move. Two such modes are possible, the fast mode in which all components move in phase, and a slow mode in which the superfluid component is out-of-phase with the flexing of the aerogel strands and the normal component [5]. The sound velocity of the slow mode, , is related to the speed of sound in aerogel, , and the superfluid fraction, ⁄ , through the following expression:

Supplementary Equation 5
where is the density of the fluid at the particular pressure. Since depends on ⁄ , the slow-mode sound velocity (and therefore the wavelength at a particular excitation frequency) changes rapidly with temperature below the superfluid transition, starting at zero at .

Supplementary Note 3: Superfluid textural differences revealed from sound resonances
While normally a nuisance, the sound resonance dissipation peaks give us some insight into the superflud state. For example, Supplementary Figure 3 shows the dissipation peak for one of the slow-mode sound resonances that is excited at a superfluid fraction of ~ 0.14. We observe that dissipation peak in the A phase is larger than the dissipation peak in the B phase.
Furthermore, there is a noticeable difference between the dissipation peak seen in the A phase