Stability of Superfluid ${}^3\mathrm{He}\text{−}\mathrm{B}$ in Compressed Aerogel

In recent work, it was shown that new anisotropic $p$-wave states of superfluid ${}^3\mathrm{He}$ can be stabilized within high-porosity silica aerogel under uniform positive strain. In contrast, the equilibrium phase in an unstrained aerogel is the isotropic superfluid $B$ phase. Here we report that this phase stability depends on the sign of the strain. For a negative strain of $\sim 20\%$ achieved by compression, the $B$ phase can be made more stable than the anisotropic $A$ phase, resulting in a tricritical point for $A$, $B$, and normal phases with a critical field of $\sim 100\mathrm{mT}$. From pulsed NMR measurements, we identify these phases and the orientation of the angular momentum.

Figure 1

(a) and (b) Schematics of the experimental arrangements for samples 2 and 3. Sample 2, $\mathit{ϵ}=-19.4\%$, has a strain axis parallel to the magnetic field. For sample 3, $\mathit{ϵ}=-22.5\%$, the strain axis is perpendicular to the magnetic field. (c) Liquid susceptibility normalized to the susceptibility of the normal state on warming (open red circles) and cooling (solid blue circles) versus reduced temperature for sample 3 in $H=196\mathrm{mT}$. Similar behavior was observed in sample 2 but was omitted for clarity. (Inset) NMR spectra for ${}^3\mathrm{He}$ in aerogel: sample 3 in the normal state (black solid curve) and $A$ phase (blue dash-dotted curve); sample 2 in the $A$ phase (red dashed curve). The normal state spectra have the same line shape. The spectra for samples 2 and 3 were obtained in$H=95.6\hspace{0.17em}$ and 79.2 mT, respectively, and have been scaled to a common field and temperature for comparison.

Figure 2

Superfluid phase diagram $T_{BA}/T_c$ versus $H^2$ for three samples at a pressure of $P=26.3\mathrm{bar}$ from warming experiments. Two of the three have negative strains, $\mathit{ϵ}<0$, and one is isotropic, $\mathit{ϵ}=0$. All three have the common feature of a quadratic dependence on magnetic field for the $B$- to $A$-phase transitions. Remarkably, for negative strains, there appears to be a critical field, $H_c$. The superfluid transition temperature, $T_c$, for samples 1, 2, and 3 at this pressure are 2.213, 2.194, and 2.186 mK, respectively.

Figure 3

NMR frequency shifts as a function of temperature and tip angle. The magnitude of the shifts presented here has been scaled to a common field of $H=196\mathrm{mT}$, and the tip angle dependence has been scaled to a common temperature of $T/T_c=0.8$ based on our results from sample 1. In all of the panels, the dipole-locked configuration, $\widehat{\mathit{l}}\perp \mathit{H}$, is shown by the black solid curves, and the black dashed curves are the dipole-unlocked configuration, $\widehat{\mathit{l}}\parallel \mathit{H}$, with ${\mathrm{\Omega }}_A$ taken from the measurements for sample 1 [2]. Comparison of frequency shifts in the $A$ phase: (a) and (b) for $\mathit{ϵ}\parallel \mathit{H}$ and (c) and (d) $\mathit{ϵ}\perp \mathit{H}$. (a) and (c) Dependence on reduced temperature with small tip angle, $\beta <20°$. (a) Warming in $H=174\mathrm{mT}$, yellow crosses; cooling in $H=174\mathrm{mT}$, green $\times$; cooling in $H=95.6\mathrm{mT}$, blue circles. (c) Cooling in $H=79.2\mathrm{mT}$, red circles. (b) and (d) Dependence on tip angle after cooling from the normal state. (b) $T/T_c=0.87$, $H=49.1\mathrm{mT}$, green squares; $T/T_c=0.82$, $H=196\mathrm{mT}$, blue circles. (d) $T/T_c=0.78$, $H=95.6\mathrm{mT}$, red circles. The blue and red solid curves are calculated for the easy-plane 2D glass model (see text).