Surface-Induced Order Parameter Distortion in Superfluid ${}^3\mathrm{He}\mathrm{\text{−}}B$ Measured by Nonlinear NMR

The $B$ phase of superfluid ${}^3\mathrm{He}$ is a three-dimensional time-reversal invariant topological superfluid, predicted to support gapless Majorana surface states. We confine superfluid ${}^3\mathrm{He}$ into a thin nanofluidic slab geometry. In the presence of a weak symmetry-breaking magnetic field, we have observed two possible states of the confined ${}^3\mathrm{He}\mathrm{\text{−}}B$ phase manifold, through the small tipping angle NMR response. Large tipping angle nonlinear NMR has allowed the identification of the order parameter of these states and enabled a measurement of the surface-induced gap distortion. The results for two different quasiparticle surface scattering boundary conditions are compared with the predictions of weak-coupling quasiclassical theory. We identify a textural domain wall between the two $B$ phase states, the edge of which at the cavity surface is predicted to host gapless states, protected in the magnetic field.

Figure 1

Calculated spatial dependence of the two components of the $B$ phase gap for diffuse (solid lines) and specular (dashed lines) boundaries (a) near a wall at $z=0$ [41] and (b) in a slab of thickness $D=10{\xi }_0$ [42].

Figure 2

NMR signatures of the $B$ phase confined in the slab at $T=0.4T_c^{\mathrm{bulk}}$. (a),(b) Spectra (Fourier transform of free-induction decay) observed after small tipping angle $\beta =2°$ pulses on separate cooldowns from the $A$ phase show either a positively shifted ($B_{+}$) or a negatively shifted ($B_{-}$) NMR line, or a combination [26]. (c),(d) Dependence of the frequency shift on the tipping angle $\beta$ in nonlinear NMR experiments on $B_{-}$ and $B_{+}$. These data provide a measurement of two gap distortion averages, $\overline{q}$ and $\overline{Q}$ (3). The $B_{+}$ data at $\beta <104°$ only (green circles) were used in the fit, since magic angle ${\beta }^{*}>104°$ for all $\overline{q}$ (see text). For comparison, the tipping angle dependence predicted for uniform planar distortion $q={\Delta }_{\perp }/{\Delta }_{\parallel }$ is shown.

Figure 3

Dependence of gap distortion on temperature and boundary conditions. (a),(b) Gap distortion predicted by quasiclassical theory [35], and results of nonlinear NMR; (c) predicted weak temperature dependence of ratio $\overline{q}/\overline{Q}$ of distortion parameters. (d) Temperature dependence of the small $\beta$ frequency shifts $\Delta f_{+}$ and $\Delta f_{-}$ in $B_{+}$ and $B_{-}$ states; (e) their ratio; (f) inferred temperature dependence of average planar distortion assuming temperature-independent $\overline{q}/\overline{Q}$, in comparison with prediction [43]. The experimental data range is limited by the transition to the $A$ phase [26].

Figure 4

Metastability of the $B_{-}$ state. (a) By preparing the sample in $B_{-}$ state, ramping the static field from 32 mT down to $H_{\min }$ and back (we can only interrogate the state of the sample in 32 mT), and measuring the relative weight of $B_{-}$ and $B_{+}$ NMR signals, we probe the stability of $B_{-}$ as a function of field $H_{\min }$. The $B_{-}\to B_{+}$ conversion is observed near $H^{*}=20\mathrm{mT}$. (b) Textural domain wall (TDW) between $B_{+}$ and $B_{-}$ and pinning mechanism. The arrows represent $\widehat{\mathbf{l}}$, free-energy density $F$ is shown schematically. The dipole energy $F_D$ difference between $B_{-}$ and $B_{+}$ pushes the wall to the right, expanding the $B_{+}$ domain; however, in a large enough field the domain wall is arrested ahead of a scratch on the cell surface by the Zeeman energy $F_H$ of the domain wall.