Evidence for a Spatially Modulated Superfluid Phase of ${}^3\mathrm{He}$ under Confinement

In superfluid ${}^3\mathrm{He}\text{−}B$ confined in a slab geometry, domain walls between regions of different order parameter orientation are predicted to be energetically stable. Formation of the spatially modulated superfluid stripe phase has been proposed. We confined ${}^3\mathrm{He}$ in a $1.1\mu \mathrm{m}$ high microfluidic cavity and cooled it into the $B$ phase at low pressure, where the stripe phase is predicted. We measured the surface-induced order parameter distortion with NMR, sensitive to the formation of domains. The results rule out the stripe phase, but are consistent with 2D modulated superfluid order.

Figure 1

Domain wall at the heart of the stripe phase in ${}^3\mathrm{He}\text{−}B$ [27]. Shown here are the elements of the energy gap matrix $\mathit{\Delta }$, such that $\mathit{A}=e^{i\varphi }\mathit{R}\mathit{\Delta }$. When the domain wall is absent $\mathit{\Delta }$ is diagonal, see Eq. (1). Crossing the domain wall lying in the $yz$ plane at $x=0$, ${\mathrm{\Delta }}_{zz}$ changes sign, and off-diagonal elements ${\mathrm{\Delta }}_{xz}$ and ${\mathrm{\Delta }}_{zx}$ emerge, while ${\mathrm{\Delta }}_{xx}$ and ${\mathrm{\Delta }}_{yy}$ (not shown) remain close to ${\mathrm{\Delta }}_{\parallel }$. The gap amplitudes were calculated $z=2.5{\xi }_0$ away from one of the surfaces in a $D=10{\xi }_0$ thick slab at $T=0.5T_c^{\text{bulk}}$; ${\xi }_0$ is the Cooper pair diameter, ${\mathrm{\Delta }}_{\text{bulk}}$ is the bulk $B$ phase gap.

Figure 2

NMR measurements on a $D=1.1\mu \mathrm{m}$ slab of superfluid ${}^3\mathrm{He}$. (a) Signatures of the $A$ and $B$ phases observed with small tipping pulses. At the second-order normal-superfluid transition a negative frequency shift develops as the slab enters the $A$ phase with the dipole-unlocked orientation. On further cooling a first-order phase transition into the $B$ phase with a planar distortion occurs, where two spin-orbit orientations $B_{+}$ and $B_{-}$ are observed. Inset: Sharp $A\text{−}B$ transition with small hysteresis. (b) Technique for applying a set of pulses with different tipping angle $\beta$ but equal heating: all pulses coincide in length and amplitude; $\beta$ is reduced by applying the initial section of the pulse with a 180° phase shift, which cancels out a similar section that follows (both light blue), so only the remainder of the pulse (red) tips the spins. (c) Initial frequency shifts in $B_{+}$ after such pulses yield $\partial \mathrm{\Delta }{f}_{+}/\partial \cos \beta$ and $\mathrm{\Delta }{f}_{+}(\beta \to 0)$. Good agreement between $\mathrm{\Delta }{f}_{+}(\beta \to 0)$ obtained here and $\mathrm{\Delta }{f}_{+}(4°)$, smoothed from (a), indicates that the heating due to the 20°–60° pulses is negligible. Open (filled) symbols show data taken on stepwise warm-ups after fast (slow) cooldown from the $A$ phase.

Figure 3

Temperature dependence of planar distortion parameters $\overline{q}$, $\overline{Q}$ (a) and their ratio (b) inferred from NMR measurements, Fig. 2. Solid lines show weak-coupling calculations for the translationally invariant $B$ phase [44]. In the absence of strong coupling the temperature of the $A\text{−}B$ transition $T_{AB}^{\mathrm{BCS}}$ is higher than $T_{AB}$ observed experimentally. While $\overline{Q}$ is in agreement with the theory, $\overline{q}$ gets progressively reduced approaching $T_{AB}$. We interpret this in terms of development of domains with opposite sign of ${\mathrm{\Delta }}_{\perp }$ on warming. The right-hand vertical axis in (b) estimates the fraction of the slab occupied by the majority domains, taken here to have positive ${\mathrm{\Delta }}_{\perp }$ [45], under a qualitative assumption of steplike energy gap profile, Eq. (4). Any systematic difference between measurements taken while warming after a fast and a slow cooldown through $A\text{−}B$ transition (open/filled symbols) is small.

Figure 4

Possible regular domain configurations in ${}^3\mathrm{He}\text{−}B$ confined to a slab, view perpendicular to the slab plane. $+$ or $-$ indicates the sign of ${\mathrm{\Delta }}_{\perp }$. (a) Translationally invariant $B$ phase. (b) Predicted stripe phase. (c),(d) Proposed polka dot phase that is easier to nucleate than the stripe phase. The values of the gap distortion parameter $\overline{q}/\overline{Q}$ are derived under a simplified assumption of steplike domain walls, Eq. (4). Pinning of domain walls in the experimental cell may introduce disorder to these structures. All features in (c) and (d) are taken to be of characteristic size similar to the pitch $W$ of the stripe phase. (e) A variant of (d) in which dot diameter $W^{\prime }$ is smaller than dot separation $W$.