Stripe order in superfluid ${}^3$He confined in narrow cylinders

We theoretically investigate pairing states of the spin-triplet $p$-wave superfluid ${}^3$He confined in narrow cylinders. The surface-induced distortion and the multiple internal degrees of freedom of the order parameter lead to the occurrence of a stripe structure along the cylinder axis in the superfluid ${}^3$He-B phase. We show that in sufficiently small cylinders with an anisotropic surface scattering, the stripe order with broken translational symmetry may be stabilized as the lowest energy state. Periodic spin-current textures caused in this inhomogeneous superfluid phase are also discussed.

Figure 1

(a) The BW-polar transition curves $T_{BP}\left(R\right)$ in cylinders with radius $R$ and the specular surface. $T_{BP}$ for the BW state with a spatial modulation along the cylinder axis (red solid curve) is higher than that for the uniform BW state (black dotted curve). The inset shows the evolution of the modulation $Q$ along the $T_{BP}\left(R\right)$ curve. (b) Sketch of the spatial profile of the order parameter in the stripe BW state with the spatial modulation along the $z$ axis. (c) Temperature dependences of ${\Delta }_1$ (blue), ${\Delta }_2$ (black), and ${\Delta }_3$ (red) at ${\xi }_0/R=0.17$ for the modulated (solid curves) and uniform (dotted curves) BW states.

Figure 2

Pairing states of superfluid ${}^3$He confined in narrow cylinders with radius $R$ and the specular surface. (a) $T\text{−}{R}^{-1}$ phase diagram obtained by numerically solving GL equations. The ABM state is realized in the blue colored area, while in the white colored region at large $R$'s, the BW state uniform along the cylinder axis is stable. For specular surface scattering, the stripe order with the axial modulation is not stable against the ABM state, and thus the lower and upper instability curves for the stripe phase (red dashed curves) as well as the $T_{BP}\left(R\right)$ curve for the uniform BW state (dotted curves) are marginal. Radial dependences of the order parameters in the stripe phase and the ABM states at ${\xi }_0/R=0.15$ and $T/T_c=0.85$ [the symbol * in (a)] are shown in (b) and (c), respectively.

Figure 3

Stripe phase realized in narrow cylinders with the anisotropic surface scattering described by Eq. (6). (a) $T\text{−}{R}^{-1}$ phase diagram obtained by numerically solving GL equations. The stripe phase with the spatial modulation along the cylinder axis is stabilized in the region sandwiched by red solid curves. The inset shows an example of a cylinder with the surface condition, Eq. (6). (b) The structure of spin currents in this inhomogeneous state at ${\xi }_0/R=0.12$ and $T/T_c=0.79$ for which ${\xi }_{0}Q=0.11$. The spatial modulation gives rise to the periodic planar spin currents $J_{r\varphi }$ and $J_{\varphi r}$ as well as the oscillating behavior in $J_{\varphi z}$ and $J_{z\varphi }$. (c) Sketch of the planar spin currents flowing in opposite directions on adjacent nodal planes of $A_{rr}\left(A_{\varphi \varphi }\right)$. Blue and pink colored regions denote positive and negative values of $A_{rr}\left(A_{\varphi \varphi }\right)$, respectively. Red (black) arrows represent the directions of the spin component (flow).