Textures in superfluidHe3-A: Hydrodynamic and magnetic effects in a cylindrical pore

Transformation of the Ginzburg-Landau free energy for superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ to cylindrical polar coordinates yields the field equations and specular boundary conditions for a cylindrical geometry. Similar transformations give the particle and spin current densities. Application to ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-$A$ near $T_c$ in a long pore of radius $R$ predicts various stable configurations. For $R\gg 6$ μm, $\hat{d}$ and $\hat{l}$ flare upward near the center, inducing an extra free energy per unit length that is independent of $R$, and an angular momentum per particle $\approx 0.782\frac{{{\rho }_s}^{\parallel }}{\rho }$. For $R\ll 6$ μm, $\hat{d}$ is uniform and $\hat{l}$ is radial, with a depaired region of radius $\approx \xi \mathrm{}\left(T\right)\mathrm{}$ near the center; the corresponding extra free energy per unit length is proportional to $\ln \left[\frac{R}{\xi \mathrm{}\left(T\right)\mathrm{}}\right]$, with no current or angular momentum. In a large cylinder ($R\gg 6$ μm), an applied axial field deforms $\hat{d}$ and $\hat{l}$, increasing the angular momentum up to a critical magnetic field (≈ 20-30 G), when $\hat{d}$ and $\hat{l}$ abruptly undergo a textural transition and become radial. In contrast, an applied axial superflow in a large cylinder decreases the angular momentum.