Stability of a half-quantum vortex in rotating superfluid ${}^3\text{He}\text{−}A$ between parallel plates

We have found the precise stability region of the half quantum vortex (HQV) for superfluid ${}^3\text{H}\text{e}-A$ phase confined in parallel plates with a narrow gap under rotation. Standard Ginzburg-Landau free energy, which is well established, is solved to locate the stability region spanned by temperature $T$ and rotation speed $\left(\Omega \right)$. This $\Omega \text{−}T$ stability region is wide enough to check it experimentally in available experimental setup. The detailed order-parameter structure of HQV characterized by $A_1$ core is given to facilitate the physical reasons of its stability over other vortices or textures.

Figure 1

(Color online) Free-energy comparison for AT, HQV, SV and $\left(0,-1,2,1\right)$ states as a function of $\Omega$ for $R=10\mu \text{m}$ and $t=T/T_c=0.97$. $\Delta f$ is the relative free energy to AT.

Figure 2

(Color online) Order parameter amplitude $\left|A\right|$ normalized by $\pi k_B{T}_c$ and $\hat{\mathbf{d}}$ vector profiles for (a) AT, (b) HQV, and (c) SV for $R=1.0\mu \text{m}$ and $t=0.97$. Left and center columns show the cross section of OP along the radial direction $r$. Right column shows $\hat{\mathbf{d}}$ vector patterns. In (b) it winds by $\pi$ around the center $r=0$.

Figure 3

(Color online) Stability region of HQV sandwiched between ${\Omega }_{c1}$ and ${\Omega }_{c2}$ as a function of $R$ ($t=0.97$ and $\Delta t=0.05$). ${\Omega }_{c1}=0.05\text{rad}/\text{s}$ is the extrapolated value for $R=1.5\text{mm}$. Inset shows the free-energy comparison for $R=20\mu \text{m}$, displaying the successive transitions from AT to HQV at ${\Omega }_{c1}$ and from HQV to SV at ${\Omega }_{c2}$.

Figure 4

(Color online) Stability region of HQV in $\Omega$ versus $T/T_c$ ($R=10\mu \text{m}$ and $\Delta t=0.05$). $A_{1}T$ denotes the $A_1$ phase texture where only $\downarrow \downarrow$ pairs exist.