Figure 1

(Color online) Calculated pressure to temperature phase diagram consisting of the corresponding VCT curve $T_{\text{VCT}}\left(P\right)$ (solid curve) in the bulk B phase or B-like phase, the superfluid transition curve $T_c\left(P\right)$ (dotted one), and the corresponding A-B transition curve $T_{\text{AB}}\left(P\right)$ (dashed one) of liquid ${}^3\text{H}\text{e}$ in bulk [thin (black) curves] and an uniaxially deformed aerogel [thick (red) curves]. For comparison, we also show the VCT curve [dashed-dotted (blue) curve] in a globally isotropic aerogel where the resulting $T_c\left(P\right)$ cannot be distinguished from that [thick (red) dotted curve] in the anisotropic ones and thus, is not drawn in the figures. The parameters ${\delta }_u=0.01$ and ${\delta }_u=-0.003$ are used for (a) compressed and (b) stretched aerogels, respectively, and ${\left(2\pi \tau \right)}^{-1}=0.135\left(\text{mK}\right)$ was assumed in both of (a) and (b). The A-phase-core vortex is stable at higher temperatures than the VCT curve. The VCT curve in the isotropic aerogel shifts to higher temperatures than that in bulk, reflecting the reduction in the SC correction. As shown in (a), the A-phase-core vortex is stabilized by the small compression and survives even in the low-pressure limit while in (b), it tends to be lost due only to a small stretch.Reuse & Permissions

Figure 2

(Color online) Radial dependences of $C_{\mu ,\nu }^{\left(Q\right)}\left(\tilde{r}\right)$ of (a) an axisymmetric vortex and (b) a nonaxisymmetric one with $\left|Q\right|=2$ components at $T=2.040\left(\text{mK}\right)$ and $P=28\left(\text{bar}\right)$ in the A-like phase in the uniaxially stretched aerogel with ${\delta }_u=-0.03$. In (a), $C_{+0}^{\left(0\right)}$ and $C_{0+}^{\left(0\right)}$ remain finite at $\tilde{r}=0$ which correspond to the $\beta$-phase component and that of MHV core, respectively. In (b), the components $C_{0,\nu }^{\left(Q\right)}\left(\tilde{r}\right)$ are identically zero and other negligibly small components are not shown. Further, the dashed curves imply $Q\ne 0$ components. At $\tilde{r}=0$, just the two components $C_{+0}^{\left(0\right)}$ and $C_{-0}^{\left(-2\right)}$ remain finite, implying a polar core $A_{\eta ,i}\left(r=0\right)=a_{x,z}{\delta }_{\eta ,x}{\delta }_{i,z}$.Reuse & Permissions

Figure 3

(Color online) Spacial variations in the order parameter $A_{\eta ,i}$ of (a) the axisymmetric vortex and (b) the nonaxisymmetric one on the $x$ axis (black curves) and $y$ axis (red dashed ones). In (a), the vortex center is occupied with the $\beta$ phase $a_{x,z}{\left(\widehat{x}+i\widehat{y}\right)}_{\eta }{\delta }_{ı,z}$ and the component of the dipole-locked MHV center $a_{z,y}{\delta }_{\eta ,z}{\left(\widehat{x}+i\widehat{y}\right)}_i$. In (b), $a_{\eta ,i}$’s start varying around $\tilde{r}=10$ on the $y$ axis, corresponding to the $\mathbf{d}$ texture (see the text below), and a polar state $a_{x,z}{\delta }_{\eta ,x}{\delta }_{i,z}$ is realized at the vortex center. The three thin solid curves in (a) with extremely small magnitudes denote $a_{y,y}$ (top), $a_{y,x}$ (middle), and $a_{x,x}$ (bottom). Negligibly small contributions on the $x$ axis are not shown in (b).Reuse & Permissions

Figure 4

Comparison of radial dependence of the free energy density between the axisymmetric nonunitary Mermin-Ho vortex and the nonaxisymmetric polar-core vortex. For both the two curves, the free energy density was normalized by $2\pi K{\left|\Delta \right|}^2$ at $T=2.040\left(\text{mK}\right)$ and $P=28\left(\text{bar}\right)$.Reuse & Permissions

Figure 5

(Color online) Calculated pressure to temperature phase diagram consisting of VCT (solid) curve in the A-like phase, $T_c\left(P\right)$ (dotted) curve, $T_{\text{AB}}\left(P\right)$ (dashed) curve, and the polar-A transition (Ref. 5) (dashed-dotted) curve in superfluid ${}^3\text{H}\text{e}$ in the uniaxially stretched aerogel with ${\delta }_u=-0.03$. The polar-core vortex is stable at higher temperatures than the solid curve and no VCT occurs in the B-like phase. In the Inset, the total free energy for ${\tilde{r}}_b=30$ is compared between the nonunitary MHV (dashed curve) and the polar-core vortex (solid one) as a function of the temperature (mK). The VCT occurs at the temperature pointed by an arrow.Reuse & Permissions

Figure 6

(Color online) Sketch of the $\mathbf{l}$ texture (blue arrows) and the $\mathbf{d}$ texture (red ones) in the polar-core vortex far from the vortex center (right) and inside the bump in $f\left(\tilde{r}\right)$ (center). Inside the bump, order-parameter field can be expressed as $A_{\eta ,i}\simeq {\delta }_{\eta ,x}{\left(a_{x,z}\widehat{z}+i{a}_{x,\varphi }\widehat{\varphi }\right)}_i$, indicating $\mathbf{l}\parallel \widehat{r}$ and $\mathbf{d}\parallel \widehat{x}$, where $a_{x,\varphi }$ corresponds to $a_{x,y}$ for $\varphi =0$ and $a_{x,x}$ for $\varphi =\pi /2$ in Eq. (10). The circle symbol and black arrow in the left figure expressing the close vicinity of the vortex center denote $a_{x,z}\widehat{z}$ and $a_{x,\varphi }\widehat{\varphi }$, respectively. In the left figure, on approaching the vortex center, $a_{x,\varphi }$ (i.e., the length of the black arrow) gradually vanishes, and at the vortex center, the polar state $A_{\eta ,i}={\delta }_{\eta ,x}{\left(a_{x,z}\widehat{z}\right)}_i$ is realized.Reuse & Permissions