Chiral superfluid phase of liquid ${}^3\mathrm{He}$ in planar aerogels

Motivated by the realization of the chiral superfluid phase of liquid ${}^3\mathrm{He}$ stabilized at lower pressures and over a wide temperature range in planar aerogels, we examine equilibrium properties of the superfluid $A$ phase suffering from impurity scatterings due to planar correlated defects. In the specular limit of scattering due to the planar defects, an analog of Anderson's theorem for the $s$-wave Fermi superfluid is well satisfied in the chiral $A$ phase in the planar aerogels just like in the polar phase in nematic aerogels, and the chiral $A$ phase region is extended by such a planar anisotropy even with no strong coupling correction. It is also pointed out that the vortex energy in the chiral $A$ phase depends on the relative sense of the vorticity to the chirality. Based on this result, it is argued that, due to the presence of chiral domains induced by the randomness, the vortex lattice structure shows broken time-reversal symmetry and may include vortex loops.

Figure 1

Schematic pictures describing scattering events of normal quasiparticles via 2D-like (planar) defects lying in the $x-y$ plane.

Figure 2

Examples of the resulting pressure ($P$) to temperature ($T$) phase diagrams. Each red curve denotes the superfluid transition curve in each case. As noted in the text, the planar (high $T$) pairing state can be identified with the chiral or ABM pairing state, and each planar to $B$ second order transition line (blue curve) is regarded as a reasonable estimate of the first order $A-B$ transition curve in each case. In the upper figure, the two transition curves at a fixed anisotropy (${\delta }_D=10$) are described by increasing the scattering strength in the way ${\left(2\pi \tau \right)}^{-1}\left(\mathrm{mK}\right)=0.5$ (solid curves), 1.0 (dotted ones), and 1.5 (dashed ones), while, in the lower figure, they are described at a fixed impurity strength ${\left(2\pi \tau \right)}^{-1}=1.2\left(\mathrm{mK}\right)$ by increasing the anisotropy in the way ${\delta }_D=5$ (solid curves), 10 (dotted ones), and 50 (dashed ones), respectively.

Figure 3

Temperature dependence of the quasiparticle energy gap $\left|\mathrm{\Delta }\right|$. Numerical data points are described in the case of the clean limit in the upper figure and, in the lower figure, in the present impure case with ${\left(2\pi \tau \right)}^{-1}=2.0\left(\mathrm{mK}\right)$ and ${\delta }_D=50$ for which the $B$ phase is absent even in $T\to 0$ limit. In both of the figures, the parameter values [13] at $P=0$ (bar) have been used. Each solid curve denotes the best-fitted $T^4$ curve with $a_{\perp }=11.73$ in (a) and the corresponding curve with $a_{\perp }=26.12$ in (b), while the $\mathrm{\Delta }\left(0\right)/T_{c0}$ value is 2.026 in (a) and 1.98 in (b). It is found that the ideal $T^4$ behavior is satisfied in the impure case only below $0.35T_c$. As noted in the text, this impurity scattering effect in the present planar aerogel seems to be remarkably different from the corresponding one in the nematic aerogel where the corresponding $T^3$ behavior is well satisfied over a wide temperature range even in the impure case [4].

Figure 4

Spatial distribution of the order parameter components accompanying a HQV lying at ($-a$, 0) in the compatible case (see the text) at $P=3$(bars) and $T=1.0$(mK). This HQV and another one lying at ($a$,0) forms one HQV pair with the size $2a=1.6$ ($\mu \mathrm{m}$). We note that $T_c\left[P=3\left(\mathrm{bar}\right)\right]=1.25$ (mK) for the parameter values ${\left(2\pi \tau \right)}^{-1}=0.118$ (mK) and $\delta =4.4$ characterizing the model aerogel used here [see Eq. (25)]. The left and right figures express $|A_{++}^{(+)}|$ and $|A_{+-}^{(+)}|$, respectively.

Figure 5

Spatial distribution of the order parameter components accompanying a HQV existing at ($a,0$) in the incompatible situation. The winding number of this HQV is $+1/2$ as well as in Fig. 4. The parameter values used to obtain this solution are the same as those in Fig. 4. The left and right figures express $|A_{--}^{(-)}|$ and $|A_{-+}^{(-)}|$, respectively.

Figure 6

Dependencies of the vortex energy of a HQV pair on the pair size $2a$. The red curve denotes the total energy $F\left(a\right)$ of one HQV pair, the blue curve is the contribution to $F\left(a\right)$ of the quadratic gradient term (24), and the green curve is the corresponding one of $O\left(\right|\mathrm{\Delta }{|}^4$) gradient energy $f_{\mathrm{FLgrad}4}$ (see the text). (a) expresses the results in the case of Fig. 4, while (b) expresses those in the case of Fig. 5, respectively. The difference in $F\left(a\right)$ between the two cases is estimated to be about $0.25F\left(a\right)$ in the present study using the system size $40\times 8$ ($\mu {\mathrm{m}}^2$). The HQV pair studied in (a) and (b) has been created in terms of the same set of parameter values as in Figs. 4 and 5.

Figure 7

Spatial distributions of the order parameter components of a PV realized in $a\to 0$ limit of Fig. 6 in the compatible (upper two figures) and the incompatible (lower two figures) configurations. The upper-left and upper-right figures express $A_{x+}^{(+)}$ and $A_{x-}^{(+)}$, respectively. The lower-left and lower-right figures express $A_{x-}^{(-)}$ and $A_{x+}^{(-)}$, respectively. We note that, in all figures, the vortex winding number is $+1$. The spin index $x$ implies that the $d$- vector around this PV is spatially uniform and directed to the $x$ axis (see the text).

Figure 8

Spatial distributions of the order parameter components defined in Eq. (32) accompanying a PV in the compatible (upper two figures) and the incompatible (lower two figures) configurations, respectively. The upper-left and upper-right figures express the results of $|A_{xr}^{(+)}|$ and $|A_{x\varphi }^{(+)}|$, respectively, while the lower-left and lower-right figures express the results of $|A_{xr}^{(-)}|$ and $|A_{x\varphi }^{(-)}|$, respectively.

Figure 9

(a) Schematic figure of a possible defective vortex lattice created by a rotation angular velocity $\mathrm{\Omega }>0$ in the globally compatible situation (i.e., with $\widehat{l}\parallel \widehat{z}$ over a large area). A symbol V denotes a rotation-induced vortex. A chiral domain wall is indicated by an ellipse. Hence, we have the incompatible situation inside the chiral domain, and thus, it was assumed that no rotation-induced vortices are present there. (b) Corresponding figure in the globally incompatible situation (i.e., with $\widehat{l}\parallel -\widehat{z}$ over a large area). In this case, the rotation-induced vortices prefer to stay in the chiral domain where $\widehat{l}\parallel \widehat{z}$. (c) Possible configuration of one vortex pair (or, vortex loop) to be realized in the situation of (b). In this case, both a rotation-induced vortex inside the chiral domain and an antivortex (denoted as the symbol A) outside it are compatible with the chirality. (d) The 3D picture drawing the vortex-antivortex pair indicated in (c) as a vortex loop. Here, the upper and lower planes denote planar defects composing the aerogel locally. The segments of the vortex loop perpendicular to the $z$ axis will be strongly pinned by the defect planes so that such a vortex loop does not easily disappear.