Half-Quantum Vortices in Nematic and Chiral Phases of ³He

Indeed the discovery of HQVs in ³He infused into Nafen was reported soon after the discovery of the polar phase based on the nuclear magnetic resonance (NMR) signature of pairs of HQVs created by rotating the polar phase of superfluid ³He-Nafen [8]. In addition, these authors observe that the NMR signature of the HQV pair persists into the polar-distorted chiral phase of ³He-Nafen. That discovery comes 40 years after HQVs were predicted theoretically as a novel class of topological defects in condensed matter [7]. The prediction of HQVs, combined with more recent theoretical ideas for developing topological condensed matter as platforms for quantum information processing [9], led to searches for HQVs in diverse condensed matter systems, from Bose-Einstein condensates of optically trapped spin S=1 ²³Na atoms [10] to spin-triplet superconductors thought to be electronic analogs of superfluid ³He-A [11].

Here we report theoretical predictions for the structure of HQVs, their stability and the pressure-temperature phase diagram for two phases of rotating superfluid ³He confined in Nafen aerogels which host HQVs. Our analysis is based on an anisotropic impurity model for Nafen combined with strong-coupling Ginzburg-Landau (GL) theory that quantitatively accounts for the relative stability of the confined equal-spin pairing (ESP) superfluid phases in ³He-Nafen reported in Ref. [1]. This strong-coupling GL formalism also accounts for the relative stability of the A and B phases of pure superfluid ³He over the entire pressure-temperature range [12], as well as the vortex phase diagram of rotating ³He-B [13]. Using the strong-coupling GL theory, with the addition of the impurity model for the Nafen described below, we predict the pressure-temperature phase diagram for superfluid ³He-Nafen. The theoretically predicted phase diagram is in excellent agreement with the experimentally reported phase transitions observed in the ESP phases of ³He-Nafen [1].

For rotating ³He-Nafen with $\boldsymbol{\Omega}\parallel\hat{\bf z}$ we find two distinct vortex phases within the polar and polar-distorted chiral phases. In the polar phase region we find a stable array of pairs of HQVs. The cores of the HQVs are found to be the spin-polarized $\beta$-phase. At lower temperatures HQVs with additional internal structure are found embedded in the polar-distorted chiral phase. This HQV phase is the equilibrium state of rotating ³He-Nafen at temperatures below the polar to chiral phase transition, which is characterized by an anisotropic chiral order parameter and spontaneous supercurrents flowing along the axis of the HQVs and confined near their cores. Observation of these effects would provide key signatures for the identification of HQVs in both the polar and chiral phases of ³He-Nafen.

A special class of equal-spin pairing (ESP) states are those for which $\Delta_{\uparrow\downarrow}=0$ and $|\Delta_{\uparrow\uparrow}|=|\Delta_{\downarrow\downarrow}|$ for all ${\bf p}$ for a fixed direction $\hat{\bf d}$ in spin space. For ³He confined in low density Nafen, there are two bulk ESP phases: (1) the polar phase described by an order parameter of the form, $A^{\text{P}}_{\alpha i}=\Delta^{\text{P}}\,\hat{\bf d}_{\alpha}\,\hat{\bf n}_{i}$, where $\hat{\bf n}$ is the direction of the orbital pair wave function, and (2) the chiral phase described by an order parameter of the form, $A^{\text{C}}_{\alpha i}=\Delta^{\text{C}}\,\hat{\bf d}_{\alpha}\,\left(\hat{\bf n}_{i}\pm i\epsilon\hat{\bf m}_{i}\right)$, where a second p-wave orbital develops with $\hat{\bf m}\perp\hat{\bf n}$ and out of phase by $\pm\pi/2$. This type of in-plane chiral phase was also found to be a stable equilibrium phase in $100\,\mbox{nm}$ cylindrical pores [2].

The orbital axis $\hat{\bf n}$ of the polar phase is locked along the anisotropy axis of Nafen, i.e. $\hat{\bf n}=\hat{\bf z}$. The direction of the spin quantization axis, $\hat{\bf d}$, is weakly coupled to the orbital state via the nuclear dipolar energy which aligns $\hat{\bf d}\perp\hat{\bf n}$, i.e. $\hat{\bf d}=\cos\alpha\hat{\bf x}+\sin\alpha\hat{\bf y}$. The angle $\alpha\in\{0,2\pi\}$ is thus a degeneracy variable for the ESP phase. The other degeneracy variable is the global phase, $\vartheta\in\{0,2\pi\}$, of the amplitude, $\Delta=|\Delta|e^{i\vartheta}$ for either the polar or chiral phase.

The Ginzburg-Landau free energy functional is expressed in terms of linearly independent invariants constructed from $A_{\alpha i}$, $A_{\alpha i}^{*}$ and their gradients, $\boldsymbol{\nabla}_{j}A_{\alpha i}$ and $\boldsymbol{\nabla}_{j}A_{\alpha i}^{*}$. In particular, the GL functional can be expressed in terms of free energy densities,

where the bulk free energy density, $f_{\mathrm{b}}$, is given by one second-order invariant and five fourth-order invariants, the nuclear Zeeman energy, $f_{\text{Z}}$, and nuclear dipole-dipole energy, $f_{\mathrm{d}}$, which are also second-order in $A$, and the gradient energy, $f_{\text{$\boldsymbol{\nabla}$}}$, which is second-order in gradients of $A$. These terms, and the pressure-dependent material coefficients that define the strong-coupling GL functional, are discussed in detail in Ref. [13] and summarized in the Appendix. The last term in Eq. 3, $f_{\mathrm{imp}}$, is the free energy density associated with pair breaking by the Nafen strands. Impurities in a p-wave superfluid are pair breaking. Elastic scattering of quasiparticles comprising Cooper pairs leads to suppression of the order parameter over a region of order the coherence length, $\xi$, and a loss in condensate energy that depends on the density of impurities and the quasiparticle-impurity cross-section. Nafen is a highly porous anisotropic material comprised of long strands of crystalline Al₂O₃, c.f. Fig. 1 of Ref. [17]. Nafen-90, with density, $\rho=90\,\mbox{mg/cm}^{3}$, has a mean inter-strand distance of $L_{s}=47.8\,\mbox{nm}$, strand lengths of order millimeters and mean strand radius of $r_{s}=4.0\,\mbox{nm}$ [17].

For our analysis of the effects of Nafen on liquid ³He infused into the Nafen structure, we model Nafen as an array of non-magnetic line impurities of local areal density $n_{s}({\bf r})$. The effects of the array of anisotropic impurities on the superfluid phases of ³He is described to leading order in $A$ by the pair-breaking free energy density,

where ${\mathrm{I}}_{ij}({\bf r})$ is a uniaxial tensor under orbital space rotations that depends on the local areal strand density, $n_{s}({\bf r})$. In general the nematic axis varies in space. Here we treat the Nafen strands as globally aligned along an axis, $\hat{\bf z}$, and neglect fluctuations in the orientation of the nematic axis ¹¹1The effects of random anisotropy in anisotropic aerogels on the phases of superfluid ³He is discussed by several authors. See Ref. [5] and references therein.. Thus, for uniaxial nematic aerogels the impurity free energy density is defined by

is a uniaxial tensor that parametrizes the anisotropic scattering of quasiparticles by Nafen strands with areal density, $n_{s}({\bf r})$. The coefficients $\sigma_{\perp}$ and $\sigma_{\parallel}$ determine the cross-radii for quasiparticle scattering normal to the impurity strands and at grazing incidence along the strands, respectively. This form for $I_{ij}$ is consistent with the pair-breaking energy resulting from scattering by impurities embedded in superfluid ³He in Refs. [19; 5]. The prefactor, $\frac{\mbox{\tiny 1}}{\mbox{\tiny 3}}\,N_{f}\,\xi_{0}$, is the scale set by the coefficient of the second-order term for the bulk condensation energy, i.e. $f_{\mathrm{b}}=\alpha(T)\mbox{Tr}\big{\{}AA^{{\dagger}}\big{\}}$ with $\alpha(T)=\frac{\mbox{\tiny 1}}{\mbox{\tiny 3}}\,N_{f}\,\ln(T/T_{c})$, and the coherence length $\xi_{0}\equiv\hbar v_{f}/2\pi k_{\text{B}}T_{c}$ that enters the GL material parameters for the gradient energy density, $f_{\text{$\boldsymbol{\nabla}$}}$ (see Eqs. 4 and 9 of Ref. [13]). Note that $T_{c}$ is the superfluid transition temperature of pure, bulk ³He, $N_{f}=\frac{\mbox{\tiny 3}}{\mbox{\tiny 4}}\,n/E_{f}$ is the normal state density of states at the Fermi level, $E_{f}=\frac{\mbox{\tiny 1}}{\mbox{\tiny 2}}p_{f}v_{f}$ is the Fermi energy, $p_{f}$ is the Fermi momentum, $v_{f}=p_{f}/m^{*}$ is the Fermi velocity, $m^{*}$ is the effective mass of the ³He quasiparticles, and $n=p_{f}^{3}/3\pi^{2}\hbar^{3}$ is the density of liquid ³He. All of these Fermi liquid properties, as well as the stiffness coefficients, $K_{1,2,3}$, defining the GL gradient energy, $f_{\text{$\boldsymbol{\nabla}$}}$ (Eqs. 4 and 9 of Ref. [13]) and the strong-coupling $\beta$ parameters (Eqs. 2,7,8 and 12 of Ref. [13]) defining the bulk free energy density, $f_{\mathrm{b}}$, depend on pressure and are given in, or can be obtained from, Table II of Ref. [13].

The second-order impurity contribution given by Eq. 4 is essential, not only because it breaks rotation symmetry, but also because it competes with $\alpha(T,p)\,A_{\mu i}\,A_{\mu i}^{*}$, where $\alpha(T,p)$ is vanishingly small in the GL limit, i.e. near the bulk $T_{c}$. Uniaxial symmetry imposed on ³He by Nafen allows for additional anisotropic impurity corrections to the the gradient energy density, $f_{\mbox{\tiny$\nabla$}}$, as well as the fourth-order bulk free energy density, $f_{\mbox{\tiny b}}$. However, the gradient coefficients, e.g. $K_{1}\sim N_{f}\xi_{0}^{2}$, are finite near $T_{c}$. Thus, Nafen with mean impurity density, $\bar{n}_{s}=1/L_{s}^{2}$, generates perturbative corrections to the gradient energies of relative order $f_{\text{$\nabla$}}^{\text{imp}}/f_{\text{$\nabla$}}=\sigma_{||,\perp}\xi_{0}/L_{s}^{2}$, which varies from $0.06$ at high pressure to $0.20$ at $p=0$ bar. Thus, the transition line $T_{c_{2}}(p)$ shown in Fig. 2, as well as the spatial variations of the order parameter, are well described by retaining the leading order impurity term in the GL functional, Eq. 4. The uniaxial corrections to the gradient terms and fourth order terms may be related to the small deviations from experiment in the calculated transition line $T_{c_{2}}(p)$ at the lowest pressures, as shown in Fig. 2, but the analysis required to check this conjecture is outside the scope of this paper.

The areal density for impurities is given by $n_{s}({\bf r})=\sum_{i=1}^{N_{s}}\delta^{(2)}({\bf r}-{\bf r}_{i})$, where ${\bf r}_{i}$ is the position in the two-dimensional plane of the $i^{\text{th}}$ line impurity and $N_{s}$ is the total number of Nafen strands. The local density of a single line impurity is reasonably represented by the two-dimensional delta function when the geometric radius of the Nafen strand is small compared to the coherence length, i.e. $r_{s}\ll\xi_{0}$, which is the case for ³He infused into Nafen aerogel ²²2See Eq. 9.14 and related text in Ref. 37..

We investigate the pair-breaking effects of Nafen aerogel on superfluid ³He by embedding ³He in a square lattice of line impurities with mean areal density $\bar{n}_{s}=1/L_{s}^{2}$ where $L_{s}$ is the average interstrand distance of Nafen, c.f. Figure 1. The second-order transition from the normal state to the polar state is obtained from linear instability analysis as described in the Appendix. The transition temperature is generally suppressed and given by

to leading order in $\sigma_{\parallel}/\xi_{0}$, where $\bar{n}_{s}\xi_{0}^{2}$ is the dominant contribution from the $Q=0$ mode, while $\beta\approx 0.19\ln(0.10\,L_{s}/r_{s})$ is a correction from the $Q\neq 0$ modes (see Appendix). Note that for $\sigma_{\parallel}=0$, corresponding to specular scattering along the strands, i.e. ${\bf p}_{z}^{\prime}={\bf p}_{z}$, there is no suppression of the superfluid transition relative to that of pure ³He; scattering from the strands leads only to suppression of pairing into the in-plane ${\bf p}_{x},{\bf p}_{y}$ orbitals [21].

The experimental data (red squares in Fig. 2) for the normal to polar transition in Nafen-90 shows that $T_{c_{1}}<T_{c}$ implying non-specular scattering along the strands. The theoretical result (black line) for $T_{c_{1}}$ from Eq. 7 is shown in Fig. 2 for a pressure independent scattering cross section of $\sigma_{\parallel}=2.565\,\mbox{nm}$. We also used the experimentally reported areal density, or strand spacing $L_{s}=47.8\,\mbox{nm}$, and strand radius $r_{s}=4.0\,\mbox{nm}$. Note that the pressure dependence of $T_{c_{1}}$ arises from the pressure dependence of $T_{c}(p)$ for pure ³He and the corresponding coherence length, $\xi_{0}(p)=\hbar v_{f}/2\pi\,k_{\text{B}}\,T_{c}$, which were taken from Table II of Ref. [13].

For $T_{c_{2}}<T<T_{c_{1}}$ an inhomogeneous polar phase is the equilibrium phase with order parameter, $A_{\alpha i}=\Delta^{\text{P}}({\bf r})\,\hat{d}_{\alpha}\hat{z}_{i}$. The inhomogeneity of the polar phase order parameter, induced by pair-breaking from the Nafen impurities, is shown in Fig. 1. For Nafen-90 we find a second transition at $T_{c_{2}}(p)$, calculated and shown in Fig. 2 for in-plane scattering cross section $\sigma_{\perp}=10.16\,\mbox{nm}=3.961\times\sigma_{||}$, at which the Cooper pairs generate $\hat{{\bf p}}_{x,y}$ orbitals in the presence of a dominant polar amplitude ³³3Measurements of spin diffusion of ³He quasiparticles in Nafen reported in Ref. [38] give a ratio of mean free paths for heat transport along ($\parallel$) or perpendicular ($\perp$) to the Nafen strands of $l_{\parallel}/l_{\perp}=960\,\mbox{nm}/290\,\mbox{nm}=3.31$. The mean free paths for uniaxial Nafen are inversely proportional to the corresponding cross sections for quasiparticle scattering off a homogeneous density of line impurities, i.e. $l_{\parallel}/l_{\perp}=\sigma_{\perp}/\sigma_{\parallel}$. Our analysis of the phase diagram for an array of line impurities yields the ratio $\sigma_{\perp}/\sigma_{\parallel}=3.96$, in rough agreement with that inferred from spin diffusion. The comparison is better if we use the value of $\sigma_{\perp}$ obtained for a homogeneous aerogel density, $\sigma_{\perp}/\sigma_{\parallel}=8.021\,\mbox{nm}/2.565\,\mbox{nm}=3.13$.. An approximate analytic result for $T_{c_{2}}(p)$ based on the dominant $Q=0$ mode of the linear instability equation is shown for comparison and discussed in the Appendix ⁴⁴4See also Ref. [32]. We find an order parameter that is an ESP state with $A_{\alpha i}({\bf r})=\hat{d}_{\alpha}\,(\Delta^{\text{P}}({\bf r})\hat{z}_{i}\,\pm i\,\Delta^{\text{$\perp$}}({\bf r})\hat{m}_{i})$, with $\hat{\bf m}\perp\hat{\bf z}$. This phase breaks time-reversal symmetry with the chiral axis in-plane, $\hat{\bf l}=\hat{\bf z}\times\hat{\bf m}$. The strong-coupling GL theory and the impurity lattice model for Nafen-90 accurately account for the relative stability of the superfluid phases of ³He-Nafen ⁵⁵5Based on NMR frequency shift the polar-distorted chiral phase is a Larkin-Imry-Ma (LIM) phase in which random transverse fluctuations of the Nafen anisotropy axis lead to finite range orientational correlations of the transverse orbital state, i.e. $\langle\hat{\bf m}({\bf r})\hat{\bf m}(\mathbf{0})\rangle=e^{-r/\xi_{\mbox{\tiny LIM}}}$. The LIM correlation length is of order $\xi_{\mbox{\tiny LIM}}\approx 1\,\mu\mbox{m}$ [1], and the characteristic wavenumber, $Q_{\mbox{\tiny LIM}}=2\pi/\xi_{\mbox{\tiny LIM}}\ll Q_{1}=2\pi/L_{s}$, where $L_{s}\approx 50\,\mbox{nm}$ is the mean spacing between Nafen impurities. Thus, the gradient energy associated with LIM fluctuations, $f_{\mbox{\tiny LIM}}\propto K_{1}Q_{\mbox{\tiny LIM}}^{2}$, which in principle contributes to suppression of $T_{c_{2}}$, is dominated by the gradient energy associated depairing by the lattice of Nafen impurities, $f_{\mbox{\tiny$\nabla$}}\propto K_{1}Q_{1}^{2}$. Thus, the phase transition line, $T_{c_{2}}(p)$ shown in Fig. 2 is accurately calculated neglecting the LIM fluctuations.. Strong-coupling corrections to the GL functional are essential for the stability of the polar-distorted chiral phase. Weak-coupling theory predicts a polar to polar-distorted B-phase and no polar-distorted chiral phase [25; 26]. Within strong-coupling GL theory, with strong uniaxial anisotropy, the polar to polar-distorted chiral transition persists over the full pressure range for strongly anisotropic Nafen-90. Strong-coupling effects weaken at pressures below the bulk triple point, but are still present down to $p=0$ bar. The polar-distorted B phase is stabilized at lower temperatures for uniaxial anisotropy corresponding to Nafen-90, but we find that the polar-distorted chiral phase always appears between the polar phase and the polar-distorted B phase, if the latter is stable. See Fig. 4.2 and the related discussion in Ref. [6].

Note that the nuclear dipolar energy (Eq. 5 in Ref. [13]) is minimized for $\hat{\bf d}\perp\hat{\bf z}$ for both the polar phase and the polar-distorted chiral phase, as is the Zeeman energy for fields parallel to the nematic axis, ${\bf H}\parallel\hat{\bf z}$. This is the configuration suitable for topologically stable half-quantum vortices.

There are two distinct HQVs, as described in Eq. 2, in which one spin component hosts a $2\pi$ phase vortex while the time-reversed spin component is vortex free. The matrix representation of the order parameter corresponding to the HQV with $2\pi$ phase winding of the $\Delta_{\uparrow\uparrow}$ component is $A^{+}_{\alpha i}({\bf r})=\genfrac{}{}{}{1}{1}{2}\left[\Delta_{+}({\bf r})\,e^{i\phi}\,(\hat{x}+i\hat{y})_{\alpha}\,\hat{z}_{i}+\Delta_{-}({\bf r})\,(\hat{x}-i\hat{y})_{\alpha}\,\hat{z}_{i}\right]$, while that for the HQV with $2\pi$ phase winding in $\Delta_{\downarrow\downarrow}$ is, $A^{-}_{\alpha i}({\bf r})=\genfrac{}{}{}{1}{1}{2}\left[\Delta_{+}({\bf r})\,(\hat{x}+i\hat{y})_{\alpha}\,\hat{z}_{i}+\Delta_{-}({\bf r})\,e^{i\phi}\,(\hat{x}-i\hat{y})_{\alpha}\,\hat{z}_{i}\right]$. Superfluid rotation via HQVs involves pairs of these spatially separated HQVs,

with $\phi_{\pm 0}=\arctan(y/(x\mp x_{0}/2))$ where $x_{0}$ is the distance between the phase singularities of the HQV pair. Note that $\Delta_{\pm}({\bf r})$ have zeroes at $\pm x_{0}/2$, and that for $x_{0}=0$ the pair of HQVs collapse to a singly quantized vortex (SQV) in the polar phase with spatially uniform $\hat{\bf d}=\hat{\bf x}$.

We calculated the equilibrium and metastable vortex solutions for corresponding pairs of HQVs, as well as SQVs, per unit cell by numerical minimization of the strong-coupling GL free energy functional for vortex phases in both the polar and chiral regions of the phase diagram shown in Fig. 2. The calculations are based on a square grid $1600\,\mbox{nm}\times 1600\,\mbox{nm}$ with grid spacing $h=0.15\xi(p,T)$, where $\xi(p,T)$ is the GL coherence length, which is $\xi\approx 110\,\mbox{nm}$ for $p=15\,\mbox{bar}$ and $T/T_{c}=0.95$. The Nafen impurity model parameters for Nafen-90 were used as described in the previous section, and the self-consistent order parameter is scaled in units of the pure polar phase amplitude, $\Delta_{P}=\sqrt{|\alpha(T)|/2\beta_{12345}}$. Stationary states of the GL functional are obtained by numerical solution of the Euler-Lagrange equations for the order parameter, $A_{\alpha i}({\bf r})$, using the L-BFGS numerical optimization algorithm described in Refs. [27; 13]. Convergence to a particular stationary solution generally depends on the initialization of the order parameter and asymptotic boundary conditions. To obtain the stable HQV pair we initialize with a starting HQV order parameter profile. For the initial profile we use $\Delta({\bf r})=\Delta_{P}\tanh(|{\bf r}|/\sqrt{2}\xi)$ for each HQV. Similarly, we obtain stationary SQVs by setting $x_{0}=0$.

In both regions of the phase diagram we find stable vortex phases that are pairs of HQVs with separation of $x_{0}\simeq 1030\,\mbox{nm}$ within a unit cell of dimension $d_{v}\approx 70\,\mu\mbox{m}$. Figure 3 shows the amplitudes and phases of the pair of HQVs, where $C_{\pm 0}=\genfrac{}{}{}{1}{1}{2}\Delta_{\pm}e^{i\phi_{\pm 0}}$ are complex amplitudes expressed in terms of the spin and orbital angular momentum tensors, $\lambda^{\mu\nu}_{\alpha i}$, $\mu,\nu\in\{-1,0,+1\}$ [13]. The $2\pi$ phase winding of $\phi_{\pm 0}$ for each HQV is correlated with the zero of the corresponding amplitude. Note that there is no difference in the vortex structure, or relative stability, for axially aligned fields ${\bf H}=0-370G\ \hat{\boldsymbol{\Omega}}$, as expected for $\hat{\bf d}\perp{\bf H}$.

Each member of the HQV pair is also spin polarized. The resulting zero-field magnetization density for the HQV pair is given by ${\bf m}({\bf r})=m_{0}(|C_{+0}|^{2}-|C_{-0}|^{2})\hat{\boldsymbol{\Omega}}$ where $m_{0}=g^{\prime}_{z}|\Delta_{P}|^{2}$. The GL material parameter $g_{z}^{\prime}$ leads to a magnetization density of order $m_{0}\approx n(\gamma\hbar)\ln(E_{f}/k_{B}T_{c})(\Delta_{P}/E_{f})^{2}$ [28; 29]. The magnetization density of the HQV pair in the ESP basis is shown in the third panel of Fig. 3. Note, that the $m^{\uparrow\uparrow}$ and $m^{\downarrow\downarrow}$ contributions to the magnetic moment exactly cancel, i.e. the net magnetic moment induced by vortex flow vanishes identically, i.e. there is no vortex-induced Barnett effect for the HQV pair in the polar phase. This result contrasts with the vortex-induced Barnett effect for vortices in ³He-B or ³P₂ vortices in neutron matter [28; 29; 30; 13]. Our interpretation is that for the polar state the intrinsic orbital angular momentum of the pair condensate vanishes ⁶⁶6c.f. Eq. 32 for ${\mathcal{L}}_{i}$ in Ref.13 and evaluate for any polar state., and thus there is no transfer of intrinsic orbital angular momentum into spin angular momentum induced by rotation.

The basic structure of HQV pairs in the polar-distorted chiral phase is a simple generalization of Eq. 8 obtained by the replacement, $\hat{z}_{i}\rightarrow{\hat{z}}_{i}+i\epsilon({\bf r})\hat{y}_{i}$, where $\epsilon({\bf r})$ defines the local chiral amplitude. Initialization of the HQV pair proceeds similarly to that for HQVs in the polar phase with the additional initialization of $\epsilon\ll 1$ to that for the equilibrium chiral phase in the absence of rotation. In addition, for $A_{\alpha i}$ with $i=x,y$, we impose a condition of zero gradient on the computational boundary.

An idea for observing the axial currents, previously proposed for the D-core vortex in ³He-B [13], is to inject electrons perpendicular to the nematic axis of ³He-Nafen from the outer cell boundary. In the presence of axial currents the electrons will be transported in ³He parallel to the nematic axis, and can be captured and detected by imaging on the top and bottom surfaces of the cell. Such a detection of axial supercurrents in the polar-distorted chiral phase of ³He-Nafen would provide strong evidence of broken time-reversal symmetry and non-trivial topology in this novel phase of ³He.