Electron bubbles and Weyl fermions in chiral superfluid ${}^3\mathrm{He}\text{−}A$

Electrons embedded in liquid ${}^3\mathrm{He}$ form mesoscopic bubbles with large radii compared to the interatomic distance between ${}^3\mathrm{He}$ atoms, voids of $N_{\text{bubble}}\approx 200$ ${}^3\mathrm{He}$ atoms, generating a negative ion with a large effective mass that scatters thermal excitations. Electron bubbles in chiral superfluid ${}^3\mathrm{He}\text{−}A$ also provide a local probe of the ground state. We develop a scattering theory of Bogoliubov quasiparticles by negative ions embedded in ${}^3\mathrm{He}\text{−}A$ that incorporates the broken symmetries of ${}^3\mathrm{He}\text{−}A$, particularly broken symmetries under time reversal and mirror symmetry in a plane containing the chiral axis $\widehat{\mathbf{l}}$. Multiple scattering by the ion potential, combined with branch conversion scattering by the chiral order parameter, leads to a spectrum of Weyl fermions bound to the ion that support a mass current circulating the electron bubble—a mesoscopic realization of chiral edge currents in superfluid ${}^3\mathrm{He}\text{−}A$ films. A consequence is that electron bubbles embedded in ${}^3\mathrm{He}\text{−}A$ acquire angular momentum, $\mathbf{L}\approx -(N_{\text{bubble}}/2)\hslash \widehat{\mathbf{l}}$, inherited from the chiral ground state. We extend the scattering theory to calculate the forces on a moving electron bubble, both the Stokes drag and a transverse force, ${\mathbf{F}}_{\text{W}}=\frac{e}{c}\mathbf{v}\times {\mathbf{B}}_{\text{W}}$, defined by an effective magnetic field, ${\mathbf{B}}_{\text{W}}\propto \widehat{\mathbf{l}}$, generated by the scattering of thermal quasiparticles off the spectrum of Weyl fermions bound to the moving ion. The transverse force is responsible for the anomalous Hall effect for electron bubbles driven by an electric field reported by the RIKEN group. Our results for the scattering cross section, drag, and transverse forces on moving ions are compared with experiments and shown to provide a quantitative understanding of the temperature dependence of the mobility and anomalous Hall angle for electron bubbles in normal and superfluid ${}^3\mathrm{He}\text{−}A$. We also discuss our results in relation to earlier work on the theory of negative ions in superfluid ${}^3\mathrm{He}$.

Figure 1

An unbounded ${}^3\mathrm{He}\text{−}A$ film with an inner boundary of radius $R\gg {\xi }_0$. A ground-state edge current of magnitude $J=\frac{1}{4}n\hslash$ circulates on the inner boundary generating an angular momentum $L_z=-(N_{\text{hole}}/2)\hslash$ with $N_{\text{hole}}=n\pi R^{2}w$, i.e. the number of ${}^3\mathrm{He}$ atoms excluded by the hole.

Figure 2

Phase shifts as a function of angular momentum channel number $l$ for the hard-sphere potential with $k_{f}R=11.17$. Note that channels with $l>12$ are effectively irrelevant.

Figure 3

Local quasiparticle density of states around an electron bubble in superfluid ${}^3\mathrm{He}\text{−}A$ at distance $r=30k_f^{-1}$ from the center of the bubble, and in the equatorial plane, i.e. polar angle $\vartheta =\pi /2$ relative to the chiral axis. The bubble is described as a hard sphere with radius $R=11.17k_f^{-1}$. The inset shows two low-energy resonances with internal structure in their spectral density.

Figure 4

The LDOS resolved into orbital angular momentum components labeled by $m$ as defined by Eq. (39). $N_m(\mathbf{r},E)$ represents a discrete spectrum of Weyl fermions bound to the electron bubble, plus hybridization with the nodal quasiparticles. Note the double degeneracy of each subgap energy level.

Figure 5

Current density $\mathbf{j}\left(\mathbf{r}\right)$ in units of $N_f{v}_f{k}_B{T}_c$ calculated at distance $k_{F}r=30.0$ from the bubble center. We used the hard sphere model with $k_{F}R=11.17$, shown in gray. The chiral axis $\widehat{\mathbf{l}}$ determines the direction of Cooper pair angular momentum. The temperature is taken as $T=0.5T_c$.

Figure 6

Radial dependence of the current density $j_{\varphi }(r,\vartheta )$ for $\vartheta =\pi /2$, in units of $v_f{N}_f{k}_B{T}_c$, calculated for the electron bubble with hard sphere radius of $k_{f}R=11.17$ and temperature $T=0.5T_c$. The current develops sharply from the bubble radius, then decays rapidly for $r>R$. Quantum oscillations on the Fermi-wave-length scale are evident at short distances. (Inset) The current decays exponentially, $j_{\varphi ,\text{av}}\sim -e^{-r/{\xi }_0}/{\left(k_{f}r\right)}^2$, at distances greater than the coherence length, ${\xi }_0=608.7k_f^{-1}$; $j_{\varphi ,\text{av}}$ is the quasiclassical envelope obtained by averaging $j_{\varphi }\left(r\right)$ over a Fermi wavelength as in Eq. (40).

Figure 7

Angular momentum of hard sphere “bubbles” embedded in ${}^3\mathrm{He}\text{−}A$ as a function of hard sphere radius, $k_{f}R$. For an electron bubble, $k_{f}R=11.17$. In the macroscopic limit, $R\gg {\xi }_0$ (red dashed line), $L_z$ scales to $L_z^{\infty }=-(N_{\text{bubble}}/2)\hslash$. (Inset) Temperature dependence of $L_z$ for the electron bubble.

Figure 8

Panel (a) shows a polar plot of the differential cross section, $d{\sigma }^{(+)}({\widehat{\mathbf{k}}}^{\prime },\widehat{\mathbf{k}};E)/d{\mathrm{\Omega }}_{{\mathbf{k}}^{\prime }}$, as a function of the in-plane scattering angle angle ${\varphi }^{\prime }-\varphi$ [Eqs. (64) and (66)], with incoming and outgoing momenta lying in the $xy$ plane, i.e. ${\theta }^{\prime }=\theta =\pi /2$ and $\widehat{\mathbf{k}}={\mathbf{e}}_x$ $\left(\varphi =0\right)$. The contours mark the magnitudes of the differential cross sections in units of $\pi R^2$ on a logarithmic scale. The quasiparticle energy is $E=1.01\mathrm{\Delta }$, and the ion-quasiparticle potential is a hard sphere with $k_{f}R=11.17$. Similarly, (b) shows the asymmetry in the angular distribution of scattered quasiparticles given by $d{\sigma }^{(-)}({\widehat{\mathbf{k}}}^{\prime },\widehat{\mathbf{k}};E)/d{\mathrm{\Omega }}_{{\mathbf{k}}^{\prime }}$, which changes sign continuously across the lines $\mathrm{\Delta }\varphi =0,\pi$. The sign change is indicated by the dashed red curve. (c) shows the sum of these two differential cross sections, highlighting the asymmetry in the angular distribution of scattering quasiparticles for $\mathbf{l}\left|\right|\mathbf{z}$. The angular distribution for quasiparticle-ion scattering in the normal state is shown as the dashed green line.

Figure 9

The longitudinal (a) and transverse (b) transport cross sections as a function of energy for $T=0.5T_c$. The peak-dip structure at energies below the maximum gap $\mathrm{\Delta }$ are resonances originating from scattering of quasiparticles by chiral fermions bound to the surface of the electron bubble associated with distinct angular momentum channels [Eqs. (67) and (68)]. The quasiparticle-ion potential is a hard sphere with $k_{f}R=11.17$. The insets highlight the low-energy region.

Figure 10

Longitudinal and transverse Stokes parameters, ${\eta }_{xx}/{\eta }_{\text{N}}$ (solid blue line) and ${\eta }_{xy}/{\eta }_{\text{N}}$ (dashed red line), as a function of $T/T_c$. Calculations are based on the hard sphere quasiparticle-ion potential with $k_{f}R=11.17$.

Figure 11

Experimental data for the longitudinal mobility normalized to the normal-state mobility is from Ref. [8] shown as blue circles. The theoretical result based on the hard sphere (HS) quasiparticle-ion potential is the black curve. Results based on the soft core (SS) potential are shown as the dashed green curve, and those for the four-parameter potential with intermediate attraction (SSWAW) are shown as the red dashed curve. (Inset) Scattering phase shifts vs angular momentum channel calculated for the three potentials. For the HS model: $k_{f}R=11.17$; SS model: $V_0=1.01E_f$ and $k_{f}R=12.48$; SSWAW model: $V_0=100E_f{V}_1=10E_f,k_f{R}^{\prime }=10.99$, and $R/R^{\prime }=0.36$, all constrained by the experimental value of ${\mu }_{\text{N}}$.

Figure 12

(a) Hall ratio for the motion of electron bubbles in ${}^3\mathrm{He}\text{−}A,tan\alpha \equiv v_y/v_x={\eta }_{xy}/{\eta }_{xx}$, as a function of temperature for the hard sphere model for the quasiparticle-ion potential with $k_{f}R=11.17$ (black line). The experimental data was from the RIKEN group [8, 9]. Theoretical results for the repulsive soft core potential (dashed green line) and repulsive potential with short-range attraction (dashed red line) are shown for comparison. The dotted blue line corresponds to the calculation based on the formulas from Salmelin et al. [23, 24] presented in Refs [8, 9]. (b) The same results presented as a function of $\mathrm{\Delta }\left(T\right)/k_{B}T$.