Electron bubbles and Weyl Fermions in chiral superfluid $^3$He-A

Electrons embedded in liquid $^3$He form mesoscopic bubbles with radii large compared to the interatomic distance between $^3$He atoms, voids of $N_{bubble}\approx 200$ $^3$He atoms, generating a negative ion with a large effective mass that scatters thermal excitations. We develop scattering theory of Bogoliubov quasiparticles by negative ions embedded in $^3$He-A that incorporates the broken symmetries of $^3$He-A, particularly time-reversal and mirror symmetry in a plane containing the chiral axis $\hat{\bf l}$. Multiple scattering by the ion potential, combined with Andreev 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 - the mesoscopic realization of chiral edge currents in superfluid $^3$He-A films. A consequence is that electron bubbles embedded in $^3$He-A acquire angular momentum, ${\bf L}\approx -(N_{bubble}/2)\hbar\,\hat{\bf l}$, 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, ${\bf F}_{W} = \frac{e}{c}{\bf v}\times{\bf B}_{W}$, defined by an effective magnetic field, ${\bf B}_{W}\propto\hat{\bf l}$, 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 $^3$He-A. We also discuss our results in relation to earlier theoretical work on negative ions in superfluid $^3$He.