Edge mass current and the role of Majorana fermions in $A$-phase superfluid ${}^3$He

The total angular momentum associated with the edge mass current flowing at the boundary in the superfluid ${}^3$He $A$ phase confined in a disk is proved to be $L=N\hslash /2$, consisting of $L^{\mathit{MJ}}=N\hslash$ from the Majorana quasiparticles (QPs) and $L^{\mathrm{cont}}=-N\hslash /2$ from the continuum state. We show this based on an analytic solution of the chiral order parameter for the quasiclassical Eilenberger equation. Important analytic expressions are obtained for mass current, angular momentum, and density of states (DOS). Notably the DOS of the Majorana QPs is exactly $N_0/2$ ($N_0$: normal state DOS) responsible for the factor 2 difference between $L^{\mathit{MJ}}$ and $L^{\mathrm{cont}}$. The current decreases as $E^{-3}$ against the energy $E$, and $L\left(T\right)\propto -T^2$. This analytic solution is fully backed up by numerically solving the Eilenberger equation. We touch on the so-called intrinsic angular momentum problem.

Figure 1

Energy profiles of $\theta$-angle-resolved LDOS (a) and mass current along the edge (b) at $x=0$ for $\theta =\pi /2$. At the zero temperature, quasiparticles fill the colored (shaded) states in (a). Mass current from the bound and continuum states is derived by integrating the blue (light gray) and pink (gray) regions in (b), respectively.

Figure 2

Profiles of mass current along the edge at $T=0.5T_c$ under the self-consistent pair potential (solid line) and uniform pair potential (dashed line). The units are ${\xi }_0=\hslash v_F/2\pi k_B{T}_c$ and $j_0=m{v}_F{N}_0\pi k_B{T}_c$. Inset: A profile of the self-consistent pair potential.

Figure 3

Left axis: Temperature dependence of angular momentum under the self-consistent pair potential (solid circles) and uniform pair potential (open circles). Right axis: Temperature dependence of the component of the superfluid density tensor ${\rho }_{s\parallel }^0$ (solid line) and ${\rho }_{s\perp }^0$ (dotted line).