Majorana Bound State in Rotating Superfluid ${}^3\mathrm{He}\mathrm{\text{−}}A$ between Parallel Plates

A concrete and experimentally feasible example for testing the putative Majorana zero-energy state bound in a vortex is theoretically proposed for a parallel plate geometry of superfluid ${}^3\mathrm{He}\mathrm{\text{−}}A$ phase. We examine the experimental setup in connection with ongoing rotating cryostat experiments. The theoretical analysis is based on the well-established Ginzburg-Landau functional, supplemented by microscopic calculations of the Bogoliubov–de Gennes equation, both of which allow the precise location of the parameter regions of the Majorana state to be found in realistic situations.

Figure 1

Spatial structures of the order parameters $|A_{+}|$ (left-hand side) and $|A_{-}|$ (right-hand side) for the (0,2) state. The sense of the phase winding and its number are shown. $R=0.8\mu \mathrm{m}$ and $\Omega =100\mathrm{rad}/\sec$.

Figure 2

Spatial structures of the order parameters, $|A_{+}|$ (left-hand side) and $|A_{-}|$ (right-hand side) for the (1,3) state. $R=0.8\mu \mathrm{m}$ and $\Omega =100\mathrm{rad}/\sec$.

Figure 3

Size $R$ dependence of the critical angular velocity ${\Omega }_c$ from (0,2) at rest to (1,3) at a higher rotation ($T/T_c=0.95$). An extrapolated value of ${\Omega }_c=0.06\mathrm{rad}/\sec$ is found at $R=1.5\mathrm{mm}$. The inset shows the energy differences (arbitrary scale) between the two states as a function of $\Omega$ for $R=100\mu \mathrm{m}$.

Figure 4

Energy spectrum $E_{\mathit{q}}$ with $q_z=0$ (a) and $q_{\theta }=1$ (b) in the (1,3) vortex under no rotation $\Omega =0$, where $\Delta \equiv \max |A_{\pm }(r)|$. The branches labeled “Core” and “Edge” in (a) are eigenstates having wave functions tightly bounded at the vortex core and the edge, respectively. The zero-energy state (ZES) appears within $|q_z|<k_F$ at $q_{\theta }=1$.