Electron-Tunneling-Assisted Non-Abelian Braiding of Rotating Majorana Bound States

It has been argued that fluctuations of fermion parity are harmful for the demonstration of non-Abelian anyonic statistics. Here, we demonstrate a striking exception in which such fluctuations are actively used. We present a theory of coherent electron transport from a tunneling tip into a Corbino geometry Josephson junction where four Majorana bound states (MBSs) rotate. While the MBSs rotate, electron tunneling happens from the tip to one of the MBSs thereby changing the fermion parity of the MBSs. The tunneling events in combination with the rotation allow us to identify a novel braiding operator that does not commute with the braiding cycles in the absence of tunneling, revealing the non-Abelian nature of MBSs. The time-averaged tunneling current exhibits resonances as a function of the tip voltage with a period that is a direct consequence of the interference between the noncommuting braiding operations. Our work opens up a possibility for utilizing parity nonconserving processes to control non-Abelian states.

Figure 1

(a) Schematic of a Corbino geometry Josephson junction formed by thin-film superconductors ($S_1$ and $S_2$) deposited on the surface of a topological insulator (TI). In the presence of four flux quanta $4{\mathrm{\Phi }}_0$, four MBSs ${\gamma }_j$ (red balls) appear in the junction. Majorana positions can move along the junction when applying a small voltage across the junction, allowing us to perform an adiabatic rotation. (b) Braiding depicted as worldlines of the four MBSs corresponding to the $\pi /2$ rotation shown in (a).

Figure 2

(a) Time-dependent electron tunneling between the rotating MBSs and a metal tip for detecting non-Abelian statistics. (b) Tunneling-assisted braiding created by the composition of the $\pi /2$ rotation shown in (a) and electron tunneling. The tunneling effect reverses the exchange direction of a Majorana pair involving ${\gamma }_1$. A signature of the interference processes involving the non-Abelian braiding operations—the tunneling-assisted braiding and the braiding shown in Fig. 1—is probed by the time-averaged tunneling current.

Figure 3

Plot of the time averaged differential conductance given in Eq. (13) with parameters $\hslash T_J^{-1}=0.1\mathrm{meV}={10}^{-1}\hslash \lambda =10k_{B}T=10\mathrm{\Gamma }$. The conductance peak spacing $h/(4T_J)$ is a consequence of the non-Abelian state evolution within the degenerate ground-state manifold.