Deterministic Creation and Braiding of Chiral Edge Vortices

Majorana zero modes in a superconductor are midgap states localized in the core of a vortex or bound to the end of a nanowire. They are anyons with non-Abelian braiding statistics, but when they are immobile one cannot demonstrate this by exchanging them in real space and indirect methods are needed. As a real-space alternative, we propose to use the chiral motion along the boundary of the superconductor to braid a mobile vortex in the edge channel with an immobile vortex in the bulk. The measurement scheme is fully electrical and deterministic: edge vortices ($\pi$-phase domain walls) are created on demand by a voltage pulse at a Josephson junction and the braiding with a Majorana zero mode in the bulk is detected by the charge produced upon their fusion at a second Josephson junction.

Figure 1

Panels (a) and (b): Josephson junction geometries to deterministically inject a pair of edge vortices in chiral edge channels at opposite boundaries of a superconductor (yellow). The injection happens in response to a $2\pi$ increment in the superconducting phase difference $\varphi (t)$, driven by a time-dependent voltage $V(t)$ or flux $\mathrm{\Phi }(t)$. In panel (a) edge vortex 1 crosses the $2\pi$ branch cut of bulk vortex $R$, resulting in a fermion parity switch. Panel (c) shows the corresponding braiding of world lines in space-time: an overpass indicates that the vortex crosses a branch cut. Two crossings jointly switch the fermion parity of the edge vortices and of the bulk vortices, so that overall fermion parity is conserved. The orientation of the branch cuts can be varied by a gauge transformation and the measurable fusion outcome does not depend on it.

Figure 2

Starting from the layout of Fig. 1, we have inserted a second Josephson junction ($J_2$) and we have added normal metal contacts ($N_1$, $N_2$) to measure the current $I(t)$ carried by the edge modes in response to the voltage $V(t)$ applied to the superconductor. A unit charge per $2\pi$ increment of $\varphi$ is transferred from the superconductor into the normal metal contact. The counterpropagating Dirac edge mode along the upper edge of the Chern insulator is decoupled from the superconductor and plays no role in the analysis.

Figure 3

Bottom panel: Scattering phase $\eta (\varphi )-\eta (0)$ according to Eq. (5) (solid curve) and as obtained numerically (blue data points) from a lattice model [28] of the system shown in Fig. 2. There are no fit parameters in the comparison, the ratio $W/{\xi }_0=4.04$ was obtained directly from the simulation [35, 37]. The grey data points show the result without vortices, when there is no net increment as $\varphi$ advances from 0 to $2\pi$. Top panel: Current density in the lattice model. The two vortices are faintly visible.