Braiding Majorana zero modes in spin space: from worldline to worldribbon

We propose a scheme to braid Majorana zero modes (MZMs) through steering the spin degree of freedom, without moving, measuring, or more generically fusing the modes. For a spinful Majorana system, we show that braiding two MZMs is achieved by adiabatically reversing the Majorana spins, which topologically corresponds to twisting two associated worldribbons, the extention of worldlines that track the braiding history of MZMs. We demonstrate the feasibility of applying the current scheme to the superconductor/2D-topological-insulator/ferromagnetic-insulator (SC/2DTI/FI) hybrid system which is currently under construction in experiment. A single braiding of two MZMs is precisely achieved by adiabatically reversing the FI magnetization, not relying on details of the reversing path, and the braiding operation is shown to be stable against local imperfections such as the static and dynamical disorder effects. The stability is a consequence of the intrinsic connection of the current scheme to topological charge pumping. The proposed device involves no auxiliary MZMs, rendering a minimal scheme for observing non-Abelian braiding and having advantages with minimized errors for the experimental demonstration.