Braiding Majorana zero modes through Majorana spin

The Majorana physics in a spin-full system is generally beyond the simplified spin-less model in the sense that the manipulation, detection and braiding of MZMs are in principle allowed through the addtional spin degree of freedom. In this work, we propose a new spin-based scheme to braid Majorana zero modes (MZMs) in a generic spin-full Majorana system. Due to the spin-triplet nature of MZMs, the worldlines, tracking the braiding history of MZMs, should be extended to worldribbons in spin space. The braiding of two MZMs is shown to be equivalent to the self-rotation of their spin by $\pi$ which can be depicted as twisting the two associated worldribbons. Although this scheme is general for any spin-full Majorana system, we explicitly show its realization based on a concrete model, in which the Fermion parity switch due to braiding MZMs is rooted from the topological charge pumping.