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Majorana and fractionally charged bound states in 1-D Rashba nanowire under spatially varying Zeeman fields

We study topological phase transitions in one dimensional (1-D) Rashba nanowire under a spatially varying Zeeman field when coupled to an $s$-wave superconductor substrate. We show that this system supports both Majorana bound states (MBS) and fractionally charged bound states (FBS) of Jackiw-Rebbi type. By disassembling Zeeman Hamiltonian into multiple helical components, we find that each helical component is relating to a corresponding topological region, characterized by the emergence of MBS. FBS arises in the overlapping gapped area created by any two helical components with topologically differing configuration, analogous to those formed at the knot in SSH model. We then develop a general criteria for the occurrence conditions of MBS and FBS. Our results suggest that systems with large Rashba spin orbit couple amplitude or in presence of weak Zeeman fields favor MBS, and otherwise FBS are more favorable. In the end, we demonstrate that spin components of zero energy bound states in topological phases are polarized in the plane perpendicular to Rashba vector, and the polarization oscillates with the variance of phases of the Zeeman field.