Topological aspects of Josephson pi-junctions in Kitaev wires

We investigate the topological properties of Josephson junctions with a phase shift of $\pi$ in one dimensional topological superconductors. We show that by imposing a reality constraint on the order parameter, the resulting zero energy junction bound state can be mapped onto a soliton solution of a Jackiw-Rebbi type Dirac equation similar to that in polyacetylene. Further, we consider $\pi$-junctions where the order parameter phase winds across the junction so that the aforementioned bound state generically acquires a gap. We relate these observations to the classification of the junctions according to their anti-unitary symmetries.