Fractional spin Josephson effect in topological spin superconductors

Topological spin superconductors are $p$-wave spin-triplet exciton insulators whose topological edge modes have been shown to obey non-Abelian braiding statistics. Based on a toy model as the spin counterpart of the Kitaev's chain, we study the spin Josephson effect adopting the $S$ matrix as well as the Green's function method. The on-site energies of these topological edge modes lead to a transition between the fractional and integer spin Josephson effects. Moreover, nonvanishing on-site energies will also induce a charge pump through the spin Josephson junction. These two effects, distinct features of topological spin superconductors and absent in Majorana systems, can be utilized for spin transport detection of topological spin superconductors.