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Jackiw-Rebbi-type bound state carrying fractional fermion parity

We find the coexistence of two kinds of non-abelian anyons, Majorana fermion at the geometric ends and Jackiw-Rebbi-type bound state (JRBS)at a domain-wall, in a topological superconducting phase in one-dimensional (1D) systems. Each localized JRBS carries a new fractional quantity, half of the parity of fermion number. This induces a topological protected crossing at the zero energy for its eigen-energy. For a chain embedded with a JRBS, one is possible to switch between the occupied and empty states of Majorana zero energy state (MZES) by varying the strength of external magnetic field across that crossing point. This enable a way to encode a quantum qubit into one MZES without breaking parity conservation. We propose that such JRBS and Majorana fermion can appear in two 1D models, one can be accomplished in an artificial lattice with staggered hopping, staggered spin-orbital interaction and staggered superconducting pairing for cold fermion atoms, the other is describing a 1D semiconductor chain sandwiched between s-wave superconductor and antiferromagnet.