Jackiw-Rebbi-type bound state carrying fractional fermion parity

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