Zero energy modes to mid-gap states in one dimensional charge conserving superconductors - Exact results

We consider one dimensional charge conserving superconductors where intrinsic attractive pairing interactions between the electrons gives rise to superconductivity. Using Bethe ansatz we show that in such systems with open geometry, when appropriate boundary conditions are applied, the system can exhibit topological phase characterized by the existence of zero energy bound states localized at the edges. For certain boundary conditions these bound states gives rise to degeneracy in the ground state structure which results in the existence of fractionalized 1/4 spins at the edges. We show that by tuning the boundary parameters appropriately, these zero energy modes can be turned into bound states who's energy is within the dynamically generated mass gap in the bulk, and hence are called midgap states. We provide physical realization of such systems.