Free Majorana Modes in Superconducting Quantum Wires

An s-wave superconducting wire with attractive interactions can admit a zero-energy bound state equation (solution) at an edge similar to the Jackiw-Rebbi model; this is a specific aspect of low-dimensional quantum systems, Dirac equation and e.g. of the Luther-Emery liquid with a spin gap. In this Letter, K. Le Hur Europhys. Lett., 49 (6), pp. 768-774 (2000), I introduced a magnetic spin-1/2 impurity interacting with such a (spin) bound state in a Luther-Emery wire representing then a Hubbard ladder in a d-wave superconducting state. This method is general and show how Majorana fermions at zero energy, i.e. Majorana zero modes, can take place in a superconducting wire model from the two-channel Kondo effect. Within these two channels (wires), the Luther-Emery form of the superconducting term can be reached within the weak-coupling attractive limit. Due to the interest in Majorana zero modes from magnetic impurities interacting with an s-wave superconductor, I take time to analyze zero-energy edge solutions in my model and present a correspondence with the p-wave superconducting wire in the topological phase through alternatives versions of the quantum field theory. I develop the relation between the edge magnetic susceptibility and the local capacitance measure in a p-wave superconducting wire. I elaborate on the idea that Majorana zero modes, i.e. free Majorana fermions, can be realized with magnetic impurities bridging the gap between two s-wave superconducting wires. The spin gap and the resonance with the impurity can protect the free Majorana solutions when including perturbations.