Majorana modes and transport across junctions of superconductors and normal metals

We study Majorana modes and transport in one-dimensional systems with a $p$-wave superconductor (SC) and normal metal leads. For a system with a SC lying between two leads, it is known that there is a Majorana mode at the junction between the SC and each lead. If the $p$-wave pairing $\Delta$ changes sign or if a strong impurity is present at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the sub-gap conductance between the leads and the SC. We derive an analytical expression as a function of $\Delta$ and the length $L$ of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the shifts oscillate and decay exponentially as $L$ is increased. The energy shifts exactly match the location of the peaks in the conductance. Using bosonization and the renormalization group method, we study the effect of interactions between the electrons on $\Delta$ and the strengths of an impurity inside the SC or the barriers between the SC and the leads; this in turn affects the Majorana modes and the conductance. Finally we propose a novel experimental realization of these systems, in particular of a system where $\Delta$ changes sign at one point inside the SC.