"Majorana and Andreev bound states in topological wires in the
proximity of superconductors"
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We study one-dimensional topological SN and SNS long junctions
obtained by placing a topological insulating nanowire in the proximity
of either one or two SC finite-size leads. Using the Majorana
Polarization order parameter we find that for a finite-size SN
junction the ABS spectrum exhibits a zero-energy extended state which
carries a full Majorana fermion, while the ABS of long SNS junctions
with a phase difference of pi transform into two zero-energy states
carrying two Majorana fermions with the same MP. We also study the
effects of finite SC penetration depths in such junctions, as well as
the effects of uniform phase gradients. Last but not least we analyze
a more realistic model for the coupling between a superconducting
substrate and a topological wire, the resulting proximity effect, and
the role played by the Andreev bound states in the topological wire in
such a setup.