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Substrate-induced Majorana renormalization in topological nanowires

We theoretically consider the substrate-induced Majorana localization length renormalization in nanowires in contact with a bulk superconductor in the strong tunnel-coupled regime, showing explicitly that this renormalization depends strongly on the transverse size of the one-dimensional nanowires. For metallic (e.g. Fe on Pb) or semiconducting (e.g. InSb on Nb) nanowires, the renormalization effect is found to be very strong and weak respectively because the transverse confinement size in the two situations happens to be 0.5nm (metallic nanowire) and 20nm (semiconducting nanowire). Thus, the Majorana localization length could be very short (long) for metallic (semiconducting) nanowires even for the same values of all other parameters (except for the transverse wire size). We also show that any tunneling conductance measurements in such nanowires, carried out at temperatures and/or energy resolutions comparable to the induced superconducting energy gap, cannot distinguish between the existence of the Majorana modes or ordinary subgap fermionic states since both produce very similar broad and weak peaks in the subgap tunneling conductance independent of the localization length involved. Only low temperature (and high resolution) tunneling measurements manifesting sharp zero bias peaks can be considered to be signatures of Majorana modes in topological nanowires.