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Majorana bound states (MBS) and Andreev bound states (ABS) in realisticMajorana nanowires setups have similar experimental signatures which make themhard to distinguishing one from the other. Here, we characterize the continuousMajorana/Andreev crossover interpolating between fully-separated,partially-separated, and fully-overlapping Majorana modes, in terms of globaland local topological invariants, fermion parity, quasiparticle densities,Majorana pseudospin and spin polarizations, density overlaps and transitionprobabilities between opposite Majorana components. We found that ininhomogeneous wires, the transition between fully-overlapping trivial ABS andnontrivial MBS does not necessarily mandate the closing of the bulk gap ofquasiparticle excitations, but a simple parity crossing of partially-separatedMajorana modes (ps-MM) from trivial to nontrivial regimes. We demonstrate thatfully-separated and fully-overlapping Majorana modes correspond to the twolimiting cases at the opposite sides of a continuous crossover: the onlydistinction between the two can be obtained by estimating the degree ofseparations of the Majorana components. This result does not contradict thebulk-edge correspondence: Indeed, the field inhomogeneities driving theMajorana/Andreev crossover have a length scale comparable with the nanowirelength, and therefore correspond to a nonlocal perturbation which breaks thetopological protection of the MBS.

Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires