Crossover between Majorana and Andreev bound states in Majorana nanowires, and the fate of the topological phase transition

Majorana bound states (MBS) are topologically protected and spatially-separated zero-energy excitations localized at the opposite ends of Majorana nanowires, i.e., proximitized semiconducting nanowires with strong spin-orbit coupling and broken time-reversal symmetry, and exhibit nonabelian exchange statistics that may lead to a major breakthrough in the field of fault-tolerant quantum computation. Quite a few experiments observed signatures compatible with the existence of MBS in Majorana nanowires. However, these signatures may also be explained by the presence of trivial Andreev bound states (ABS) with zero or near-zero energies, induced by random disorder, impurities, or smooth confinement. In general, trivial ABS can be described as a pair of partially-separated Majorana modes, as opposed to the fully-separated MBS. In our work, we study the continuous crossover interpolating between fully-separated MBS, partially-separated ABS, and fully-localized ABS. We will characterize this crossover in terms of the global and local topological invariants, fermion parity, Majorana polarization, and density of states. We found that the topological phase transition between trivial ABS and nontrivial MBS does not correspond to the closing of the bulk gap, but to a simple parity crossing of the ABS from trivial to nontrivial regime. This result does not contradict the bulk-edge correspondence: Indeed, the field inhomogeneities driving the Majorana/Andreev crossover have a length scale comparable with the nanowire length, and therefore correspond to a nonlocal perturbation which breaks the topological protection of the MBS.