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

TL;DR

This paper investigates the transition between Majorana and Andreev bound states in inhomogeneous nanowires, revealing that the topological phase transition can occur without bulk gap closing, and characterizing the crossover through various invariants.

Contribution

It introduces a detailed characterization of the Majorana/Andreev crossover in inhomogeneous nanowires, highlighting the role of partial separation and nonlocal perturbations in topological transitions.

Findings

Transition between trivial and nontrivial states can occur without bulk gap closing.

Partially-separated Majorana modes undergo a parity crossing during the crossover.

Inhomogeneities can break topological protection by acting as nonlocal perturbations.

Abstract

Majorana bound states (MBS) and Andreev bound states (ABS) in realistic Majorana nanowires setups have similar experimental signatures which make them hard to distinguishing one from the other. Here, we characterize the continuous Majorana/Andreev crossover interpolating between fully-separated, partially-separated, and fully-overlapping Majorana modes, in terms of global and local topological invariants, fermion parity, quasiparticle densities, Majorana pseudospin and spin polarizations, density overlaps and transition probabilities between opposite Majorana components. We found that in inhomogeneous wires, the transition between fully-overlapping trivial ABS and nontrivial MBS does not necessarily mandate the closing of the bulk gap of quasiparticle excitations, but a simple parity crossing of partially-separated Majorana modes (ps-MM) from trivial to nontrivial regimes. We demonstrate that fully-separated and fully-overlapping Majorana modes correspond to the two limiting cases at the opposite sides of a continuous crossover: the only distinction between the two can be obtained by estimating the degree of separations of the Majorana components. 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.