Possibility of an adiabatic transport of an edge Majorana through an extended gapless region

In the context of slow quenching dynamics of a p-wave superconducting chain, it has been shown that a Majorana edge state can not be adiabatically transported from one topological phase to the other separated by a quantum critical line. On the other hand, the inclusion of a phase factor in the hopping term, that breaks the effective time reversal invariance, results in an extended gapless region between two topological phases. We show that for a finite chain with an open boundary condition there exists a non-zero probability that an edge Majorana can be adiabatically transported from one topological phase to the other across this gapless region following a slow quench of the superconducting term; this happens for an optimum transit time, that is proportional to the system size and diverges for a thermodynamically large chain. We attribute this phenomenon to the mixing of the Majorana only with low-lying inverted bulk states. Remarkably, the Majorana state always persists with the same probability even after the quenching is stopped. For a periodic chain, on the other hand, we find a Kibble-Zurek scaling of the defect density with a renormalized rate of quenching.