Behaviour of Andreev states for topological phase transition

We consider three one-dimensional superconducting structures: 1) the one with $p$-wave superconductivity; 2) the main experimental model of a nanowire with $s$-wave superconductivity generated by the bulk superconductor due to the proximity effect in an external magnetic field and Rashba spin–orbit interaction; 3) the boundary of a two-dimensional topological insulator with an $s$-wave superconducting order in an external magnetic field. We obtain precise analytic results for the “superconductor–magnetic impurity–superconductor” model. Using the Bogoliubov–de Gennes Hamiltonian, we study the behavior of stable states arising in these structures, with energies near the edges of the energy gap of “electron” (“hole”) type for the first model and “electron plus hole” type for the other two models in the case where the system passes from the topological phase to the trivial one. For the topological phase transition, resonance (decaying) states turn out to play a major role; the spin flip and the change of sign of the charge occur due to the transition of bound states to resonance ones and vice versa with their energy changing to the opposite ones as the gap closes. The results are consistent with the absence of a zero-bias conductance peak in the trivial topological phase observed in a recent experiment.