Xiao-Li Huang and Yuli V. Nazarov
Phys. Rev. B 100, 085408 – Published 7 August 2019
ABSTRACT: 
We show theoretically that in the superconducting nanostructures the gapped states of different topology are not always protected by separating gapless states. Depending on the structure design parameters, they can be either protected or not, with a protection–unprotection transition separating these two distinct situations. We build up a general theoretical description of the transition vicinity in the spirit of Landau theory. We speculate that similar protection–unprotection transitions may also occur for other realizations of topological protection in condensed matter systems.