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We show theoretically that in the superconducting nanostructures the gappedstates of different topology are not always protected by separating gappedstates. Depending on the structure design parameters, they can be eitherprotected or not, with a protection-unprotection transition separating thesetwo distinct situations. We build up a general theoretical description of thetransition vicinity in the spirit of Landau theory. We speculate that similarprotection-unprotection transitions may also occur for other realizations oftopological protection in condensed matter systems.

Topology protection-unprotection transition: an example from multi-terminal superconducting nanostructures