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.@en

Recently, we have proposed an unusual mechanism of superconducting current that is specific for quantum Hall edge channels connected to superconducting electrodes. We have shown that the supercurrent can be mediated by a nonlocal electron-electron interaction that provides an opportunity for a long-distance information transfer in the direction opposite to the electron flow. A convenient model for such interaction is that of an external circuit. The consideration has been performed for the case of a single channel. In order to facilitate the experimental verification and the observation of peculiar features of the effect, in this paper, we provide a more detailed description of the phenomenon and extend the results to more sophisticated setups. We establish that the dynamical phase contributes to superconducting interference; this being the manifestation of the channel chirality. We consider setups that include the scattering between quantum Hall channels of opposite direction and multiple superconducting contacts. For a single quantum Hall constriction, we derive a general and comprehensive relation for the interaction-induced supercurrent in terms of scattering amplitudes and demonstrate the nonlocal nature of the current by considering its sensitivity to scattering. In multiterminal setups, we reveal the characteristic phase dependences of the supercurrents explaining those in terms of interference of Andreev reflection processes. For more complex setups encompassing, at least, two constrictions, we find an interplay between noninteracting and interaction-induced currents and contributions of more complex interference processes.@en