Drastic effect of weak interaction near special points in semiclassical multiterminal superconducting nanostructures

A generic semiclassical superconducting nanostructure connected to multiple terminals hosts a quasi-continuous spectrum of Andreev states. The is sensitive the phases terminals. It can be either gapped or gapless depending on point in multi-dimensional parametric space these phases. Special points this correspond setting some phase 0 and rest $\pi$. For nanostructure, three distinct spectra come together vicinity special point: two different topology separating by virtue topological protection. In paper, we show that weak interaction manifesting as quantum fluctuations drastically changes narrow point. We develop an model derive universal action describes situation. complicated incorporating non-local time matrix order parameter, its full analysis beyond scope present paper. Here, identify address limits: one one, concentrating first-order correction last case. both cases, find squeezes domain so tend contact each other immediately, defying domains strong coupling where perturbation theory does not work. phase, logarithmic divergence corrections. This leads us interesting hypothesis: might induce exponentially small gap formerly phase.