Spontaneous PT-symmetry breakdown in superconducting weak links

We formulate a description of transport in a superconducting weak link in terms of the non-Hermitian quantum mechanics. We find that the applied electric field exceeding a certain critical value changes the topological structure of the effective non-Hermitian Hamiltonian of the weak link in the Hilbert space causing the parity reflection -- time reversal symmetry (PT-symmetry) breakdown. We derive the expression of the critical electric field and show that the PT-symmetry breakdown gives rise to the switching instability in the current-voltage characteristic of the weak link. Taking into account superconducting fluctuations we quantitatively describe the experimentally observed differential resistance of the weak link in the vicinity of the critical temperature.