Josephson phase transition influenced by the presence of quantum impurities in the superconductors

By considering quantum impurities in the superconductors, we mimic their influences on the phase transition in a Josephson junction. It is found that the impurities in the superconductors can take nontrivial effects to the Josephson phase transition, and they are able to promote or hinder the 0 → π phase-transition phenomenon. Namely, for the 0 ( π ) junction with one impurity, it has an opportunity to become a π (0) junction, following the adjustment of the impurity-superconductor coupling. However, when each superconductor has an embedded impurity, their effect on the phase transition tends to become weak. We believe that this work can be helpful in understanding the contribution of impurities to the Josephson phase transition.