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Universal power-law decay of electron-electron interactions due to nonlinear screening in a Josephson junction array

Josephson junctions are the most prominent nondissipative and at the same time nonlinear elements in superconducting circuits allowing Cooper pairs to tunnel coherently between two superconductors separated by a tunneling barrier. Due to this, physical systems involving Josephson junctions show highly complex behavior and interesting novel phenomena. Here, we consider an infinite one-dimensional chain of superconducting islands where neighboring islands are coupled by capacitances. We study the effect of Josephson junctions shunting each island to a common ground superconductor. We treat the system in the regime where the Josephson energy exceeds the capacitive coupling between the islands. For the case of two offset charges on two distinct islands, we calculate the interaction energy of these charges mediated by quantum phase slips due to the Josephson nonlinearities. We treat the phase slips in an instanton approximation and map the problem onto a classical partition function of interacting particles. Using the Mayer cluster expansion, we find that the interaction potential of the offset charges decays with an universal inverse-square power law behavior.