Inhomogeneous superconductivity in comb-shaped Josephson Junction Networks

We show that some of the Josephson couplings of junctions arranged to form an inhomogeneous network undergo a non-perturbative renormalization provided that the network's connectivity is pertinently chosen. As a result, the zero-voltage Josephson critical currents I turn out to be enhanced along directions selected by the network's topology. This renormalization effect is possible only on graphs whose adjacency matrix admits a hidden spectrum (i.e. a set of localized states disappearing in the thermodynamic limit). We provide a theoretical and experimental study of this effect by comparing the superconducting behaviour of c a comb-shaped Josephson junction network and a linear chain made with the same junctions: we show that the Josephson critical currents of the junctions located on the comb's backbone are bigger than those of the junctions located on the chain. Our theoretical analysis, based on a discrete version of the Bogoliubov-de Gennes equation, leads to results which are in good quantitative agreement with experimental results.