Inhomogeneous superconductivity in comb-shaped Josephson junction networks

Abstract

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 Ic 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 an 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 behavior of 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 the ones 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.

Quantum spins on star graphs and the Kondo model

“…In general, spatial inhomogeneities may be random, due to the presence of disorder or noise, as well as non-random, as a result of an external control on the geometry of the system: in a broad sense, their formation can be dynamically generated or induced through a suitable engineering of the system. As a consequence the effects of spatial inhomogeneities have been investigated in a variety of systems, ranging from pattern formation in systems with competing interactions [1] to Josephson networks with non-random, yet non-translationally invariant architecture [2,3].…”

Spatially Inhomogeneous Superconducting and Bosonic Networks With Emergent Complex Behaviors

“…In this section we review how the pertinent choice of the network's connectivity leads to the emergence of new phenomena in quantum devices realized with superconducting JNNs [22] and cold atoms in optical lattices [30,32]. Our subsequent analysis well evidences that new emerging phenomena are possible only if the network's connectivity is described by an adjacency matrix supporting an hidden spectrum.…”

“…An hidden spectrum of the adjacency matrix emerges, for instance, when one analyzes bundled graphs [29] (i.e., those obtained by grafting a fiber graph to every point of a base graph) while, for graphs with constant coordination number (such as the Sierpinski gasket and the ladder graph), the adjacency matrix does not support any hidden spectrum [31]. In the following, we shall analyze the simple paradigmatic case of comb networks showing how the hidden spectrum of the adjacency matrix leads to unusual quantum behaviors such as the emergence of the spatial BEC on the comb's backbone for a Bose gas living on a combshaped optical lattice [30,32] and of the enhanced responses observed for classical combshaped JJNs made of Niobium grains [21,22][see Fig.1]. To better clarify our arguments, we find instructive to compare our results with those obtainable if the same devices were defined on a chain, since the latter is, after all, the simplest graph of euclidean dimension 1.…”

Exactly solvable tight-binding model on the RAN: fractal energy spectrum and Bose–Einstein condensation

“…Recent studies suggest the possibility that the network topology may act as a catalyst for Bose-Einstein condensation allowing condensation even if d ≤ 2. The presence of BEC has been, in fact, proven, for simple d ≤ 2 non-translationally invariant networks as comb, star and wheel lattices [8][9][10][11][12][13], as well as in complex networks as the Apollonian one [14][15][16][17].…”