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Interdependent superconducting networks

Abstract

Cascades are self-amplifying processes triggered by feedback mechanisms that may cause a substantial part of a macroscopic system to change its phase in response to a relatively small local event. The theoretical background for these phenomena is rich and interdisciplinary, with interdependent networks providing a versatile framework to study their multiscale evolution. However, laboratory experiments aimed at validating this ever-growing volume of predictions have not been accomplished, mostly because of the lack of a physical mechanism that realizes interdependent couplings. Here we demonstrate an experimental realization of an interdependent system as a multilayer network of two disordered superconductors separated by an electric insulating film. We show that Joule heating effects due to large driving currents act as dependency links between the superconducting layers, igniting overheating cascades via adaptive and heterogeneous back-and-forth electrothermal feedback. We characterize the phase diagram of mutual superconductive transitions and spontaneous microscopic critical processes that physically realize interdependent percolation and generalize it beyond structural dismantling. This work establishes a laboratory manifestation of the theory of interdependent systems, enabling experimental studies to control and to further develop the multiscale phenomena of complex interdependent materials.

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Fig. 1: Design and experimental setup of thermally interdependent superconducting networks.
Fig. 2: Thermally interdependent networks of RSJJs.
Fig. 3: Microscopic kinetics and their lifetime at the first-order transition thresholds.
Fig. 4: Mutual phase diagram showing theory versus experiments.

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Data availability

Source data are provided with this paper. All data supporting our findings are available from the corresponding author upon reasonable request.

Code availability

Source codes and videos showing the states of resistors, their currents and the power dissipated in both layers during the transition can be freely accessed at the GitHub repository: https://github.com/BnayaGross/Interdependent-SC-networks.

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Acknowledgements

S.H. acknowledges financial support from the Israel Science Foundation (ISF), the China–Israel Science Foundation, the Office of Naval Research (ONR), the Bar-Ilan University Center for Research in Applied Cryptography and Cyber Security, the EU project RISE, the US-Israel Binational Science Foundation (NSF-BSF) grant no. 2019740 and the Defense Threat Reduction Agency (DTRA) grant no. HDTRA-1-19-1-0016. I.B., A.F. and S.H. acknowledge partial support from the Italy-Israel grant ‘EXPLICS’. A.F. acknowledges partial support from the ISF Israel–China grant no. 3192/19. B.G. acknowledges the support of the Mordecai and Monique Katz Graduate Fellowship Program. We thank A. Bashan, M. M. Danziger and S. Boccaletti for stimulating discussions.

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Contributions

I.B., A.F. and S.H. initiated and designed the research. M.L., I.V. and A.F. fabricated the samples, carried on the experiments and collected the data. I.B. and B.G. developed the modelling and the adaptive algorithm for solving the thermally coupled Kirchhoff equations. B.G. designed the codes. B.G. and I.B. carried out the numerical simulations. I.B. designed and created the figures. I.B. developed the mean-field theory. I.B. was the leading writer of the paper with contributions from B.G., S.H. and A.F. A.F. and S.H. supervised the research. All authors critically reviewed and approved the paper.

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Correspondence to I. Bonamassa.

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Nature Physics thanks Juan Rocha, Simon Levin and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–8.

Supplementary Video 1

Evolution of the current flow during the metastable plateau stage both for the heating and cooling directions.

Supplementary Video 2

Evolution of the power dissipated during the metastable plateau stage both for the heating and cooling directions.

Source data

Source Data Fig. 1

Data plotted in Fig. 1c,f.

Source Data Fig. 2

Data plotted in Fig. 2c,d.

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Bonamassa, I., Gross, B., Laav, M. et al. Interdependent superconducting networks. Nat. Phys. 19, 1163–1170 (2023). https://doi.org/10.1038/s41567-023-02029-z

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