Magnetic-field tuned DC transport and complex ac conductance of a two-dimensional granular In/InOx composites, where the distribution of intergrain coupling was continuously tuned, were performed to explore the transition from a robust superconductor to an anomalous metal and then an insulating state. Resistivity saturation is found on both superconducting and insulating sides of an “avoided” magnetic-field-tuned superconductor-to-insulator transition (H-SIT) in a two-dimensional In/InOx composite, where the anomalous metallic behaviors cut off conductivity or resistivity divergence in the zero-temperature limit. The granular morphology of the material implies a system of Josephson junctions (JJ) with a broad distribution of Josephson coupling E_J and charging energy E_C, with a H-SIT determined by the competition between E_J and E_C. By virtue of self-duality across the true H-SIT, we invoke macroscopic quantum tunneling effects to explain the temperature-independent resistance where the “failed superconductor” side is a consequence of phase fluctuations and the “failed insulator” side results from charge fluctuations. While true self-duality is lost in the avoided transition, its vestiges are argued to persist, owing to the incipient duality of the percolative nature of the dissipative path in the underlying random JJ system [1].

Further tuning of the inter-grain coupling to eliminate the anomalous metallic phase, the complex ac conductance revealed a T→0 saturating dissipative response in a finite magnetic field, coexisting with a robust superfluid density. The anomalous power- law spectra for the dissipative response appear to indicate quantum critical behavior proximate to a quantum superconductor to anomalous-metal transition as probed in the kilo-Hertz range, and point to signatures of gapless superconductivity in this granular superconducting system [2].

[1] Xinyang Zhang, Alexander Palevski, Aharon Kapitulnik, “Anomalous metals: from “failed superconductor” to “failed insulator”,” PNAS 119, e2202496119 (2022).

[2] Xinyang Zhang, Jinze Wu, Alexander Palevski, Aharon Kapitulnik, “Gapless superconductivity in the low-frequency electrodynamic response of two-dimensional granular In/InOx composites,” Phys. Rev. Research (2025). Phys. Rev. Research 7, 013183 (2025).