Anomalous fractal scaling in two-dimensional electric networks

Much of the qualitative nature of physical systems can be predicted from the way it scales with system size. Contrary to the continuum expectation, we observe a profound deviation from logarithmic scaling in the impedance of a two-dimensional LC circuit network. We find this anomalous impedance contribution to sensitively depend on the number of nodes N in a curious erratic manner and experimentally demonstrate its robustness against perturbations from the contact and parasitic impedance of individual components. This impedance anomaly is traced back to a generalized resonance condition reminiscent of Harper's equation for electronic lattice transport in a magnetic field, even though our circuit network does not involve magnetic translation symmetry. It exhibits an emergent fractal parametric structure of anomalous impedance peaks for different N that cannot be reconciled with a continuum theory and does not correspond to regular waveguide resonant behavior. This anomalous fractal scaling extends to the transport properties of generic systems described by a network Laplacian whenever a resonance frequency scale is simultaneously present. Scaling theory governs much of the physics around us. Here, the authors present experimental evidence showcasing the breakdown of logarithmic scaling in electric circuits and the emergence of fractal-like patterns in impedance scaling relationships.

Automated summary

The qualitative nature of physical systems can be predicted from their scaling relationship with system size. In a two-dimensional LC circuit network, there is a profound deviation from logarithmic scaling in impedance, which depends sensitively on the number of nodes N and is robust against perturbations. This anomalous impedance behavior is due to a generalized resonance condition and exhibits a fractal-like structure of impedance peaks for different N. It is not explained by continuum theory or regular waveguide resonant behavior.