Anomalous fractal scaling in two-dimensional electric networks

TL;DR

This paper uncovers a surprising fractal scaling anomaly in the impedance of 2D electric networks, revealing complex resonance behaviors that deviate from classical continuum predictions and are robust against perturbations.

Contribution

It demonstrates the existence of anomalous fractal impedance scaling in 2D LC networks, linking it to a generalized resonance condition similar to Harper's equation, and extends the phenomenon to generic network transport.

Findings

Impedance exhibits erratic, fractal-like peaks with system size N.

Anomalous impedance is robust against component perturbations.

The phenomenon relates to a generalized resonance condition akin to Harper's equation.

Abstract

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 the 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 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.