Fig. 1: Origin of logarithmic impedance scaling in continuum media and its violation in lattices.

a In a continuous sample such as a square plate of length N and resistivity ρ, the diagonal-to-diagonal impedance necessarily scales like \(\rho \log N\). This is easily seen by slicing the sample into strips perpendicular to the diagonal and noticing that each strip approximately contributes a serial impedance that is inversely proportional to its width. This is because each successive “shell” in the sample scales with its linear dimension l as l^D−1 = l, such that the total impedance scales like \({\int}^{N}{l}^{-1}dl \sim \log N\). b Behavior of \(\beta =-\frac{d\log | Z| }{d\log N}\), the fractional rate of change of impedance Z diagonally with the system size N, across circuit lattices with and without an AC frequency scale. While a purely resistive 2D circuit (blue) exhibits a smoothly vanishing β consistent with the continuum approximation in (a), our 2D LC circuit (red) with an illustrative frequency scale of ω_r = 1.95 exhibits anomalous scaling behavior with pronounced and erratically located peaks. The dashed lines represent the constant or saturated scaling of other dimensions derived from β of scaling theory applicable to non-resonant reactive media.