Exploring the interplay between fractionality and PT symmetry in magnetic metamaterials

We study a nonlinear magnetic metamaterial modeled as a split-ring resonator array, where the standard discrete laplacian is replaced by its fractional form. We find a closed-form expression for the dispersion relation as a function of the fractional exponent s and the gain/loss parameter {\gamma} and examine the conditions under which stable magneto-inductive waves exist. The density of states is computed in closed form and suggests that the main effect of fractionality is the flattening of the bands, while gain/loss increase tends to reduce the bandgaps. The spatial extent of the modes for a finite array is computed by means of the participation ratio R, which is also obtained in closed form. For a fixed fractionality exponent, an increase in gain/loss {\gamma} decreases the overall R, from the number of sites N towards N/2 at large {\gamma}. The nonlinear dynamics of the average magnetic energy on an initial ring during a cycle shows a monotonic increase with {\gamma}, and it is qualitatively similar for all fractional exponents. This is explained as mainly due to the interplay of nonlinearity and PT symmetry.