Fractionality and PT-symmetry in an electrical transmission line

We examine the stability of a 1D electrical transmission line in the simultaneous presence of PT-symmetry and fractionality. The array contains a binary gain/loss distribution $\gamma_{n}$ and a fractional Laplacian characterized by a fractional exponent $\alpha$. For an infinite periodic chain, the spectrum is computed in closed form, and its imaginary sector is examined to determine the stable/unstable regions as a function of the gain/loss strength and fractional exponent. In contrast to the non-fractional case where all eigenvalues are complex for any gain/loss, here we observe that a stable region can exist when gain/loss is small, and the fractional exponent is below a critical value, $0 < \alpha < \alpha_{c1}$ . As the fractional exponent is decreased further, the spectrum acquires a gap with two nearly-flat bands. We also examined numerically the case of a finite chain of size N. Contrary to what happens in the infinite chain, here the stable region always lies above a critical value $\alpha_{c2} < \alpha < 1$. An increase in gain/loss or $N$ always reduces the width of this stable region until it disappears completely.