PT -symmetric direct electrical transmission lines: Localization behavior

In this paper, we study a finite direct electrical transmission line when we distribute resistors Rn according to a parity-time (PT) distribution composed of a gain (-R) and loss (+R) sequence. Considering zero boundary conditions, we find analytical results for the frequency spectrum ω(R,kd) as a function of resistance R and the wave number kd. A frequency spectrum analysis shows a phase transition from real to complex eigenvalues as a function R for fixed kd=N, where 2N is the size of the transmission line. Numerically, we study localization properties through the normalized localization length Λ(R,kd). This measure shows good agreement with the analytical results and gives an account of the PT-phase transition. Our results pave a solid way toward studying the interplay between parity-time symmetry concepts and one-dimensional electrical transmission lines, aiming to find another generation of electronic devices capable of controlling the flow of energy.