PT-symmetric direct electrical transmission lines: Localization behavior.

In this paper, we study a finite direct electrical transmission line when we distribute resistors R_{n} 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,k_{d}) as a function of resistance R and the wave number k_{d}. A frequency spectrum analysis shows a phase transition from real to complex eigenvalues as a function R for fixed k_{d=N}, where 2N is the size of the transmission line. Numerically, we study localization properties through the normalized localization length Λ(R,k_{d}). 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.