$\mathcal{PT}$-symmetric tight-binding model with asymmetric couplings

Abstract: We study spectral and transport properties of one-dimensional tight-binding $\mathcal{PT}$-symmetric chains with alternating couplings. Based on the transfer matrix method, we have analytically developed the expressions for the transmission and reflection coefficients for any values of control parameters. These expressions are obtained in a very compact form which separately imbed the generic energy dependence valid for any periodic structure, as well as specific properties of a unit cell composing the scattering setup. Out main interest is in specific properties of the left/right reflections that are due to the $\mathcal{PT}$ symmetric structure of the model. We have found that for the case of asymmetric couplings between dimers, a new type of specific points emerge in the spectrum, which are responsible for quite specific properties of the unidirectional reflectivity.