Stable directional amplification in one-dimensional non-Hermitian single-band systems

We investigate the directional amplification of non-Hermitian single-band systems in one dimension by using two models. The first model is the Hatano-Nelson model with added cavity decay and the second model is a linear lattice with fully asymmetric next-nearest-neighbor hopping. Based on the Dyson equation, we calculate the Green's functions of the two models analytically and investigate the directional amplification of a weak probe field. We find that, although the periodic-boundary spectrum of the two models determines the frequency interval in which the probe field can be directionally amplified, the open-boundary spectrum influences not only the bandwidth of the amplification curve but also the shape of the transmission curves. The bandwidth and the gain can be tuned independently by the parameters of the system. We also find that, for lattices with finite size, the frequency interval of the directional amplification shrinks due to the finite-size effects. Our thorough analysis is expected to provide a deeper insight into the directional amplification in non-Hermitian single-band systems and a tool for designing directional amplifiers and sensors.