Non-Hermitian band theory of directional amplification

Owing to its open-system nature, amplification of electromagnetic and other signals is naturally described in terms of non-Hermitian operators. Recently, it has been shown that the non-Hermitian band theory is dramatically shaped by the non-Hermitian skin effect (NHSE), namely the exponential accumulation of eigenstates to the boundary of the system. In view of the NHSE, non-Bloch band theory based on the generalized Brillouin zone has been formulated to predict the topological edge modes. Here, we show that the non-Bloch band theory provides a natural framework for the intriguing phenomenon of directional amplification (or nonreciprocal amplification) that is useful in many applications. The magnitude of directional amplification is expressed in terms of the quantities of the generalized Brillouin zone. Our results provide a natural and quantitative theoretical formulation for the directional amplification.