Impurity-induced counter skin-effect and linear modes in the Hatano–Nelson model

Non-reciprocal lattice systems are among the simplest non-Hermitian systems, exhibiting several key features absent in their Hermitian counterparts. In this study, we investigate the Hatano–Nelson model with impurity and unveil how the impurity influences the intrinsic non-Hermitian skin effect of the system. We present an exact analytical solution to the problem under open and periodic boundary conditions, irrespective of the impurity’s position and strength. Numerical simulations thoroughly validate this exact solution. Our analysis reveals a distinctive phenomenon where a specific impurity strength, determined by the non-reciprocal hopping parameters, induces a unique skin state at the impurity site. This impurity state exhibits a skin effect that counterbalances the boundary-induced skin effect, a phenomenon we term the impurity-induced counter skin-effect. These findings offer insights into the dynamics of non-Hermitian systems with impurities, elucidating the complex interplay between impurities and the system’s non-reciprocal nature. We propose a possible implementation of this system for a non-Hermitian discrete-time quantum walk, and we demonstrate that an impurity-induced counter skin-effect also exists in multi-band models.