Exact solution of single impurity problem in non-reciprocal lattices: impurity induced size-dependent non-Hermitian skin effect

TL;DR

This paper provides exact solutions for single impurity effects in non-Hermitian non-reciprocal lattices, revealing size-dependent transitions from non-skin to skin states and spectrum changes, and clarifying impurity-induced topological transitions.

Contribution

It offers an exact analytical approach to the impurity problem in non-Hermitian models, highlighting size-dependent impurity strength effects on spectral and localization transitions.

Findings

Impurity strength induces transition from non-skin to skin states.

Spectrum changes from ellipse to real segment with impurity increase.

Critical impurity strength scales exponentially with system size.

Abstract

Non-Hermitian non-reciprocal systems are known to be extremely sensitive to boundary conditions, exhibiting diverse localizing behaviors and spectrum structures when translational invariance is locally broken, either by tuning the boundary coupling strength, or by introducing an effective boundary using impurities or defects. In this work, we consider the single impurity problem in the Hatano-Nelson model and the Su-Schreieffer-Heeger model, which can be exactly solved with the single impurity being treated as an effective boundary of the system. From our exact solutions for finite-size systems, we unveil that increasing the impurity strength can lead to a transition of the bulk states from non-skin states to skin states, accompanied by the change of the spectrum structure from an ellipse in the complex plane to a segment along the real axis. These exact results indicate that the critical value of impurity strength is size-dependent, and increases exponentially with the lattice size when the impurity is strong or the system is large enough. OUr exact solutions are also useful for determining the point-gap topological transition in the concerned models.