Fate of magnetic impurity induced states in a non-Hermitian s-wave superconductor

Recently, the non-Hermitian system has attracted great attention. However, the interplay between the non-Hermiticity and impurities in superconductors remains to be explored. In this work, we investigate the magnetic impurity chain on a three-dimensional non-Hermitian $s$-wave superconductor with the imbalanced pairing term. For the Yu-Shiba-Rusinov (YSR)-type state, we find that their fate is determined by the generalized in-gap nature of these states: These in-gap states and related topological phases survive only when the effective superconducting gap exists. With the increasing strength of the non-Hermitian part of the pairing term, the host superconductor is lead to a non-Hermitian gapless phase with exceptional surfaces, while these in-gap states and topological phases disappear. Our work generalizes the in-gap state nature of the YSR state to the non-Hermitian superconductor and reveals the relation between the exceptional surfaces and the YSR-type state, which provides a minimal starting point to explore the interplay between the non-Hermiticity and magnetic impurities in the $s$-wave superconductor.