Point-gap bound states in higher-dimensional non-Hermitian systems

The non-Hermitian skin effect exerts a unique influence on impurity-induced bound states in one dimension, including the presence of a non-zero threshold of impurity strength and boundary-sensitive bound states. In higher dimensions, the behavior of skin modes becomes significantly intricate owing to the presence of geometry-dependent skin effects, leading to the emergence of novel characteristics associated with bound states.

In this work, we analyze the impurity-induced bound state in a 2D non-Hermitian system. We find the threshold of impurity strength required to generate a bound state decreases to zero in the 2D non-Hermitian system with geometry-dependent skin effect (GDSE) due to the existence of Bloch saddle points. The Newton Polygon of the two-dimensional amoeba directly determines the anisotropy in the localization length of bound states induced by large-strength impurities. Additionally, the connection between the central hole of the amoeba and the localization length of bound states induced by weak-strength impurity is established.