Abstract
We study the characterization of the non-Hermitian skin effect (NHSE) in non-Hermitian systems with on-site disorders. We extend the applications of generalized Brillouin zone (GBZ) theory to these systems. By proposing a modified GBZ theory, we give a faithful description of the NHSE. For applications, we obtain a unified $\ensuremath{\beta}$ for system with long-range hopping and explain the GBZ irrelevance of the magnetic suppression of the NHSE in the previous study.
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