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This paper introduces scalar measures that adequately characterize the non-Hermitian skin effects under open boundary condition. Using these measures, the authors reveal that the topological properties of the non-Hermitian skin effect give a macroscopic enhancement of non-normality under open boundary condition. The topological enhancement of non-normality governs the perturbation sensitivity of the spectra and the anomalous time-evolution dynamics intrinsic to non-Hermiticity. They also show that these measures correctly describe the disorder-induced topological phase transitions of the skin effect.
Nakai et al. (2024) studied this question.