The non-Hermitian skin effect is a nonreciprocity-induced localization phenomenon in which a macroscopic number of eigenstates accumulate anomalously at the boundary, accompanied by the extreme sensitivity to boundary conditions. Here, we develop a geometric characterization of the non-Hermitian skin effect. We demonstrate that the localization length scale associated with the skin effect is encoded in the quantum metric defined solely from right eigenstates, but not in the biorthogonal quantum metric. Moreover, we show that quantum metrics can exhibit power-law divergences at gapless points that depend on different boundary conditions. We also reveal that cusps of the generalized Brillouin zone in non-Bloch band theory are signaled by discontinuities in the quantum metrics. We illustrate these behaviors using prototypical non-Hermitian models, such as the Hatano-Nelson model and the non-Hermitian, nonreciprocal Su-Schrieffer-Heeger model.

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• Geometric & topological phases
• Open quantum systems & decoherence
• Phase transitions
• Skin effect
• Non-Hermitian systems
• Nonequilibrium systems
• Su-Schrieffer-Heeger model