The dynamics of non-Hermitian quantum systems have taken on an increasing relevance in light of quantum devices which are not perfectly isolated from their environment. The interest in them also stems from their fundamental differences from their Hermitian counterparts, particularly with regard to their spectral and eigenvector correlations. These correlations form the fundamental building block for understanding the dynamics of quantum systems as all other correlations can be reconstructed from it. In this Letter, we study such correlations across a localization transition in non-Hermitian quantum systems. As a concrete setting, we consider non-Hermitian power-law banded random matrices which have emerged as a promising platform for studying localization in disordered, non-Hermitian systems. We show that eigenvector correlations show marked differences between the delocalized and localized phases. In the delocalized phase, the eigenvectors are strongly correlated as evinced by divergent correlations in the limit of vanishingly small complex eigenvalue spacings. On the contrary, in the localized phase, the correlations are independent of the eigenvalue spacings. We explain our results in the delocalized phase by appealing to the Ginibre random-matrix ensemble. On the other hand, in the localized phase, an analytical treatment sheds light on the suppressed correlations, relative to the delocalized phase. Given that eigenvector correlations are fundamental ingredients towards understanding real- and imaginary-time dynamics with non-Hermitian generators, our results open an avenue for characterizing dynamical phases in non-Hermitian quantum many-body systems.

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• Anderson localization
• Localization
• Many-body localization
• Disordered systems
• Non-Hermitian systems
• Exact diagonalization
• Perturbation theory
• Random matrix theory