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The dynamics of non-Hermitian quantum systems have taken on an increasingrelevance in light of quantum devices which are not perfectly isolated fromtheir environment. The interest in them also stems from their fundamentaldifferences from their Hermitian counterparts, particularly with regard totheir spectral and eigenvector correlations. These correlations form thefundamental building block for understanding the dynamics of quantum systems asall other correlations can be reconstructed from it. In this work, we studysuch correlations across a localisation transition in non-Hermitian quantumsystems. As a concrete setting, we consider non-Hermitian power-law bandedrandom matrices which have emerged as a promising platform for studyinglocalisation in disordered, non-Hermitian systems. We show that eigenvectorcorrelations show marked differences between the delocalised and localisedphases. In the delocalised phase, the eigenvectors are strongly correlated asevinced by divergent correlations in the limit of vanishingly small complexeigenvalue spacings. On the contrary, in the localised phase, the correlationsare independent of the eigenvalue spacings. We explain our results in thedelocalised phase by appealing to the Ginibre random matrix ensemble. On theother hand, in the localised phase, an analytical treatment sheds light on thesuppressed correlations, relative to the delocalised phase. Given thateigenvector correlations are fundamental ingredients towards understandingreal- and imaginary-time dynamics with non-Hermitian generators, our resultsopen a new avenue for characterising dynamical phases in non-Hermitian quantummany-body systems.

Eigenvector Correlations Across the Localisation Transition in non-Hermitian Power-Law Banded Random Matrices