Experimental measurement of non-Hermitian left eigenvectors

Abstract

The duality of left and right eigenvectors underpins the comprehensive understanding of many physical phenomena. In Hermitian systems, left and right eigenvectors are simply Hermitian-conjugate pairs. In contrast, non-Hermitian eigenstates have left and right eigenvectors that are distinct from each other. However, despite the tremendous interest in non-Hermitian physics in recent years, the roles of non-Hermitian left eigenvectors (LEVs) are still inadequately explored. Their physical consequences and observable effects remain elusive, so much so that LEVs seem largely like objects of primarily mathematical purpose. In this study, we present a method based on the non-Hermitian Green’s function for directly retrieving both LEVs and right eigenvectors (REVs) from experimentally measured steady-state responses. We validate the effectiveness of this approach in two separate acoustic experiments: one characterizes the non-Hermitian Berry phase, and the other measures extended topological modes. Our results not only unambiguously demonstrate observable effects related to non-Hermitian LEVs but also highlight the under-appreciated role of LEVs in non-Hermitian phenomena.

Graphical abstract

Keywords

non-Hermitian systems / Green’s function / left eigenvectors / acoustic measurements / Berry phase / topological states

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Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

G. M. thanks Zhao-Qing Zhang and Henning Schomerus for discussions. X. W. thanks Junjie Lu for helpful discussions. This work was supported by the National Key R&D Program (No. 2022YFA1404400), the Hong Kong Research Grants Council (Nos. RFS2223-2S01 and 12301822), and the Hong Kong Baptist University (Nos. RC-RSRG/23-24/SCI/01 and RC-SFCRG/23-24/R2/SCI/12).

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