Detecting bulk and edge exceptional points in non-Hermitian systems through generalized Petermann factors

Abstract

Non-orthogonality in non-Hermitian quantum systems gives rise to tremendous exotic quantum phenomena, which can be fundamentally traced back to non-unitarity. In this paper, we introduce an interesting quantity (denoted as \eta) as a new variant of the Petermann factor to directly and efficiently measure non-unitarity and the associated non-Hermitian physics. By tuning the model parameters of underlying non-Hermitian systems, we find that the discontinuity of both \eta and its first-order derivative (denoted as \mathrm{\partial }\eta) pronouncedly captures rich physics that is fundamentally caused by non-unitarity. More concretely, in the 1D non-Hermitian topological systems, two mutually orthogonal edge states that are respectively localized on two boundaries become non-orthogonal in the vicinity of discontinuity of \eta as a function of the model parameter, which is dubbed “edge state transition”. Through theoretical analysis, we identify that the appearance of edge state transition indicates the existence of exceptional points (EPs) in topological edge states. Regarding the discontinuity of \mathrm{\partial }\eta, we investigate a two-level non-Hermitian model and establish a connection between the points of discontinuity of \mathrm{\partial }\eta and EPs of bulk states. By studying this connection in more general lattice models, we find that some models have discontinuity of \mathrm{\partial }\eta, implying the existence of EPs in bulk states.

Graphical abstract

Keywords

non-Hermitian / Su−Schrieffer−Heeger (SSH) model / exceptional point

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Declarations

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

Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC) Grant No. 12074438, the Guangdong Basic and Applied Basic Research Foundation under Grant No. 2020B1515120100, the Open Project of Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices under Grant No. 2022B1212010008, and the Fundamental Research Funds for the Central Universities, Sun Yat-sen University (No. 23ptpy05).

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