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

Non-orthogonality in non-Hermitian quantum systems gives rise to tremendous exotic quantum phenomena, which can be fundamentally traced back to non-unitarity and is much more fundamental and universal than complex energy spectrum. 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 $\partial \eta$) 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 $\partial\eta$, we investigate a two-level non-Hermitian model and establish a connection between the points of discontinuity of $\partial \eta$ and EPs of bulk states. By studying this connection in more general lattice models, we find that some models have discontinuity of $\partial\eta$, implying the existence of EPs in bulk states.