Geometric origin of self-intersection points in non-Hermitian energy spectra

Unlike Hermitian systems, non-Hermitian energy spectra under periodic boundary conditions can form closed loops in the complex energy plane, a phenomenon known as the point-gap topology. In this paper, we investigate the self-intersection points of such non-Hermitian energy spectra and reveal their geometric origins. We rigorously demonstrate that these self-intersection points result from the intersection of the auxiliary generalized Brillouin zone and the Brillouin zone in one-band systems, as confirmed by an extended Hatano-Nelson model. This finding is further generalized to multiband systems, illustrated through a non-Hermitian Su-Schrieffer-Heeger model. Moreover, we address multiple self-intersection points and derive the geometric conditions for general $n$-fold self-intersection points. Our results enhance the fundamental understanding of generic non-Hermitian quantum systems and provide theoretical support for further experimental investigations of energy self-intersection points.