Topology of pseudo-Hermitian line of multiband non-Hermitian systems

Topology in non-Hermitian systems arises not only from Berry phases but also from intrinsic properties unique to non-Hermitian systems, such as complex eigenvalues and non-orthogonal eigenstates. These properties lead to phenomena like the non-Hermitian skin effect and exceptional topology in non-separable bands. We propose a permutation group approach to describe state exchanges induced by exceptional points in two-dimensional spaces. Additionally, we show that the topology of non-separable bands must be reconsidered when addressing the nontriviality of two-dimensional parameter spaces. Furthermore, we highlight the role of pseudo-Hermitian lines in multiband systems, revealing that non-separable bands can exhibit topological non-triviality even without exceptional points in 2D spaces.