Physical Exceptional Points without Degeneracy of Energy Levels.

Exceptional Point (EP) is an interesting physical phenomenon in non-Hermitian physics, at which for a singular matrix the eigenvalues and the eigenvectors coalesce. In this paper, we generalize the concept of usual EPs to that of physical EPs, an issue in certain protected subsystems, for example, the defective edge states of non-Hermitian topological insulator, or the defective degenerate ground states in non-Hermitian systems with spontaneously symmetry breaking, et.al. For these physical EPs, the coalescence of eigenvectors may occur without requiring the eigenvalues degeneracy. In addition, for these physical EPs, more subtle structures are explored, including basis defectiveness, non-Hermitian weight vectors, and hidden quantum phase transitions. By taking the topologically protected edge states in non-Hermitian Su-Schrieffer-Heeger model as an example, we show the physical properties of different types of physical EPs.