Defective Edge States and Anomalous Bulk-Boundary Correspondence in non-Hermitian Topological Systems

Abstract: Non-Hermitian topological systems show quite different properties as their Hermitian counterparts. An important, puzzled issue on non-Hermitian topological systems is the existence of defective edge states beyond usual bulk-boundary correspondence (BBC) that localize either on the left edge or the right edge of the one-dimensional system. In this paper, to understand the existence of the defective edge states, the theory of anomalous bulk-boundary correspondence (A-BBC) is developed that distinguishes the non-Bloch bulk-boundary correspondence (NB-BBC) from non-Hermitian skin effect. By using the one-dimensional non-Hermitian Su-Schrieffer-Heeger model as an example, the underlying physics of defective edge states is explored. The defective edge states are physics consequence of boundary exceptional points of anomalous edge Hamiltonian. In addition, with the help of a theorem, the number anomaly of the edge states in non-Hermitian topological systems become a mathematic problem under quantitative calculations by identifying the Abelian/non-Abelian non-Hermitian condition for edge Hamiltonian and verifying the deviation of the BBC ratio from 1. In the future, the theory for A-BBC can be generalized to higher dimensional non-Hermitian topological systems (for example, 2D Chern insulator).