Defective edge states and number-anomalous bulk-boundary correspondence in non-Hermitian topological systems

Non-Hermitian topological systems exhibit properties very different from those of their Hermitian counterparts. An important puzzling issue for non-Hermitian topological systems is the existence of defective edge states beyond the usual bulk-boundary correspondence (BBC). In this Rapid Communication, to understand the existence of these defective edge states, the number-anomalous bulk-boundary correspondence (NA-BBC) theory, which distinguishes the non-Bloch BBC, is developed. With the one-dimensional non-Hermitian Su-Schrieffer-Heeger model taken as an example, the underlying physics of the defective edge states is explored. The defective edge states are a consequence of non-Hermitian coalescence from the anomalous edge Hamiltonian. In addition, with the help of a theorem, the number anomaly of the edge states in non-Hermitian topological systems becomes a mathematical problem in quantitative calculations when identifying the normal/non-normal non-Hermitian condition for the edge Hamiltonian and verifying the deviation of the BBC ratio from 1. In the future, NA-BBC theory can be generalized to higher-dimensional non-Hermitian topological systems (for example, the two-dimensional Chern insulator).