Deterministic bulk-boundary correspondences for skin and edge modes in a general two-band non-Hermitian system

In isolated Hermitian systems, the bulk-boundary correspondence is generally thought of as a compelling principle guiding the occurrence of robust edge states through the topological invariants of the bulk. However, in open non-Hermitian systems that support non-Hermitian skin effects, the universality of the bulk-boundary correspondence has so far remained disputable in spite of being subjected to intensive studies. Here we provide analytical and numerical evidence of bulk-boundary correspondences for both skin and edge modes in 1D two-band non-Hermitian systems within the framework of a prototypical non-Hermitian Su-Schrieffer-Heeger model. Two different topological winding numbers, one on a Brillouin zone and the other on a generalized Brillouin zone, are defined to fully characterize the topological phases of matter and to understand such deterministic correspondences. In addition to the exact solutions for bulk states, we also obtain explicit solution forms for edge modes along with analytical conditions for their existence and localization, all exhibiting remarkable agreement with numerical results. Our results solve the elusive non-Hermitian topology and may facilitate experimental investigations of bulk-boundary correspondence in a wide range of non-Hermitian systems.