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The breakdown of the bulk-boundary correspondence in non-Hermitian (NH)topological systems is an open, controversial issue. In this paper, to resolvethis issue, we ask the following question: Can a (global) topological invariantcompletely describe the topological properties of a NH system as its Hermitiancounterpart? Our answer is no. One cannot use a global topological invariant(including non-Bloch topological invariant) to accurately characterize thetopological properties of the NH systems. Instead, there exist a new type oftopological invariants that are absence in its Hermitian counterpart -- thestate dependent topological invariants. With the help of the state-dependenttopological invariants, we develop a new topological theory for NH topologicalsystem beyond the general knowledge for usual Hermitian systems and obtain anexact formulation of the bulk-boundary correspondence, includingstate-dependent phase diagram, state-dependent phase transition and anomaloustransport properties (spontaneous topological current). Therefore, theseresults will help people to understand the exotic topological properties ofvarious non-Hermitian systems.

State-dependent Topological Invariants and Anomalous Bulk-Boundary Correspondence in non-Hermitian Topological Systems