  Non-Hermitian systems as theoretical models of open or dissipative systems
exhibit rich novel physical properties and fundamental issues in condensed
matter physics.We propose a generalized local-global correspondence between the
pseudo-boundary states in the complex energy plane and topological invariants
of quantum states. We find that the patterns of the pseudo-boundary states in
the complex energy plane mapped to the Brillouin zone are topological
invariants against the parameter deformation. We demonstrate this approach by
the non-Hermitian Chern insulator model. We give the consistent topological
phases obtained from the Chern number and vorticity. We also find some novel
topological invariants embedded in the topological phases of the Chern
insulator model, which enrich the phase diagram of the non-Hermitian Chern
insulators model beyond that predicted by the Chern number and vorticity. We
also propose a generalized vorticity and its flipping index to understand
physics behind this novel local-global correspondence and discuss the
relationships between the local-global correspondence and the Chern number as
well as the transformation between the Brillouin zone and the complex energy
plane. These novel approaches provide insights to how topological invariants
may be obtained from local information as well as the global property of
quantum states, which is expected to be applicable in more generic
non-Hermitian systems.
