  It has been known that the bulk-boundary correspondence (BBC) of the
non-Hermitian skin effect is characterized by the topology of the complex
eigenvalue spectra, while the topology of the wave function gives rise to
Hermitian BBC with conventional boundary modes. In this work, we go beyond the
known description of the non-Hermitian topological phase by discovering a new
type of BBC that appears in generalized boundary conditions. The generalized
Brillouin zone (GBZ) possesses non-trivial topological structures in the
intermediate boundary condition between open and periodic boundary conditions.
Unlike the conventional BBC, the topological phase transition is characterized
by the generalized momentum touching of GBZ, which manifests as exceptional
points. As a realization of our proposal, we suggest the non-reciprocal
Kuramoto oscillator lattice, where the phase slips accompany the exceptional
points as a signature of such topological phase transition. Our work
establishes an understanding of non-Hermitian topological matter by
complementing the non-Hermitian BBC as a general foundation of the
non-Hermitian topological systems.
