  Non-Hermitian systems can exhibit extraordinary sensitivity to boundary
conditions. Given that topological boundary modes and non-Hermitian skin
effects can either coexist or individually appear in non-Hermitian systems, it
is of great value to present a comprehensive non-Hermitian phase diagram, for
further flexible control in realistic non-Hermitian systems. Here, we reveal
four-fold non-Hermitian phase transitions at a mathematically level, where
phase I exhibits only topological boundary modes, phase II displays both
topological boundary modes and skin modes, phase III exhibits only skin modes,
and phase IV cannot manifest any boundary modes. By deriving non-Hermitian
winding numbers, the existence or non-existence condition of topological
boundary modes are analytically expressed, consistent with the numerical
results obtained through the iterative Green's function method. Combining with
the study on non-Hermitian skin effects, we rigorously establish the four-fold
phase diagram. We also design an array of coupled resonator optical waveguides.
The introduction of non-Hermiticity in the photonic structure induces a
phenomenon similar to band inversion in topological insulators, indicating the
presence of topological boundary modes in the photonic bands.
