  We investigate a uniformly coupled non-Hermitian system with asymmetric
coupling amplitude. The asymmetric coupling equals to a symmetric coupling
threaded by an imaginary gauge field. In a closed configuration, the imaginary
gauge field leads to an imaginary magnetic flux, which induces a non-Hermitian
phase transition. For an open boundary, the imaginary gauge field results in an
eigenstate localization. The eigenstates under Dirac and biorthogonal norms and
the scaling laws are quantitatively investigated to show the affect of
asymmetric coupling induced one-way amplification. However, the imaginary
magnetic flux does not inevitably induce the non-Hermitian phase transition for
systems without translation invariance, this is elucidated from the
non-Hermitian phase transition in the non-Hermitian ring with a single coupling
defect. Our findings provide insights into the non-Hermitian phase transition
and one-way localization.
