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

Non-Hermitian phase transition and eigenstate localization induced by asymmetric coupling