  Asymmetric coupling amplitudes effectively create an imaginary gauge field,
which induces a non-Hermitian Aharonov-Bohm (AB) effect. Nonzero imaginary
magnetic flux invalidates the bulk-boundary correspondence and leads to a
topological phase transition. However, the way of non-Hermiticity appearance
may alter the system topology. By alternatively introducing the non-Hermiticity
under symmetry to prevent nonzero imaginary magnetic flux, the bulk-boundary
correspondence recovers and every bulk state becomes extended; the bulk
topology of Bloch Hamiltonian predicts the (non)existence of edge states and
topological phase transition. These are elucidated in a non-Hermitian
Su-Schrieffer-Heeger model, where chiral-inversion symmetry ensures the
vanishing of imaginary magnetic flux. The average value of Pauli matrices under
the eigenstate of chiral-inversion symmetric Bloch Hamiltonian defines a vector
field, the vorticity of topological defects in the vector field is a
topological invariant. Our findings are applicable in other non-Hermitian
systems. We first uncover the roles played by non-Hermitian AB effect and
chiral-inversion symmetry for the breakdown and recovery of bulk-boundary
correspondence, and develop new insights for understanding non-Hermitian
topological phases of matter.
