  Unlike a Chern number in $2$D and $3$D topological system, Zak phase takes a
subtle role to characterize the topological phase in $1$D. On the one hand, it
is not a gauge invariant, on the other hand, the Zak phase difference between
two quantum phases can be used to identify the topological phase transitions. A
non-Hermitian system may inherit some characters of a Hermitian system, such as
entirely real spectrum, unitary evolution, topological energy band, etc. In
this paper, we study the influence of non-Hermitian term on the Zak phase for a
class of non-Hermitian systems. We show exactly that the real part of the Zak
phase remains unchanged in a bipartite lattice. In a concrete example, $1$D
Su-Schrieffer-Heeger (SSH) model, we find that the real part of Zak phase can
be obtained by an adiabatic process. To demonstrate this finding, we
investigate a scattering problem for a time-dependent scattering center, which
is a magnetic-flux-driven non-Hermitian SSH ring. Owing to the nature of the
Zak phase, the intriguing features of this design are the wave-vector
independence and allow two distinct behaviors, perfect transmission or
confinement, depending on the timing of a flux impulse threading the ring. When
the flux is added during a wavepacket travelling within the ring, the
wavepacket is confined in the scatter partially. Otherwise, it exhibits perfect
transmission through the scatter. Our finding extends the understanding and
broaden the possible application of geometric phase in a non-Hermitian system.

Quantum Physics