Partial topological Zak phase and dynamical confinement in a non-Hermitian bipartite system

Unlike a Chern number in a two-dimensional (2D) and 3D topological system, the Zak phase takes a subtle role to characterize the topological phase in 1D. On the one hand, it is not gauge invariant, and, 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 a non-Hermitian term on the Zak phase. We show exactly that the real part of the Zak phase remains unchanged in a class of 1D bipartite lattice systems even in the presence of the on-site imaginary potential. The non-Hermitian term only gives rise to the imaginary part of Zak phase. Such a complex quantity has a physical implication when we consider the dynamical realization in which the Zak phase can be obtained through adiabatic evolution. In this context, its imaginary part represents the amplification and/or attenuation of the Dirac norm of the evolved state. Based on this finding, we investigate a scattering problem for a time-dependent scattering center, which is a magnetic-flux-driven non-Hermitian Su-Schrieffer-Heeger ring. Due to the topological 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 wave packet travelling within the ring, the wave packet is confined in the scatter partially. Otherwise, it exhibits perfect transmission through the scatter. Our finding extends the understanding and broadens the possible application of the Zak phase in a non-Hermitian system.