  In this work, the interplay between non-Hermiticity, quasi-disorder, and
repulsive interaction is studied for hard-core bosons confined in a
one-dimensional optical lattice, where non-Hermiticity is induced by the
non-reciprocal hoppings and the on-site gain and loss breaking the
time-reversal symmetry. Although the energy spectra of the static system are
fully complex, with the evolution of the initial state, the real part of the
expectation value of the Hamiltonian under the time-evolved wave function
changes stably. By means of the entanglement entropy and its dynamical
evolution, as well as the inverse participation ratio, the many-body
localization (MBL) is found to play the key role in the stability of the
dynamical behavior of the real part of the expectation value, independent of
whether the spectrum of the static Hamiltonian is real or complex. In the
delocalization phase, the dynamical evolution of the real part of the
expectation value is unstable. Meanwhile, the nearest-neighbor level spacings
statistics shows the MBL transition accompanied by the transition from the
Ginibre distribution to the complex Poisson distribution, different from the
one in the time-reversal invariant system. In addition, the dynamical stability
of the real part of the energy and the MBL transition can be characterized by
the winding number, indicating that the MBL transition and the topological
transition occur simultaneously, and the realization of the Hamiltonian is
discussed.
