  In this paper, we study periodically modulated $s=1/2$ spin chain in a linear
gradient potential (LP) that is generated by an external magnetic field. In the
absence of the LP, the system has topological states that exhibit a
magnetization plateau for a uniform external magnetic field. These topological
states have a finite integer Chern number and their stability is clarified by
an equivalent spinless fermion system derived by a Jordan-Wigner
transformation. We show that the LP, which is nothing but a constant electric
field in the spinless fermion system, destabilizes the topological states,
because it induces localization called Wannier-Stark (WS) localization. We
clarify the phase diagram in the presence of the LP and on-site diagonal
disorder. To this end, we carefully study edge excitations under the open
boundary condition, which are a hallmark of the topological order. We find a
very interesting phenomenon indicating existence of a quasi-edge modes that
take the place of the genuine edge modes in certain parameter regions. This is
a precursor of the WS localization realized in topological states. Finally, we
investigate many-body localization induced by a sufficiently strong LP or
disorder. To this end, we study the energy-level statistics for whole energy
levels, and find unexpected extended-state regimes located in intermediate
potential-gradient and weak on-site disorder regimes. We verify this phenomenon
by calculating variance of the entanglement entropy. The present system is
closely related to quantum Hall state in two dimensions, and therefore our
findings can be observed not only in experiments on ultra-cold atomic gases but
also quantum Hall physics.
