  We demonstrate that sliding motion between two layers of the newly discovered
ferroelectric and topologically trivial bismuth (Bi) monolayer [Nature 617, 67
(2023)] can induce a sequence of topological phase transitions, alternating
between trivial and nontrivial states. Interestingly, a lateral shift, even
when preserving spatial symmetry, can still switch the quantum spin Hall state
on and off. The substantial band-gap modulation and band inversion due to
interlayer sliding arise primarily from the intralayer in-plane charge transfer
processes involving Bi atoms at the outermost atomic layers, rather than the
interlayer charge redistribution. We map out the topological phase diagram and
the geometric Berry curvature-dipole induced nonlinear anomalous Hall response
resulting from sliding, highlighting the potential for robust mechanical
control over the edge current and the Hall current. Bilayer configurations that
are $\mathbb{Z}_2$ nontrivial can produce drastically different transverse
currents orthogonal to the external electric field. This occurs because both
the direction and magnitude of the Berry curvature dipole at the Fermi level
depend sensitively on the sliding displacement. Our results suggest that
bilayer bismuth could serve as a platform to realize power-efficient ``Berry
slidetronics" for topology memory applications.
