Recently, the topological flat bands and spin Hall effect have been
experimentally observed in the AB-stacked MoTe2β/WSe2β heterostructures. In
this work, we systematically study the Berry curvature effects in moir\'{e}
transition metal dichalcogenide (TMD) heterobilayers. We point out that the
moir\'{e} potential of the remote conduction bands would induce a sizable
periodic pseudo-magnetic field (PMF) on the valence band. This periodic PMF
creates net Berry curvature flux in each valley of the moir\'{e} Brillouin
zone. The combination of the effect of the Berry curvature and the spin-valley
locking can induce the spin Hall effect being observed in the experiment.
Interestingly, the valley-contrasting Berry curvature distribution generated by
the PMF can be probed through shift currents, which are DC currents induced by
linearly polarized lights through nonlinear responses. Our work shed lights on
the novel quantum phenomena induced by Berry curvatures in moir\'e TMD
heterobilayers.Comment: 10 pages, 7 figure