  The topological superconducting state is a highly sought-after quantum state
hosting topological order and Majorana excitations. In this work, we explore
the mechanism to realize the topological superconductivity (TSC) in the doped
Mott insulators with time-reversal symmetry (TRS). Through large-scale density
matrix renormalization group study of an extended triangular-lattice $t$-$J$
model on the six- and eight-leg cylinders, we identify a $d+id$-wave chiral TSC
with spontaneous TRS breaking, which is characterized by a Chern number $C=2$
and quasi-long-range superconducting order. We map out the quantum phase
diagram by tuning the next-nearest-neighbor (NNN) electron hopping and spin
interaction. In the weaker NNN-coupling regime, we identify a pseudogaplike
phase with a charge stripe order coexisting with fluctuating superconductivity,
which can be tuned into $d$-wave superconductivity by increasing the doping
level and system width. The TSC emerges in the intermediate-coupling regime,
which has a transition to a $d$-wave superconducting phase with larger NNN
couplings. The emergence of the TSC is driven by geometrical frustrations and
hole dynamics which suppress spin correlation and charge order, leading to a
topological quantum phase transition.
