  A robust theory of the mechanism of pair density wave (PDW) superconductivity
(i.e. where Cooper pairs have nonzero center of mass momentum) remains elusive.
Here we explore the triangular lattice $t$-$J$-$V$ model, a low-energy
effective theory derived from the strong-coupling limit of the Holstein-Hubbard
model, by large-scale variational Monte Carlo simulations. When the electron
density is sufficiently low, the favored ground state is an s-wave PDW,
consistent with results obtained from previous studies in this limit.
Additionally, a PDW ground state with nematic d-wave pairing emerges in
intermediate range of electron densities and phonon frequencies. For these
s-wave and d-wave PDWs arising in states with spontaneous breaking of
time-reversal and inversion symmetries, PDW formation derives from
valley-polarization and intra-pocket pairing.
