  A pair-density-wave (PDW) is a novel superconducting state with an
oscillating order parameter. A microscopic mechanism that can give rise to it
has been long sought but has not yet been established by any controlled
calculation. Here we report a density-matrix renormalization group (DMRG) study
of an effective $t$-$J$-$V$ model, which is equivalent to the Holstein-Hubbard
model in a strong-coupling limit, on long two-, four- and six-leg triangular
cylinders. While a state with long-range PDW order is precluded in one
dimension, we find strong quasi-long-range PDW order with a divergent PDW
susceptibility as well as spontaneous breaking of time-reversal and inversion
symmetries. Despite the strong interactions, the underlying Fermi surfaces and
electron pockets around the $K$ and $K^\prime$ points in the Brillouin zone can
be identified. We conclude that the state is valley-polarized and that the PDW
arises from intra-pocket pairing with an incommensurate center of mass
momentum. In the two-leg case, the exponential decay of spin correlations and
the measured central charge $c\approx 1$ are consistent with an unusual
realization of a Luther-Emery liquid.
