  In our previous work [arXiv:1803.00999, Phys. Rev. Lett. 121, 046401 (2018)],
we found a quantum spin liquid phase with a spinon Fermi surface in the two
dimensional spin-1/2 Heisenberg model with four-spin ring exchange on a
triangular lattice. In this work we dope the spinon Fermi surface phase by
studying the $t$-$J$ model with four-spin ring exchange. We perform density
matrix renormalization group calculations on four-leg cylinders of a triangular
lattice and find that the dominant pair correlation function is that of a pair
density wave; i.e., it is oscillatory while decaying with distance with a power
law. The doping dependence of the period is studied. This is the first example
where pair density wave is the dominant pairing in a generic strongly
interacting system where the pair density wave cannot be explained as a
composite order and no special symmetry is required.

Strongly Correlated Electrons;Superconductivity