Pair Density Wave in the doped t-J model with ring exchange on triangular lattice

In our former work (PRL 121 046401 (2018)), we found a quantum spin liquid with spinon Fermi surface in the two dimensional spin-1/2 Heisenberg model with four-spin ring exchange on 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 triangular lattice and find clear signatures of pair density wave. The pair correlation function is oscillatory, even though it decays rapidly with distance with a power law larger than 2. The doping dependence of the decay and the period are studied. This is the first example where the pairing density wave is the dominate pairing in a system without a spin gap.