Interplay between pair- and charge-density-wave orders in underdoped cuprates

We analyze the interplay between charge-density-wave (CDW) and pair-density-wave (PDW) orders within the spin-fermion model for the cuprates. We specifically consider CDW order with transferred momenta , as seen in experiments on the cuprates, and PDW order with total momenta . Both orders have been proposed to explain the pseudogap phase in the cuprates. We show that both emerge in the spin-fermion model near the onset of antiferromagnetism. We further show that the two orders are nearly degenerate due to an approximate particle-hole symmetry of the model. We discuss the ground-state properties of a pure PDW state and a pure CDW state, and show in particular that the PDW checkerboard state yields a vortex-antivortex lattice. For the checkerboard state, we considered a situation when both CDW and PDW orders are present at low and show that at small but finite Fermi surface curvature the presence of both condensates induces a long sought chiral s + id[subxy] superconductivity.