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 (+/- Q,0)/(0,+/- Q), as seen in experiments on the cuprates, and PDW order with total momenta (0,+/- Q)/(+/- Q,0). 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. Each order parameter is constructed out of pairs of fermions in "hot" regions on the Fermi surface, breaks U(1) translational symmetry, and changes sign when the momenta of the fermions change by (pi,pi). We further show that the two orders are nearly degenerate due to an approximate SU(2) particle-hole symmetry of the model. This near degeneracy is similar in origin to that relating conventional d-wave superconducting order and bond charge order with momentum (Q,+/- Q). The SU(2) symmetry becomes exact if one neglects the curvature of the Fermi surface in hot regions, in which case U(1), CDW, and PDW order parameters become components of an SO(4)-symmetric PDW/CDW "supervector." We develop a Ginzburg-Landau theory for four PDW/CDW order parameters and find two possible ground states: a "stripe" state in which both CDW and PDW orders develop with either (+/- Q,0) or (0,+/- Q), and a "checkerboard" state, where each order can develop with (+/- Q,0) and (0,+/- Q). We show that the SO(4) symmetry between CDW and PDW can be broken by two separate effects. One is the inclusion of Fermi surface curvature, which selects a PDW order immediately below the instability temperature. Another is the overlap between different hot regions, which favors CDW order at low temperatures. For the stripe state, we show that the competition between the two effects gives rise to a first-order transition from PDW to CDW inside the ordered state. We also argue that beyond mean-field theory, the onset temperature for CDW order is additionally enhanced due to feedback from a preemptive breaking of Z(2) time-reversal symmetry. 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 T and show that at small but finite Fermi surface curvature the presence of both condensates induces a long sought chiral s + id(xy) superconductivity.