Superconductivity, Pair Density Wave, and Néel Order in Cuprates

We investigate in underdoped cuprates possible coexistence of the superconducting order at zero momentum and pair density wave (PDW) at momentum \boldsymbol Q=(\pi, \pi) in the presence of a Néel order. By symmetry, the d-wave uniform singlet pairing dS_0 can coexist with the d-wave triplet PDW dT_\boldsymbol Q, and the p-wave singlet PDW pS_\boldsymbol Q can coexist with the p-wave uniform triplet pT_0. At half filling, we find that the novel pS_\boldsymbol Q+pT_0 state is energetically more favorable than the dS_0+dT_\boldsymbol Q state. At finite doping, however, the dS_0+dT_\boldsymbol Q state is more favorable. In both types of states, the variational triplet parameters dT_\boldsymbol Q and pT_0 are of secondary significance. Our results point to a fully symmetric Z_2 quantum spin liquid with spinon Fermi surface in proximity to the Néel order at zero doping, which may not be adiabatically connected to the d-wave singlet superconductivity at finite doping with intertwining d-wave triplet PDW fluctuations and spin moment fluctuations. The results are obtained by variational quantum Monte Carlo simulations.