Superconductivity, pair density wave, and Neel order in cuprates

We investigate in underdoped cuprates possible coexistence of the superconducting (SC) order at zero momentum and pair density wave (PDW) at momentum $\mathbf{Q}=(\pi ,\pi )$ in the presence of a Neel order. By symmetry, the $d$-wave uniform singlet pairing $d{S}_0$ can coexist with the $d$-wave triplet PDW $d{T}_{\mathbf{Q}}$, and the $p$-wave singlet PDW $p{S}_{\mathbf{Q}}$ can coexist with the $p$-wave uniform triplet $p{T}_0$. At half filling, we find the novel $p{S}_{\mathbf{Q}}+p{T}_0$ state is energetically more favorable than the $dS_0+dT_{\bf Q}$state\mathrm{.}Atfinitedoping,however,the$dS_0+dT_{\bf Q}$ state is more favorable. In both types of states, the variational triplet parameters, $d{T}_{\mathbf{Q}}$ and $p{T}_0$, are of secondary significance. Our results point to a fully symmetric ${\mathrm{Z}}_2$ quantum spin liquid with spinon Fermi surface in proximity to the Neel order at zero doping, and to intertwined $d$-wave triplet PDW fluctuations and spin moment fluctuations along with the dominant $d$-wave singlet SC at finite doping. The results are obtained by variational quantum Monte Carlo simulations.