| Summary: | Analogs of the high-T_{c} cuprates have been long sought after in transition metal oxides. Because of the strong spin-orbit coupling, the 5d perovskite iridates Sr_{2}IrO_{4} exhibit a low-energy electronic structure remarkably similar to the cuprates. Whether a superconducting state exists as in the cuprates requires understanding the correlated spin-orbit entangled electronic states. Recent experiments discovered hidden order in the parent and electron-doped iridates, some with striking analogies to the cuprates, including Fermi surface pockets, Fermi arcs, and pseudogap. Here, we study the correlation and disorder effects in a five-orbital model derived from the band theory. We find that the experimental observations are consistent with a d-wave spin-orbit density wave order that breaks the symmetry of a joint twofold spin-orbital rotation followed by a lattice translation. There is a Berry phase and a plaquette spin flux due to spin procession as electrons hop between Ir atoms, akin to the intersite spin-orbit coupling in quantum spin Hall insulators. The associated staggered circulating J_{eff}=1/2 spin current can be probed by advanced techniques of spin-current detection in spintronics. This electronic order can emerge spontaneously from the intersite Coulomb interactions between the spatially extended iridium 5d orbitals, turning the metallic state into an electron-doped quasi-2D Dirac semimetal with important implications on the possible superconducting state suggested by recent experiments.
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