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Analogs of the high-T$_c$ cuprates have been long sought after in transitionmetal oxides. Due to the strong spin-orbit coupling (SOC), the $5d$ perovskiteiridates Sr$_2$IrO$_4$ exhibit a low-energy electronic structure remarkablysimilar to the cuprates. Whether a superconducting state exists as in thecuprates requires understanding the correlated spin-orbit entangled electronicstates. Recent experiments discovered hidden order in the parent and electrondoped iridates, some with striking analogies to the cuprates, including Fermisurface pockets, Fermi arcs, and pseudogap. Here, we study the correlation anddisorder effects in a five-orbital model derived from the band theory. We findthat the experimental observations are consistent with a $d$-wave spin-orbitdensity wave order that breaks the symmetry of a joint two-fold spin-orbitalrotation followed by a lattice translation. There is a Berry phase and aplaquette spin flux due to spin procession as electrons hop between Ir atoms,akin to the intersite SOC in quantum spin Hall insulators. The associatedstaggered circulating $J_\text{eff}=1/2$ spin current can be probed by advancedtechniques of spin-current detection in spintronics. This electronic order canemerge spontaneously from the intersite Coulomb interactions between thespatially extended iridium $5d$ orbitals, turning the metallic state into anelectron doped quasi-2D Dirac semimetal with important implications on thepossible superconducting state suggested by recent experiments.

Correlation effects and hidden spin-orbit entangled electronic order in parent and electron-doped iridates Sr$_2$IrO$_4$