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

Analogs of the high-T$_c$ cuprates have been long sought after in transition metal oxides. Due to the strong spin-orbit coupling (SOC), 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 two-fold 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 SOC in quantum spin Hall insulators. The associated staggered circulating $J_\text{eff}=1/2$ 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.