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. 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 phenomena. We study the correlation and disorder effects on the spin-orbit entangled electronic states using 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 are a Berry phase and a staggered plaquette spin flux associated with circulating $J_{\rm eff}=1/2$ spin current as electrons hop between Ir atoms. 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.