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The $d$-wave pseudospin current order ($d$PSCO) with staggered circulatingpseudospin current has been proposed as the hidden electronic order to describethe unexpected breaking of spatial symmetries in stoichiometricSr$_{2}$IrO$_{4}$ and the unconventional pseudogap phenomena in electron dopedSr$_{2}$IrO$_{4}$. However, a microscopic model for the emergence of $d$PSCO isstill lacking. The nearest neighbor Coulomb repulsion $V$, which is expected tobe significant in Sr$_{2}$IrO$_{4}$ due to the large spatial extension of theIr $5d$ orbitals, is capable of driving $d$PSCO on the mean-field level, albeitthe latter is energetically degenerate to the staggered flux phase withcirculating charge current. We find the in-plane anisotropy $\Gamma_2$ in theeffective superexchange interaction between $J_\text{eff}={1\over 2}$pseudospins, originating from the cooperative interplay between Hund's rulecoupling and spin-orbit coupling of Ir $5d$ electrons, is able to lift thedegeneracy and stabilize the pseudospin currents. The effective single-orbitalmodel of $J_\text{eff}={1\over 2}$ electrons, including onsite Coulombrepulsion $U$, nearest neighbor Coulomb repulsion $V$, and the in-planeanisotropy $\Gamma_2$, is then studied. We obtain the mean-field ground states,analyze their properties, and determine the phase diagram of stoichiometricSr$_{2}$IrO$_{4}$ in the plane spanned by $U$ and $V$ at a fixed $\Gamma_2$. Wedemonstrate the realization of $d$PSCO, and its competition and coexistencewith antiferromagnetism. Remarkably, we find the coexistence of $d$PSCO andantiferromagnetism naturally leads to spin bond nematicity, with the spindirections of these three orders forming nontrivial chirality. Furthermore, weshow that the emergence of the coexistent state and its chirality can be tunedby carrier doping.

Microscopic model realization of $\boldsymbol{d}$-wave pseudospin current order in Sr$_{\boldsymbol{2}}$IrO$_{\boldsymbol{4}}$