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