  {\it Ab initio} analyses of A$_2$IrO$_4$ (A=Sr, Ba) are presented. Effective
Hubbard-type models for Ir 5$d$ $t_{2g}$ manifolds downfolded from the global
band structure are solved based on the dynamical mean-field theory. The results
for A=Sr and Ba correctly reproduce paramagnetic metals undergoing continuous
transitions to insulators below the N\'eel temperature $T_N$. These compounds
are classified not into Mott insulators but into Slater insulators. However,
the insulating gap opens by a synergy of the Neel order and significant band
renormalization, which is also manifested by a 2D bad metallic behavior in the
paramagnetic phase near the quantum criticality.
