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

{\it Ab initio} Studies on the Interplay between Spin-Orbit Interaction and Coulomb Correlation in Sr$_2$IrO$_4$ and Ba$_2$IrO$_4$