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

{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 Neel 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.