Electronic and magnetic properties of iridium ilmenites <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:msub><mml:mi>IrO</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mi>Mg</mml:mi><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi>Zn</mml:mi><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi>and</mml:mi><mml:mo> </mml:mo><mml:mi…

We theoretically investigate the electronic band structures and magnetic properties of ilmenites with edge-sharing IrO$_6$ honeycomb layers, $A$IrO$_3$ $A=$ Mg, Zn, Mn, in comparison a collinear antiferromagnet MnTiO$_3$. The compounds Mg Zn were recently reported Y.~Haraguchi {\it et al.}, Phys. Rev. Materials {\bf 2}, 054411 (2018), while MnIrO$_3$ has not been synthesized yet but stacking structure was elaborated superlattice MnTiO$_3$ K.~Miura Commun. Mater. 1}, 55 (2020). find that, contrast to MnTiO$_3$, where an energy gap opens Ti $3d$ bands by antiferromagnetic ordering high-spin $S=5/2$ moments, MgIrO$_3$ ZnIrO$_3$ have Ir $5d$ under influence both spin-orbit coupling electron correlation. Their are similar those coupled Mott insulators $j_{\rm eff}=1/2$ pseudospin degree freedom, as found monoclinic $A_2$IrO$_3$ Na Li which studied candidates for Kitaev spin liquid. Indeed, we that effective exchange interactions between pseudospins dominated Kitaev-type bond-dependent interaction symmetric off-diagonal interactions. On other hand, MnIrO$_3$, show local lattice is largely deformed, Mn appear near Fermi level complicated manner, makes qualitatively different from ZnIrO$_3$. Our results indicate network would offer good platform exotic magnetism spin-orbital entangled moments like