We have performed inelastic neutron scattering (INS) experiments to investigate the magnetic excitations in the weakly distorted face-centered-cubic (fcc) iridate double perovskites La$_2$ZnIrO$_6$ and La$_2$MgIrO$_6$, which are characterized by A-type antiferromagnetic ground states. The powder inelastic neutron scattering data on these geometrically frustrated $j_{rm eff}=1/2$ Mott insulators provide clear evidence for gapped spin wave excitations with very weak dispersion. The INS results and thermodynamic data on these materials can be reproduced by conventional Heisenberg-Ising models with significant uniaxial Ising anisotropy and sizeable second-neighbor ferromagnetic interactions. Such a uniaxial Ising exchange interaction is symmetry-forbidden on the ideal fcc lattice, so that it can only arise from the weak crystal distortions away from the ideal fcc limit. This may suggest that even weak distortions in $j_{rm eff}=1/2$ Mott insulators might lead to strong exchange anisotropies. More tantalizingly, however, we find an alternative viable explanation of the INS results in terms of spin models with a dominant Kitaev interaction. In contrast to the uniaxial Ising exchange, the highly-directional Kitaev interaction is a type of exchange anisotropy which is symmetry-allowed even on the ideal fcc lattice. The Kitaev model has a magnon gap induced by quantum order-by-disorder, while weak anisotropies of the Kitaev couplings generated by the symmetry-lowering due to lattice distortions, can pin the order and enhance the magnon gap. Our findings highlight how even conventional magnetic orders in heavy transition metal oxides may be driven by highly-directional exchange interactions rooted in strong spin-orbit coupling.
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