Non-Kitaev vs. Kitaev Honeycomb Cobaltates

Recently, honeycomb cobaltates with 3$d^7$ were proposed to display Kitaev physics despite weak spin-orbit coupling. However, other theoretical and experimental works found leading XXZ Heisenberg and negligible Kitaev interactions in BaCo$_2$(AsO$_4$)$_2$ (BCAO), which calls for a further study to clarify the origin of the discrepancies. Here we derive the analytical expressions of the spin model using strong-coupling perturbation theory. With tight binding parameters obtained by {\it ab-initio} calculations for idealized honeycomb BCAO, we find that the largest intraorbital $t_{2g}-t_{2g}$ exchange path, which was assumed to be small in the earlier theory proposal, leads to a ferromagnetic (FM) Heisenberg interaction. This becomes the dominant interaction, as other $t_{2g}-e_g$ and $e_g-e_g$ contributions almost cancel each other. Exactly the same assumed-to-be-small channel also generates an antiferromagnetic Kitaev interaction, which then cancels a FM Kitaev interaction from $t_{2g}-e_g$ paths, resulting in a small Kitaev interaction. Under the trigonal distortion, the preeminent isotropic Heisenberg becomes an anisotropic XXZ model as expected, which is the case for BCAO. However, in Na$_3$Co$_2$SbO$_6$ the intraorbital $t_{2g}-t_{2g}$ hopping is smaller and comparable to the $t_{2g}-e_g$ hopping, leading to a delicate competition between enhanced Kitaev and reduced Heisenberg interactions.