Ab initio guided minimal model for the "Kitaev" material BaCo$_2$(AsO$_4$)$_2$: Importance of direct hopping, third-neighbor exchange and quantum fluctuations

We present a simple three-parameter exchange model to describe the interactions of the lowest doublet of the honeycomb cobaltate BaCo$_2$(AsO$_4$)$_2$, which has been proposed as a possible candidate for Kitaev physics. Remarkably, it is the third-neighbor interactions, both isotropic and anisotropic, that are responsible for the unique ground state of BaCo$_2$(AsO$_4$)$_2$, stabilized by quantum fluctuations. By considering two {\it ab initio}-based complementary approaches, we analyze the electronic structure of BaCo$_2$(AsO$_4$)$_2$ and extract effective spin models that justify the minimal model. Both methods show that the dominant direct hopping makes the bond-dependent Kitaev term negligible moving the material away from the sought-after spin-liquid regime. Moreover, a significantly large third-nearest neighbor hopping supports the observed importance of the third-neighbor interactions in the stabilization of the standout double-zigzag ground state of BaCo$_2$(AsO$_4$)$_2$.