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By considering two {\it ab initio}-based complementary approaches, we analyzethe electronic structure and extract effective spin models ofBaCo$_2$(AsO$_4$)$_2$, a honeycomb material which has been proposed as acandidate for Kitaev physics. Both methods show that the dominant directhopping makes the bond-dependent Kitaev term negligible averting the materialaway from the sought-after spin-liquid regime. As a result, we present a simplethree-parameter exchange model to describe the interactions of the lowestdoublet of the honeycomb cobaltate BaCo$_2$(AsO$_4$)$_2$. Remarkably, it is thethird-neighbor interactions, both isotropic and anisotropic, that areresponsible for the standout double-zigzag ground state ofBaCo$_2$(AsO$_4$)$_2$, stabilized by quantum fluctuations. A significantlylarge third-nearest neighbor hopping, observed in {\it ab initio}, supports theimportance of the third-neighbor interactions in the stabilization of theunique ground state of BaCo$_2$(AsO$_4$)$_2$.

physical review. b./physical review. b

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