Selection of direction of the ordered moments in Na$_2$IrO$_3$ and $α-$RuCl$_3$

Abstract: The magnetic orders in Na$_2$IrO$_3$ and $\alpha-$RuCl$_3$, honeycomb systems with strong spin-orbit coupling and correlations, have been recently described by models with the dominant Kitaev interactions. In this work we discuss how the orientation of magnetic order parameter is selected in this class of models. We show that while the order-by-disorder mechanism in the models with solely Kitaev anisotropies always select cubic axes as easy axes for magnetic ordering, the additional effect of other small bond-dependent anisotropies, such as, e.g., $\Gamma$-terms, lead to a deviation of the order parameter from the cubic directions. We show that both the zigzag ground state and the face-diagonal orientation of magnetic moments in Na$_2$IrO$_3$ can be obtained within the $J_1-K_1-J_2-K_2-J_3$ model in the presence of perturbatively small $\Gamma$-terms. We also show that the zigzag phase found in the nearest neighbor Kitaev-Heisenberg model, relevant for $\alpha-$RuCl$_3$, has some stability against the $\Gamma$-term.