Kitaev-Heisenberg-$J_2$-$J_3$ model for the iridates $A_2$IrO${}_3$

A Kitaev-Heisenberg-$J_2$-$J_3$ model is proposed to describe the Mott-insulating layered iridates $A_2$IrO${}_3$ ($A=\text{Na}$, Li). The model is a combination of the Kitaev honeycomb model and the Heisenberg model with all three nearest-neighbor couplings $J_1$, $J_2$, and $J_3$. A rich phase diagram is obtained at the classical level, including the experimentally suggested zigzag ordered phase; as well as the stripy phase, which extends from the Kitaev-Heisenberg limit to the $J_1$-$J_2$-$J_3$ one. Combining the experimentally observed spin order with the optimal fitting to the uniform magnetic susceptibility data gives an estimate of possible parameter values, which in turn reaffirms the necessity of including both the Kitaev and farther neighbor couplings.

Figure 1

The honeycomb lattice, with the Kitaev label $\gamma$ for nearest-neighbor bonds, and including second and third neighbor bonds with Heisenberg couplings $J_2$ and $J_3$.

Figure 2

(a) Sample ($J_2,J_3$) slice of the classical phase diagram, with phases (I), (II), and (IV) represented in (b), (c), and (d), respectively. Region (III) contains various noncollinear spiral configurations. (b) (I) Néel. (c) (II) Zigzag. (d) (IV) Stripy.

Figure 3

Fixed $\alpha$ slices in ($J_2,J_3$) showing the magnetically ordered phases $(\text{I},\text{II},\text{III},\text{IV})=\text{Neel}$, zigzag, spiral, stripy) and shading corresponding to the ED $\chi \left(T\right)$ fit goodness. Increasing the Kitaev term (i.e., increasing $\alpha$) enlarges the extent of the zigzag and stripy phases, which occur at both small and large $\alpha$. Fits to ${\text{Na}}_2\text{Ir}{\text{O}}_3$ are shaded in orange (with dotted contour lines) and fits to ${\text{Li}}_2\text{Ir}{\text{O}}_3$ are in blue (with dashed contour lines); darker shading corresponds to good fitting with $\mu /{\mu }_B\approx 1$ and $J_2\gtrsim J_3$, while lighter shading corresponds to poor agreement. Given a magnetically ordered ground state for each of the materials, the range of allowed parameters is found by intersecting the darker shaded region with the magnetically ordered phase.