Magnetic orders proximal to the Kitaev limit in frustrated triangular systems: Application to ${\mathrm{Ba}}_3{\mathrm{IrTi}}_2{\mathrm{O}}_9$

Frustrated transition-metal compounds in which spin-orbit coupling (SOC) and electron correlation work together have attracted much attention recently. In the case of $5d$ transition metals, where SOC is large, $j_{\text{eff}}=1/2$ bands near the Fermi level are thought to encompass the essential physics of the material, potentially leading to a concrete realization of exotic magnetic phases such as the Kitaev spin liquid. Here, we derive a spin model on a triangular lattice based on $j_{\text{eff}}=1/2$ pseudospins that interact via antiferromagnetic Heisenberg ($J$) and Kitaev ($K$) exchanges, and crucially, an anisotropic $\left(\mathrm{\Gamma }\right)$ exchange. Our classical analysis of the spin model reveals that, in addition to small regions of ${120}^{\circ }, {\mathbb{Z}}_2$/dual-${\mathbb{Z}}_2$ vortex crystal and nematic phases, the stripy and ferromagnetic phases dominate the $J-K-\mathrm{\Gamma }$ phase diagram. We apply our model to the $5d$ transition-metal compound, ${\mathrm{Ba}}_3{\mathrm{IrTi}}_2{\mathrm{O}}_9$, in which the ${\mathrm{Ir}}^{4+}$ ions form layered two-dimensional triangular lattices. We compute the band structure and nearest-neighbor hopping parameters using ab initio calculations. By combining our ab initio and classical analyses, we predict that ${\mathrm{Ba}}_3{\mathrm{IrTi}}_2{\mathrm{O}}_9$ has a stripy ordered magnetic ground state.

Figure 1

(a) The major nearest-neighbor hopping channels in the limit of ideal octahedra. The parameters $t_1$ and $t_3$ describe two distinct $d$-orbital overlaps, while $t_2$ includes both $d$-orbital overlap and oxygen mediated hoppings. The green, blue, and red dotted lines label the $x, y$, and $z$ bond types according to the convention $\alpha \beta \left(\gamma \right)$ in Eq. (1).

Figure 2

Combined Luttinger-Tisza (LT) and classical Monte Carlo (CMC) phase diagrams of the triangular lattice $J-K-\mathrm{\Gamma }$ model for $\mathrm{\Gamma }>0$ (a) and $\mathrm{\Gamma }<0$ (b). The angles $\theta$ and $\varphi$ denote the radial and azimuthal angles, respectively. There are six phases represented: ferromagnet (FM), stripy (ST), 120 order (120), nematic (N), ${\mathbb{Z}}_2$ vortex ($Z_2$), and dual-${\mathbb{Z}}_2$ ($d{Z}_2$). Real-space spin representations of FM, ST and 120 orders are shown in the insets, while nematic, ${\mathbb{Z}}_2$ vortex crystal and dual-${\mathbb{Z}}_2$ vortex crystal phases are shown in Appendix. Phases bounded toward the edge ($\mathrm{\Gamma }=0$) by a dotted line are computed using CMC. The region marked by the white star is the relevant parameter regime for ${\mathrm{Ba}}_3{\mathrm{IrTi}}_2{\mathrm{O}}_9$ predicted by our combined ab initio and classical analysis.

Figure 3

(a) Half of the ${\mathrm{Ba}}_3{\mathrm{IrTi}}_2{\mathrm{O}}_9$ primitive unit cell (bounded by the rectangular box) translated once in the $a$ direction featuring Ba (pale green), Ti (blue), O (red), and Ir (yellow) atoms. There are two ${\mathrm{Ir}}^{4+}$ ions in the unit cell, each belonging to different layered triangular lattices as in (b), with each site surrounded by oxygen octahedra. Nearest-neighbor Ir atoms belonging to different unit cells are connected by the black double-arrowed line. The ${\mathrm{Ir}}^{4+}$ octahedra are face-shared with ${\mathrm{Ti}}^{4+}$ octahedra. (c) Representation of distorted octahedra featuring two new hopping parameters, $t_2$ and $t_2^{\prime }$, leading to a Dzyaloshinksii-Moriya (DM) exchange interaction in Eq. (2).

Figure 4

Band structure calculations of ${\mathrm{Ba}}_3{\mathrm{IrTi}}_2{\mathrm{O}}_9$ performed using openmx. (a) Band structure without SOC and projected density of states including Ir $t_{2g}$, Ti $d$, and O $p$ orbitals. The bands near the Fermi level are composed of contributions from ${\mathrm{Ir}}^{4+}{t}_{2g}$ orbitals and O $p$ orbitals. (b) Band structure with SOC and $j_{\text{eff}}$ projected density of states. The bands are projected onto $j_{\text{eff}}=1/2$ and $3/2$ states using maximally localized Wannier orbitals. The relative contribution of each state is represented by the intensity of the color. The $t_{2g}$ bands are well separated into $j_{\text{eff}}=1/2$ (pink) and $3/2$ (teal) bands.

Figure 5

Real space spin configuration examples of nematic (a), ${\mathbb{Z}}_2$ vortex crystal (b), and dual-${\mathbb{Z}}_2$ vortex crystal phases. The orientation of each spin into (out of) the triangular plane is represented by the blueness (redness) of the color of the spins. The colored boundary around each phase corresponds to the color used to label the phase in Fig. 2 of the main text.