Spin-orbit physics of $j=\frac{1}{2}$ Mott insulators on the triangular lattice

The physics of spin-orbital entanglement in effective $j=\frac{1}{2}$ Mott insulators, which have been experimentally observed for various $5d$ transition-metal oxides, has sparked an interest in Heisenberg-Kitaev (HK) models thought to capture their essential microscopic interactions. Here, we argue that the recently synthesized ${\mathrm{Ba}}_3{\mathrm{IrTi}}_2{\mathrm{O}}_9$ is a prime candidate for a microscopic realization of the triangular HK model, a conceptually interesting model for its interplay of geometric and exchange frustration. We establish that an infinitesimal Kitaev exchange destabilizes the ${120}^{\circ }$ order of the quantum Heisenberg model. This results in the formation of an extended ${\mathbb{Z}}_2$-vortex crystal phase in the parameter regime most likely relevant to the real material, which can be experimentally identified with spherical neutron polarimetry. Moreover, using a combination of analytical and numerical techniques, we map out the entire phase diagram of the model, which further includes various ordered phases as well as an extended nematic phase around the antiferromagnetic Kitaev point.

Figure 1

(a) Crystal structure of ${\mathrm{Ba}}_3{\mathrm{IrTi}}_2{\mathrm{O}}_9$. (b) View of single iridium layers from two different perspectives. Within the plane, the $x,y$, and $z$ exchange paths are indicated by the gray planes. The planes labeled by $x$ $(y,z)$ are normal to the coordinate axis $\widehat{x}$ $(\widehat{y},\widehat{z})$. (c) The exchange between the iridium moments (blue) is mediated by two coplanar exchange paths.

Figure 2

(a) The triangular lattice with the three lattice vectors ${\mathbf{a}}_{\gamma }$. Solid, dashed, and dotted bonds carry the three distinct Kitaev interactions (see text). (b) First Brillouin zone of the triangular lattice. The position and size of the colored dots indicate the position and weight of Bragg peaks, respectively, expected in the static spin structure factor for the ${\mathbb{Z}}_2$-vortex crystal. Each color corresponds to a different spin component as listed in panel (a).

Figure 3

${\mathbb{Z}}_2$-vortex crystal stabilized for $J_H>0$ in the presence of a small but finite Kitaev interaction $J_K$ revealed by the chirality vectors of Eq. (14) which were computed from the classical ground state (13) in the Luttinger-Tisza approximation. The color code shows the length of the chirality vector $\left|\kappa \right(\mathbf{r}\left)\right|$, normalized to one, that becomes minimal at the ${\mathbb{Z}}_2$-vortex cores. The arrows in the closeup of the left panel correspond to projections of $\kappa \left(\mathbf{r}\right)$ onto the $x\text{−}y$ plane.

Figure 4

Phase diagram of the Hamiltonian (1) with parametrization $(J_H,J_K)=(cos\alpha ,sin\alpha )$ as obtained from exact diagonalization data. Solid lines show the mapping between two Klein-dual points. Red lines mark the location of the four SU(2)-symmetric points. Yellow diamonds mark the two Kitaev points.

Figure 5

(a), (b) Spin configurations for the four SU(2)-symmetric points of the HK model (1). The gray diamonds indicate the unit cells of the order. (c) Snapshots of spin configurations in the ${\mathbb{Z}}_2$-vortex crystal (left) and its dual ${\mathbb{Z}}_2$-vortex crystal (right). For clarity, only one of the three sublattices of the triangular lattice is shown. Yellow arrows point upwards out of the plane, while blue arrows point downwards out of the plane.

Figure 6

(a) Ground-state energy of the quantum model for ferromagnetic $J_H<0$ in an external Zeeman field as a function of the direction of the applied magnetic field $\mathbf{B}$, where we have subtracted the ground-state energy for $\mathbf{B}=\left|\mathbf{B}\right|\widehat{\mathbf{z}}$. The Kitaev coupling strength is $J_K/\left|J_H\right|=tan(11\pi /10)\approx 0.32$. The energy is minimal when the magnetization is pinned along one of the three axes, and maximal when pointing along the space diagonals. (b) The same results shown for the cut along the yellow line in (a). Each line in (b) corresponds to a different value of $J_K/J_H$. While for $J_K=0$ the ground-state energy is independent of the direction of the magnetic field, the directional dependence becomes increasingly pronounced upon increasing $J_K/\left|J_H\right|$. The dashed line is a fit of Eq. (20).

Figure 7

Histogram of the spin expectation value obtained with the help of finite-temperature Monte Carlo simulations of the classical HK model close to the ferromagnetic Heisenberg point. Whereas for $J_K=0$ in panel (a) the spin covers the full $S^2$ sphere, thermal fluctuations in the presence of a finite $J_K\ne 0$ favor the alignment along one of the six $\langle 100\rangle$ directions.

Figure 8

Energy gaps of a $3\times L$ triangular lattice strip with open boundary conditions. All values are given in relation to the ground-state energy $E_0$, i.e., $\Delta E_1=E_1-E_0$. The figures on the right show numerical results for $\langle S_{{\mathbf{r}}_0}^x{S}_{\mathbf{r}}^x\rangle$ spin correlations, where the black disk with the white dot indicates the position ${\mathbf{r}}_0$, the diameter of the disks indicates the strength of the correlation, and the color indicates the sign, with red corresponding to negative (antiferromagnetic) and black to positive (ferromagnetic) correlations. For details, see the main text.

Figure 9

Spin-spin correlations in the ground state of the antiferromagnetic Kitaev model on the triangular lattice. Black circles indicate positive correlations $\langle S_i^{\gamma }{S}_j^{\gamma }\rangle >0$, whereas the red circles denote negative correlations. The small white dot indicates the position ${\mathbf{r}}_0$. The geometry of the lattice clusters lifts the degeneracy of the lattice direction, favoring chains antiferromagnetically coupled with their $x$ component along the $x$ direction while correlations along the $y$ and $z$ directions are suppressed. Whereas adjacent chains remain uncoupled, next-nearest-neighbor chains couple antiferromagnetically.

Figure 10

Upper panel (a): Ground-state energy $E_0$ (black) and its second derivative $-d^2{E}_0/d{\alpha }^2$ (red) for the HK quantum model obtained from exact diagonalization of small clusters. Peaks in the second derivative indicate the position of phase transitions. The black and red arrows indicate the corresponding axis for each data set. Lower panel (b): Classical energies (gray dots, Monte Carlo) and quantum energies (dashed, ED). The colored solid lines show analytical estimates for the classical ground-state energies of the respective ordered phases, namely, Eq. (B1) for the ${\mathbb{Z}}_6$ ferromagnet and its dual, Eq. (12) after minimization for the ${\mathbb{Z}}_2$-vortex crystal and its dual, and Eq. (22) for the nematic phase. The classical and quantum energies touch at the two fluctuation-free points: the Heisenberg FM at $\alpha /\pi =1$ and its dual point. The upper and lower rings summarize in the spirit of Fig. 4 the extension of the various phases of the quantum and the classical models, respectively.

Figure 11

(a) 24-site cluster with periodic boundary conditions containing all symmetries except for the rotational $C_3$ symmetry. The different symbols for the lattice sites indicate the four sublattices needed in the basis transformation underlying the Klein duality (A1). (b) Circle parametrization of the Heisenberg-Kitaev interactions $J_H=Jcos\alpha$ and $J_K=Jsin\alpha$ with the magenta lines indicating points on the left- and right-hand sides of the circle related by the Klein duality (A1). The filled yellow and green circles indicate the points at which the Hamiltonian (1) is SU(2) symmetric.