Large off-diagonal exchange couplings and spin liquid states in $C_3$-symmetric iridates

Iridate oxides on a honeycomb lattice are considered promising candidates for realization of quantum spin liquid states. We investigate the magnetic couplings in a structural model for a honeycomb iridate ${\mathrm{K}}_2{\mathrm{IrO}}_3$, with $C_3$ point-group symmetry at the Ir sites, which is an end member of the recently synthesized iridate family ${\mathrm{K}}_x{\mathrm{Ir}}_y{\mathrm{O}}_2$. Using ab initio quantum chemical methods, we elucidate the subtle relationship between the real space symmetry and magnetic anisotropy and show that the higher point-group symmetry leads to high frustration with strong magnetic anisotropy driven by the unusually large off-diagonal exchange couplings ($\mathrm{\Gamma }$'s) as opposed to other spin-liquid candidates considered so far. Consequently, large quantum fluctuations imply lack of magnetic ordering consistent with the experiments. Exact diagonalization calculations for the fully anisotropic $K-J-\mathrm{\Gamma }$ Hamiltonian reveal the importance of the off-diagonal anisotropic exchange couplings in stabilizing a spin liquid state and highlight an alternative route to stabilize spin liquid states for ferromagnetic $K$.

Figure 1

Crystal structure of ${\mathrm{K}}_2{\mathrm{IrO}}_3$: (a) shows the alternating layers formed of K ions, and hexagonal network formed by ${\mathrm{IrO}}_6$ octahedra with K ions at the center of the hexagons. (b) Top view of the honeycomb plane spanned by the ${\mathrm{IrO}}_6$ octahedra and K ions at center. Manifestation of the $C_3$ PG symmetry: the in-plane projections of the O-O links in each ${\mathrm{Ir}}_2{\mathrm{O}}_2$ plaquette are oriented ${120}^{\circ }$ with respect to each other (shown by arrow from the O atoms above the Ir plane to O atoms below the Ir plane).

Figure 2

Effects of $C_3$ point-group symmetry on magnetic couplings: magnetic exchange couplings (in meV) as a function of the relative twist angle $\varphi$ of the O atoms away from the central ${\mathrm{Ir}}_2{\mathrm{O}}_2$ plaquette (open symbols). The inset shows the definition of $\varphi$, where Ir and the O atoms are represented by filled blue and red circles, respectively. The vertical dashed lines represent the approximate $\varphi$ values for ${\mathrm{K}}_2{\mathrm{IrO}}_3$ and bond B1 of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, and the corresponding magnetic couplings are shown by filled symbols. The solid lines are guides to the eye.

Figure 3

Ground-state phase diagram by ED with a 24-site cluster: (a) in the ${\mathrm{\Gamma }}_{xy}-{\mathrm{\Gamma }}_{yz}$ plane using the MRCI values of $J$ and $K$; (b) in the $J_2-J_3$ plane using the MRCI values of $J, K, {\mathrm{\Gamma }}_{xy}$, and ${\mathrm{\Gamma }}_{yz}$. Schematic spin configurations are also shown. The star symbol in (a) indicates the position of MRCI parameter set and it corresponds to the origin in (b). A realistic range for ${\mathrm{K}}_2{\mathrm{IrO}}_3$ is located with shaded area in (b). Specific heat for the (c) threefold SDW and (d) zigzag phases, obtained by the ED calculations with a 12-site cluster. (e) Inverse magnetic susceptibility for the most likely realistic values of the extended range Heisenberg couplings $J_2$ and $J_3$. The dotted line is $\chi =2.1/(T-\theta )$ with $\theta =-135\mathrm{K}$.

Figure 4

Total (spin-up $+$ spin-down) density of states (DOS) per formula unit (f.u.) for the spin-orbit driven Mott insulator ${\mathrm{K}}_2{\mathrm{IrO}}_3$ along with the contributions of Ir-$5d$ and O-$2p$ states as obtained within the $\mathrm{GGA}+U$ scheme and considering spin-orbit effects, with $U_{\mathrm{Ir}-5d}=1.2\mathrm{eV}$ and $J_{\mathrm{Ir}-5d}=0.3\mathrm{eV}$. For comparison, the total DOS for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ with the same $U$ and $J$ values is also shown. The Fermi energy is shown by dashed vertical line.

Figure 5

Spin-structure factor $S\left(\mathbf{Q}\right)$ for representative momenta in different phases.

Figure 6

(a) Ground-state phase diagram around ${\mathrm{\Gamma }}_{xy}={\mathrm{\Gamma }}_{yz}=0$, extracted from Fig. 3 in the main text. (b) Representative physical quantities for the 24-site periodic cluster as a function of ${\mathrm{\Gamma }}_{yz}$ along red dashed line (${\mathrm{\Gamma }}_{yz}=\frac{8}{7}{\mathrm{\Gamma }}_{xy}+\frac{2}{7}$) in (a). Top: total spin. Middle: expectation value of the hexagonal plaquette operator. Bottom: ground-state energy per site $E_0/N$ and its second derivative $-{\partial }^2{E}_0/\partial {\mathrm{\Gamma }}_{yz}^2$.

Figure 7

Spin-spin correlation function $\langle {\tilde{\mathbf{S}}}_i·{\tilde{\mathbf{S}}}_j\rangle$ for non-Kitaev and Kitaev SL states in our phase diagram [Fig. 6]. The reference site $i$ is denoted by open square and the value of $\langle {\tilde{\mathbf{S}}}_i·{\tilde{\mathbf{S}}}_j\rangle$ at site $j$ is expressed by a circle. The filled and open circles mean positive and negative values of $\langle {\tilde{\mathbf{S}}}_i·{\tilde{\mathbf{S}}}_j\rangle$, respectively. The diameter of each circle is proportional to the magnitude, i.e., $|\langle {\tilde{\mathbf{S}}}_i·{\tilde{\mathbf{S}}}_j\rangle |$. (b) Expectation value of the hexagonal plaquette operator $|\langle O_{\mathrm{h}}\rangle |$ as a function of ${\mathrm{\Gamma }}_{xy}$ at ${\mathrm{\Gamma }}_{yz}=1$ fixed [along blue dotted line in Fig. 6].

Figure 8

(a) Spin-structure factor $S\left(\mathbf{Q}\right)$ for ${\mathrm{\Gamma }}_{xy}=0.2, {\mathrm{\Gamma }}_{yz}=-0.2$ and ${\mathrm{\Gamma }}_{xy}=2, {\mathrm{\Gamma }}_{yz}=-2$, where the system is in the threefold SDW state. (b) Real-space spin-spin correlation function $\langle {\tilde{\mathbf{S}}}_i·{\tilde{\mathbf{S}}}_j\rangle$ for a threefold SDW state (${\mathrm{\Gamma }}_{xy}=5, {\mathrm{\Gamma }}_{yz}=-5$). The reference site $i$ is denoted by open square and the value of $\langle {\tilde{\mathbf{S}}}_i·{\tilde{\mathbf{S}}}_j\rangle$ at site $j$ is expressed by a circle. The filled and open circles mean positive and negative values of $\langle {\tilde{\mathbf{S}}}_i·{\tilde{\mathbf{S}}}_j\rangle$, respectively. The diameter of each circle is proportional to the magnitude, i.e., $|\langle {\tilde{\mathbf{S}}}_i·{\tilde{\mathbf{S}}}_j\rangle |$. To detect the symmetry-broken state, the reference site is pinned by an infinite local magnetic field.