Kitaev magnetism in honeycomb ${\text{RuCl}}_3$ with intermediate spin-orbit coupling

Intensive studies of the interplay between spin-orbit coupling (SOC) and electronic correlations in transition-metal compounds have recently been undertaken. In particular, $j_{\mathrm{eff}}=1/2$ bands on a honeycomb lattice provide a pathway to realize Kitaev's exactly solvable spin model. However, since current wisdom requires strong atomic SOC to make $j_{\mathrm{eff}}=1/2$ bands, studies have been limited to iridium oxides. Contrary to this expectation, we demonstrate how Kitaev interactions arise in $4d$-orbital honeycomb $\alpha \text{−}{\mathrm{RuCl}}_3$, despite having significantly weaker SOC than the iridium oxides, via assistance from electron correlations. A strong-coupling spin model for these correlation-assisted $j_{\mathrm{eff}}=1/2$ bands is derived, in which large antiferromagnetic Kitaev interactions emerge along with ferromagnetic Heisenberg interactions. Our analyses suggest that the ground state is a zigzag-ordered phase lying close to the antiferromagnetic Kitaev spin liquid. Experimental implications for angle-resolved photoemission spectroscopy, neutron scattering, and optical conductivities are discussed.

Figure 1

Schematic diagrams depicting the DOS and change in the electronic structure of ${\mathrm{RuCl}}_3$ as SOC and the on-site Coulomb interactions are included. Red and blue colors represent the weights of $j_{\mathrm{eff}}=1/2$ and 3/2 states, respectively. Panel (a) displays the DOS without SOC, and panel (b) shows the DOS with SOC, which shows no clear separation between $j_{\mathrm{eff}}=1/2$ and 3/2 bands. On including $U$ and fixing a paramagnetic state, as shown in panel (c), the bands near the Fermi level acquire $j_{\mathrm{eff}}=1/2$ character and are separated from the 3/2 bands. Panel (d) is the DOS in a magnetic ground state realized in ${\mathrm{RuCl}}_3$.

Figure 2

(a) Electronic structure of ${\mathrm{RuCl}}_3$ without SOC and electron interactions. Red and gray curves depict the PDOS for Cl and Ru $t_{2\mathrm{g}}$, respectively. The $j_{\mathrm{eff}}$-projected band structures and density of states are laid out in the presence of SOC in (b), SOC and the on-site Coulomb interaction of $U_{\mathrm{eff}}=1.5$ eV while fixing a nonmagnetic state in (c), and with the lowest energy ZZ magnetic order in (d), respectively.

Figure 3

(a) Collinear magnetic configurations considered in the LDA+SOC+$U$ calculations. (b) Relative energy difference per Ru atom for each configuration plotted with respect to $U_{\mathrm{eff}}$. The ZZ ordered state has the lowest energy except when $U_{\mathrm{eff}}=1.0$ eV, but FM is competitive and the 120 ordered state approaches both states in energy when $U_{\mathrm{eff}}$ is large.

Figure 4

(a) First (solid), second (dashed), and third (dotted) NN bonds on the honeycomb lattice with the bond labels. Red, blue, and green colors depict the $\alpha \beta \left(\gamma \right)=xy\left(z\right),yz\left(x\right)$, and $zx\left(y\right)$ bonds, respectively, where $\alpha ,\beta$, and $\gamma$ denote the spin components interacting on the specified bond. Further neighbor hoppings with only $xy\left(z\right)$ type are depicted in the figure. (b) shows the Luttinger-Tisza phase diagram at $J_{\mathrm{H}}/U=0.2$ for fixed second and third NN exchanges. Gray shading within the 120 order phase depicts the trace of incommensurate (I) order occurring in that area. The red diamond marks the estimated parameters for ${\mathrm{RuCl}}_3$. See the main text for a description of the exchange parameters.