Deriving models for the Kitaev spin-liquid candidate material $\alpha \text{−}{\mathrm{RuCl}}_3$ from first principles

We use the constrained random phase approximation to derive from first principles the Ru-$t_{2g}$ Wannier-function-based model for the Kitaev spin-liquid candidate material $\alpha \text{−}{\mathrm{RuCl}}_3$. We find the nonlocal Coulomb repulsion to be sizable compared to the local one. In addition we obtain the contribution to the Hamiltonian from the spin-orbit coupling and find it to also contain non-negligible nonlocal terms. We invoke strong-coupling perturbation theory to investigate the influence of these nonlocal elements of the Coulomb repulsion and the spin-orbit coupling on the magnetic interactions. We find that the nonlocal Coulomb repulsions cause a strong enhancement of the magnetic interactions, which deviate from experimental fits reported in the literature. Our results contribute to the understanding and design of quantum spin-liquid materials via first-principles calculations.

Figure 1

(a) Flow diagram of methods used: density functional theory (DFT), constrained random phase approximation (cRPA), Wannier functions, and strong-coupling perturbation theory. (b) Screening processes excluded/included in the cRPA denoted by $P_t$/$P_r$, with $t$ labeling the bands in the target space (i.e., the Ru-$t_{2g}$ bands) and $r$ labeling the rest of the bands (adapted from Ref. [35]).

Figure 2

Definition of the first ($X_1,Y_1,Z_1$), second ($X_2,Y_2,Z_2$), and third ($X_3,Y_3,Z_3$) nearest-neighboring Ru-Ru bonds (cyan lines) and the local coordinates $x,y,z$ (red arrows) relative to the primitive lattice vectors $a_1,a_2,a_3$ (black arrows) of the $C2/m$ unit cell of $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 3

Left: comparison of the band structure from scalar-relativistic density functional theory (srDFT) and the noninteracting scalar-relativistic part of the Wannier-function-based Hubbard model: $H_{cf}+H_{hop}$. Right: one of the corresponding Ru-$t_{2g}$ Wannier functions.

Figure 4

Comparison of low- and high-energy states in simplified model with 1 orbital per site. Arrows indicate spin-up and spin-down holes and cyan lines indicate nearest-neighboring holes.