First-Principles Study of the Honeycomb-Lattice Iridates ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ in the Presence of Strong Spin-Orbit Interaction and Electron Correlations

An effective low-energy Hamiltonian of itinerant electrons for iridium oxide ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ is derived by an ab initio downfolding scheme. The model is then reduced to an effective spin model on a honeycomb lattice by the strong coupling expansion. Here we show that the ab initio model contains spin-spin anisotropic exchange terms in addition to the extensively studied Kitaev and Heisenberg exchange interactions, and allows us to describe the experimentally observed zigzag magnetic order, interpreted as the state stabilized by the antiferromagnetic coupling of the ferromagnetic chains. We clarify possible routes to realize quantum spin liquids from existing ${\mathrm{Na}}_2{\mathrm{IrO}}_3$.

Figure 1

Left panel: Crystal structure of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$. Right panel: Honeycomb lattice with $X$, $Y$, and $Z$ bonds. Same colored bonds indicate the same group. The $x$, $y$, and $z$ axes in defining the $t_{2g}$ orbitals are illustrated as directions out of the honeycomb plane. The honeycomb plane is then perpendicular to $(x,y,z)=(1,1,1)$. The dashed boundary represents a 24-site cluster used later for the exact diagonalization.

Figure 2

Ground state and finite temperature properties of the generalized Kitaev-Heisenberg model for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ calculated for the 24-site cluster by using the Lanczos method and thermal pure quantum states [26]. (a) Ground state magnetic order determined by applying tiny local magnetic fields ($\sim 10^{-2}\mathrm{meV}$) at a single site. (b) Temperature dependence of specific heat $C$ and entropy $S$, which are consistent with an experiment [29]. Shaded area shows uncertainty due to finite size effects [26]. (c) Temperature-dependence of in-plane and out-of-plane magnetic susceptibilities, which are also consistent with the experiment [29] at high temperatures.

Figure 3

(a) $\mathrm{\Delta }$ dependence of matrix elements of ${\mathcal{J}}_Z$, ${\mathcal{J}}_X$ as functions of $\mathrm{\Delta }$. Around the ab initio values at $\mathrm{\Delta }$ ($\sim -28\mathrm{meV}$) listed in Table 2, $K<0$, $K^{\prime }<0$, $J>0$, $J^{\prime }>0$, and $J^{″}>0$ are stably satisfied with gradual dependences on $\mathrm{\Delta }$. (b) Ground state phase diagram for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ with lattice distortions represented by changes in $\mathrm{\Delta }$. The phase boundaries are determined by anomalies (peaks signaling continuous transitions) in second derivatives of the exact energy for the 24-site cluster with respect to $\mathrm{\Delta }$. Around the ab initio parameter $\mathrm{\Delta }=-28\mathrm{meV}$, the zigzag order appears. By increasing $\mathrm{\Delta }$, a 6-site unit cell order (or 120° structure [30]) illustrated in the lower left panel and a 24-site unit cell long-period order [19], appear. (c) $\mathrm{\Delta }$ dependence of the ground state of the generalized Kitaev-Heisenberg model for “expanded lattices.” Here we neglect the small hopping parameters other than $t$ and take a larger Hund’s rule coupling $J_H=0.3\mathrm{eV}$. Spin liquid phases compete with ferromagnetic states and 12-site unit cell orders illustrated in the upper right panel [19], where the phase transitions among them are also interpreted as continuous ones.