Zigzag Magnetic Order in the Iridium Oxide ${\mathrm{Na}}_2{\mathrm{IrO}}_3$

We explore the phase diagram of spin-orbit Mott insulators on a honeycomb lattice, within the Kitaev-Heisenberg model extended to its full parameter space. Zigzag-type magnetic order is found to occupy a large part of the phase diagram of the model, and its physical origin is explained as due to interorbital $t_{2g}-e_g$ hopping. The magnetic susceptibility, spin wave spectra, and zigzag order parameter are calculated and compared to the experimental data, obtaining thereby the spin coupling constants in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and ${\mathrm{Li}}_2{\mathrm{IrO}}_3$.

Figure 1

(a) Phase diagram of the Kitaev-Heisenberg model containing 2 spin-liquid and 4 spin-ordered phases. The transition points (open dots on the $\phi$ circle) are obtained by an exact diagonalization. The gray lines inside the circle connect the points related by the exact mapping (see text). Open and solid circles in the insets indicate up and down spins. The rectangular box in the zigzag pattern (top-left) shows the magnetic unit cell. (b) Ground-state energy $E_{\mathrm{GS}}$ and its second derivative $-d^2{E}_{\mathrm{GS}}/d{\phi }^2$ revealing the phase transitions.

Figure 2

Schematics of four different exchange processes (see text for details), arranged around the $\phi$ phase diagram of Fig. 1a. Taken separately, the Hamiltonians $H_1$, $H_2$, $H_3$, and $H_4$ would favor “pure” Néel-AF, zigzag-AF, Kitaev-SL, and stripy-AF states, respectively, as indicated by arrows connecting $H_i$ with the dots on the $\phi$ circle. The circle is divided into the phase sectors by gray lines; SL phases are shaded.

Figure 3

Magnon spectra in the zigzag phase calculated using Eq. (4) with $(J,K)=(-4.0,10.5)\mathrm{meV}$. The inset shows the path along the symmetry directions in the reciprocal space; the notation of Ref. 4 is used.

Figure 4

Experimental magnetic susceptibilities $\chi (T)$ for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ [1, 6] (squares) and ${\mathrm{Li}}_2{\mathrm{IrO}}_3$ [6] (circles) fitted by the theoretical calculations. Exact $\chi$ of the 8-site (14-site) cluster is shown as solid (dashed) lines. Lanczos results for the 24-site cluster are indicated by shading [37]. Their comparison suggests that the calculated $\chi$ gives the thermodynamic limit down to $T\approx 100\mathrm{K}$ where the finite-size effects become significant.