Dynamical properties of the honeycomb-lattice iridates ${\mathrm{Na}}_2{\mathrm{IrO}}_3$

We investigate the dynamical properties of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$. For five effective models proposed for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, we numerically calculate dynamical structure factors (DSFs) with an exact diagonalization method. An effective model obtained from ab initio calculations explains inelastic neutron scattering experiments adequately. We further calculate excitation modes based on linearized spin-wave theory. The spin-wave excitation of the effective models obtained by ab initio calculations disagrees with the low-lying excitation of DSFs. We attribute this discrepancy to the location of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ in a parameter space close to the phase boundary with the Kitaev spin-liquid phase.

Figure 1

(a) Honeycomb-lattice model. Red, blue, and green lines denote $\mathcal{Z}, \mathcal{X}$, and $\mathcal{Y}$ bonds, respectively. Dotted (dashed) lines represent the second (third) neighbor interactions. The colors of these lines are the same as those for the corresponding nearest neighbor lines. Different lattice geometries for $N=24$ in (b) the $\pi /3$-rotational symmetric case (${\mathrm{C}}_3$), (c) 3 $\times$ 2 cells ($3\times 2$), and (d) 2 $\times$ 3 cells ($2\times 3$). The $x, y$, and $z$ axes in (b) correspond to the orthogonal axes for spin operators that are defined from the $5d{t}_{2g}\mathrm{orbitals}$ of ${\mathrm{Ir}}^{4+}$ [9] and the honeycomb plane is perpendicular to the $(1,1,1)$ direction. Black bold dashed lines enclose the 24 sites, and shaded rectangles indicate four sublattice unit cells. In (d), a schematic of the zigzag order is presented. For all geometries, we apply periodic boundary conditions and boundaries with the same symbol are connected.

Figure 2

(a) Four sublattice unit cells and schematic spin configuration of the Z-type collinear zigzag order. Solid (open) circles represent up (down) spins and the dotted rectangle corresponds to the shaded ones in Fig. 1. (b) Reciprocal vectors $\alpha$ and $\beta$ for the four sublattice unit cells in (a). (c)–(l) SSFs for Models I–V for $N=24$ with three lattice geometries in Figs. 1–1. The area of each circle is proportional to the amplitude. Upper (lower) panels are results for the longitudinal (transverse) component.

Figure 3

Averaged intensity, $\mathcal{I}\left(\right|\mathit{Q}|,\omega )$, for $N=24$ with three lattice geometries. The area of each circle is proportional to the scattering intensity. All results are normalized by the largest value for each data set. Dotted lines are the low-energy cutoff at about $2\mathrm{meV}$ in INS experiments [4]. The horizontal axis corresponds to distance from the $\mathrm{\Gamma }$ point. The labels on the horizontal axis denote the corresponding distance from the $\mathrm{\Gamma }$ point to the label [15].

Figure 4

DSFs, $S({\mathit{Q}}_i,\omega )$, for $N=24$. Results for different lattice geometries are included. The area of each circle corresponds to the logarithmic amplitude of the scattering intensity. The solid curves are dispersion curves estimated from the linearized spin-wave analysis. The excitation energy is normalized by the largest Heisenberg interaction $|J_{NN}|$ in each model; $|J_{NN}|$ = 4.17 meV, 4.0 meV, 4.4 meV, 3.0 meV, and 5.8 meV for Models I–V, respectively. The horizontal axis in (a)–(e) runs along the arrows shown in Fig. 2. The excitation energy is normalized. (f) Temperature dependence of specific heat for Model III numerically obtained by a diagonalization method. Red (blue) symbols are the results for $N=16$ (8).