Hidden symmetries of the extended Kitaev-Heisenberg model: Implications for the honeycomb-lattice iridates $A_2{\mathrm{IrO}}_3$

We have explored the hidden symmetries of a generic four-parameter nearest-neighbor spin model, allowed in honeycomb-lattice compounds under trigonal compression. Our method utilizes a systematic algorithm to identify all dual transformations of the model that map the Hamiltonian on itself, changing the parameters and providing exact links between different points in its parameter space. We have found the complete set of points of hidden SU(2) symmetry at which a seemingly highly anisotropic model can be mapped back on the Heisenberg model and inherits therefore its properties such as the presence of gapless Goldstone modes. The procedure used to search for the hidden symmetries is quite general and may be extended to other bond-anisotropic spin models and other lattices, such as the triangular, kagome, hyperhoneycomb, or harmonic-honeycomb lattices. We apply our findings to the honeycomb-lattice iridates ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and ${\mathrm{Li}}_2{\mathrm{IrO}}_3$, and illustrate how they help to identify plausible values of the model parameters that are compatible with the available experimental data.

Figure 1

(a) Top view of the honeycomb ${\mathrm{NaIr}}_2{\mathrm{O}}_6$ plane, the definition of global $X,Y,Z$ axes, and the $xyz$ reference frame for the spin components. The $X$ and $Y$ directions coincide with the crystallographic $\mathbf{a}$ and $\mathbf{b}$ axes. The three bond directions of the honeycomb lattice are labeled as $a$, $b$, and $c$; its two sublattices are labeled by $A$ and $B$. (b) Two edge-shared ${\mathrm{IrO}}_6$ octahedra of a $c$ bond and the definition of the local spin axes $\tilde{x}$, $\tilde{y}$, $\tilde{z}$ [used in Eq. (1)]. (c) Simultaneous cyclic permutation of the Ir-Ir bond directions $a$, $b$, $c$ and the spin components $x$, $y$, $z$ when applying a $C_3$ rotation to the model.

Figure 2

(a), (b) Unit cells for the four- and six-sublattice transformations. (c), (d) Stripy and zigzag patterns related to the FM and AF order of a hidden Heisenberg magnet via the four-sublattice transformation ${\mathcal{T}}_4$. The spins take the $z$-axis direction. (e), (f) “Vortex”-like patterns generated by the six-sublattice transformation ${\mathcal{T}}_6$. The spins are lying in the lattice plane in the case presented. The colors of the arrows in panels (d) and (f) indicate the sublattices of the hidden AF order. (g) Brillouin zones of the honeycomb (inner hexagon) and the completed triangular lattice (outer hexagon). The characteristic vectors ${\mathit{Q}}_{a,b,c}$ of the four-sublattice transformation and ${\mathit{Q}}_{1,2,3}$ of the six-sublattice transformation are shown in red and blue, respectively. (h) Bragg spots of the patterns in panels (c)–(f). The dot size is proportional to $|S_{\mathit{Q}}|$.

Figure 3

(a) Depiction of the SU(2) points using the parametrization of Ref. [37], $J=sin\theta cos\varphi$, $K=sin\theta sin\varphi$, $\mathrm{\Gamma }=cos\theta$. The distance from the center of the circle corresponds to $\theta$ going from 0 (center) through $\pi /4$ (dashed circle) to $\pi /2$ (solid circle). The polar angle is $\varphi$. Filled squares show the SU(2) points with ${\mathrm{\Gamma }}^{\prime }=0$, open squares those with nonzero ${\mathrm{\Gamma }}^{\prime }$ values given on the right along with the transformation label. The color of the points indicates their hidden FM (blue) or AF (red) nature. The green square (circle) shows the parameter values specified in Sec. 5 when discussing ${\mathrm{Na}}_2{\mathrm{IrO}}_3\left({\mathrm{Li}}_2{\mathrm{IrO}}_3\right)$. (b) The same as in panel (a) but with $\mathrm{\Gamma }=-cos\theta$.

Figure 4

(a) LSW dispersion of the Heisenberg FM (blue) and AF (red) on the honeycomb lattice. The width of the lines indicates the trace of the spin susceptibility tensor, ${\sum }_{\alpha }{\chi }_{\alpha \alpha }^{\prime \prime }(\mathit{q},\omega )$, calculated in the LSW approximation. (b) The same for the stripy and zigzag state presented in Figs. 2 and 2. Energy is scaled by $J_0$ of the hidden Heisenberg magnet. (c) The same for the “vortex”-like patterns presented in Figs. 2 and 2.

Figure 5

(a) Pseudospin angle $\alpha$ relative to the $XY$ plane (see inset) as a function of the parameter $r=D/(K+C/\sqrt{2})=-\mathrm{\Gamma }/(K+{\mathrm{\Gamma }}^{\prime })$. Dashed lines show the “magic” angle ${\alpha }_0\simeq {35}^{\circ }$ and its complement ${\overline{\alpha }}_0\simeq {55}^{\circ }$, determined by the $z$ axis and $xy$ plane, respectively, as sketched in the inset. (b) The phase diagram as a function of long-range couplings $J_2=J_3$ and anisotropy parameter $r$. Starting with the “bare,” ${\mathcal{T}}_1$-derived values of $J=5.3\mathrm{meV}$, $K=-7.0\mathrm{meV}$, $D=-9.3\mathrm{meV}$, and $C=-6.6\mathrm{meV}$, we have scaled $D$ and $C$ simultaneously to vary $r$. To stay within the zigzag phase at the smaller values of $r\lesssim 0.59$, one needs to have finite $J_{2,3}$ couplings. Otherwise, the NN-only extended KH model with negative $K<0$ switches to the incommensurate and “stripy” [11, 32, 69] ground states. The inset shows the exchange bonds $J_2$ and $J_3$.

Figure 6

(a) Map of the $\mathit{q}$-dependent classical energy (per site, in units of meV) obtained by Luttinger-Tisza method [71, 72] for the “bare,” i.e., ${\mathcal{T}}_1$-derived parameters $J=5.3\mathrm{meV}$, $K=-7.0\mathrm{meV}$, $D=-9.3\mathrm{meV}$, $C=-6.6\mathrm{meV}$ relevant for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$. The hexagon indicates the first Brillouin zone. (b) The same for the parameters $K=D=-10\mathrm{meV}$, $J=C=0$ relevant to ${\mathrm{Li}}_2{\mathrm{IrO}}_3$. (c) Length of the ordering vector for varying $C$, keeping the other parameter values unchanged. The dashed (solid) line was calculated using the above parameters $JKD$ relevant to ${\mathrm{Na}}_2{\mathrm{IrO}}_3\left({\mathrm{Li}}_2{\mathrm{IrO}}_3\right)$. Points $a$ and $b$ show the $C$ values used in panels (a) and (b), respectively. (d) LSW dispersions for the parameters used in panel (a) (left) and panel (b) (right). In the latter case, we have taken $C\simeq -1.2\mathrm{meV}$ instead of $C=0\mathrm{meV}$ to stay in the zigzag phase at the border to the spiral phase.