Magnetic structure and excitation spectrum of the hyperhoneycomb Kitaev magnet $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$

We present a theoretical study of the static and dynamical properties of the three-dimensional, hyperhoneycomb Kitaev magnet $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. We argue that the observed incommensurate order can be understood in terms of a long-wavelength twisting of a nearby commensurate period-3 state, with the same key qualitatively features. The period-3 state shows very different structure when either the Kitaev interaction $K$ or the off-diagonal exchange anisotropy $\mathrm{\Gamma }$ is dominant. A comparison of the associated static spin structure factors with reported scattering experiments in zero and finite fields gives strong evidence that $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ lies in the regime of dominant Kitaev coupling, and that the Heisenberg exchange $J$ is much weaker than both $K$ and $\mathrm{\Gamma }$. Our predictions for the magnon excitation spectra, the dynamical spin structure factors, and their polarization dependence provide additional distinctive fingerprints that can be checked experimentally.

Figure 1

Lattice structure and orthorhombic unit cell of $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. The bonds are color coded according to the type $t$ of the Kitaev interaction: red, green and blue for $t=(x,x^{\prime })$, $(y,y^{\prime })$, and $z$, respectively. The $\pm$ signs denote the sign of ${\sigma }_t=\pm 1$ in Eq. (4).

Figure 2

The shaded region shows the portion of the parameter space $(r,\varphi )$ that is believed [31, 32] to be relevant for $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. The two main subregions discussed in the text, the $K$ and the $\mathrm{\Gamma }$ regions, are separated by the boundary which begins at the hidden, isotropic SO(3) point (red star). On this boundary, the classical energies of the $K$ state (14) and of the $\mathrm{\Gamma }$ state (20) are degenerate. The two representative points shown, $P_K$ and $P_{\mathrm{\Gamma }}$, correspond to $(r,\varphi )=(\frac{3\pi }{8},\frac{25\pi }{16})$ and $(\frac{\pi }{8},\frac{97\pi }{64})$, respectively. For the phases outside the shaded region (including the white region shown), see detailed analysis in Refs. [31, 32].

Figure 3

Schematic structure of the (a) $K$ state and (b) $\mathrm{\Gamma }$ state discussed in Secs. 3a and 3b. The spins on $xy$- and $x^{\prime }{y}^{\prime }$-zigzag chains are denoted, correspondingly, by $\mathbf{A},\mathbf{B},\mathbf{C}$ and ${\mathbf{A}}^{\prime },{\mathbf{B}}^{\prime },{\mathbf{C}}^{\prime }$ in the $K$ state and $\mathbf{D},\mathbf{F},\mathbf{G}$ and ${\mathbf{D}}^{\prime },{\mathbf{F}}^{\prime },{\mathbf{G}}^{\prime }$ in the $\mathrm{\Gamma }$ state. The side panels show the corresponding Cartesian components.

Figure 4

Evolution of the components $x_1$, $y_1$, $z_1$, $x_2$, and $z_2$ of Eq. (9), and the components $x_3$, $y_3$, and $z_3$ of Eq. (15), as we change the parameter $r$ for (a) $\varphi =1.515625\pi$, (b) $\varphi =1.5625\pi$, and (c) $\varphi =1.625\pi$. Data are shown up to the point $r_c\left(\varphi \right)$, where we exit the $K$ region, see Fig. 2.

Figure 5

Direction of the vectors $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{F}$, $\mathbf{G}$, $\mathbf{D}$ at one of the points on the boundary between the $K$ and $\mathrm{\Gamma }$ states ($\varphi =25\pi /16$, $r=0.41065\pi /2$). $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are the orthorhombic lattice vectors.

Figure 6

Evolution of various angles between different sublattices as we change the parameter $r$ for (a) $\varphi =1.515625\pi$, (b) $\varphi =1.5625\pi$, and (c) $\varphi =1.625\pi$. Data are shown up to the point $r_c\left(\varphi \right)$ where we exit the $K$ region, see Fig. 2.

Figure 7

The two-parameter family of classical ground states along the special line $\varphi =3\pi /2$ of Fig. 2, where $\mathrm{\Gamma }<0$ and $K<0$. The vectors shown at each site are the Cartesian components.

Figure 8

Evolution of the absolute values of the various components of the static structure factor as a function of $r$, as we cross the boundary between the $\mathrm{\Gamma }$ and $K$ states, for (a) $\varphi =1.515625\pi$, (b) $\varphi =1.5625\pi$, and (c) $\varphi =1.625\pi$.

Figure 9

The LSW spectra computed at (a) $P_K$ and (b) $P_{\mathrm{\Gamma }}$ points of Fig. 2 are shown by white solid lines along the high-symmetry paths. The inset shows the magnetic Brillouin zone (in red) along with the high-symmetry points. The vectors ${\mathbf{B}}_1=\frac{2\pi }{a}\widehat{\mathbf{a}}$, ${\mathbf{B}}_2=\frac{2\pi }{a}\widehat{\mathbf{b}}$, and ${\mathbf{B}}_3=\frac{2\pi }{c}\widehat{\mathbf{c}}$ are the reciprocal vectors of the orthorhombic BZ.

Figure 10

Lowest branches ${\omega }_{\nu ,\mathbf{q}=0}$ as we change the parameter $r$ for (a) $\varphi =1.515625\pi$, (b) $\varphi =1.5625\pi$, and (c) $\varphi =1.625\pi$. Magenta and blue lines show the branches inside the $\mathrm{\Gamma }$ and $K$ regions, respectively. Data are shown up to $r_c\left(\varphi \right)$ where we exit the $K$ region, see Fig. 2.

Figure 11

Total intensity $I(\mathbf{Q},\omega )$ shown along $\mathrm{\Gamma }\text{−}X\text{−}{\mathrm{\Gamma }}^{\prime }$ direction of the orthorhombic BZ for [(a) and (b)] $(r,\varphi )=(\frac{3\pi }{8},\frac{25\pi }{16})$, which is the $P_K$ point of Fig. 2, [(c) and (d)] $(r,\varphi )=(0.239\pi ,\frac{97\pi }{64})$, which lies inside the $K$ region of Fig. 2, but close to the boundary with the $\mathrm{\Gamma }$ state, and [(e) and (f)] $(r,\varphi )=(\frac{\pi }{8},\frac{97\pi }{64})$, which is the $P_{\mathrm{\Gamma }}$ point of Fig. 2. The panels in the lower row show the intensity of the four lowest branches only. The LSW spectra are shown by white solid lines. The colors and the width indicate the magnitude of the intensity after convolving the structure factor with a gaussian of finite width to emulate finite experimental resolution. The color scale runs from “blue” color corresponding to the minimum to “red” color corresponding to the maximum of the intensity, and it is independently normalized for each plot.

Figure 12

The diagonal components of the dynamical spin structure factor ${\mathcal{S}}^{aa}(\mathbf{Q},\omega )$, ${\mathcal{S}}^{bb}(\mathbf{Q},\omega )$, and ${\mathcal{S}}^{cc}(\mathbf{Q},\omega )$, along the $\mathrm{\Gamma }\text{−}X\text{−}{\mathrm{\Gamma }}^{\prime }$ direction of the orthorhombic BZ, for (a)–(c) $(r,\varphi )=(\frac{3\pi }{8},\frac{25\pi }{16})$, which is the $P_K$ point of Figs. 2–2 $(r,\varphi )=(0.239\pi ,\frac{97\pi }{64})$, which lies inside the $K$ region of Fig. 2, but close to the boundary with the $\mathrm{\Gamma }$ state, and (g)–(i) $(r,\varphi )=(\frac{\pi }{8},\frac{97\pi }{64})$, which is the $P_{\mathrm{\Gamma }}$ point of Fig. 2. The LSW spectra are shown by white solid lines. The colors and the width indicate the magnitude of the associated component after convolving with a gaussian of finite width to emulate finite experimental resolution. The color scale runs from “blue” color corresponding to the minimum to “red” color corresponding to the maximum of the intensity, and it is independently normalized for each plot.

Figure 13

[(a) and (c)] The numerical values of the local energies ${\mathbf{S}}_i·{\mathbf{h}}_i$ along individual zigzag chains (with $i$ being the site index along the chain), obtained at $P_K$ and $P_{\mathrm{\Gamma }}$ points of Fig. 2, respectively. [(b) and (d)] Corresponding spin-spin correlations ${\mathbf{S}}_i·{\mathbf{S}}_{i+2}$ between odd (even) spins of the zigzag chain. All results are obtained from the nonlinear LLG simulation initialized from the commensurate $\mathbf{Q}=\frac{2}{3}\mathbf{a}$ state.

Figure 14

The modulation of $S^x,S^y$, and $S^z$ spin components along a single $xy$-zigzag chain computed at (a) $P_K$ and (b) $P_{\mathrm{\Gamma }}$ points of Fig. 2. White and gray circles represent spins on even and odd sites of the zigzag chain, respectively. Red and green line segments denote $x$ and $y$ type of NN bonds, respectively. The bottom panels show the chain configuration as in Fig. 3. The FM (AF) dimers in the $K$ state ($\mathrm{\Gamma }$ state) are highlighted by yellow ovals.