Chiral spin order in some purported Kitaev spin-liquid compounds

We examine recent magnetic torque measurements in two compounds, $\gamma -{\mathrm{Li}}_2{\mathrm{IrO}}_3$ and ${\mathrm{RuCl}}_3$, which have been discussed as possible realizations of the Kitaev model. The analysis of the reported discontinuity in torque, as an external magnetic field is rotated across the $c$ axis in both crystals, suggests that they have a translationally invariant chiral spin order of the form $\langle {\mathbf{S}}_i·\left({\mathbf{S}}_j\times {\mathbf{S}}_k\right)\rangle \ne 0$ in the ground state and persisting over a very wide range of magnetic field and temperature. An extraordinary $\left|B\right|B^2$ dependence of the torque for small fields, beside the usual $B^2$ part, is predicted by the chiral spin order. Data for small fields are available for $\gamma -{\mathrm{Li}}_2{\mathrm{IrO}}_3$ and are found to be consistent with the prediction upon further analysis. Other experiments such as inelastic scattering and thermal Hall effect and several questions raised by the discovery of chiral spin order, including its topological consequences, are discussed.

Figure 1

Magnetic field evolution of the angle-dependent torque at low temperatures in $\gamma -{\mathrm{Li}}_2{\mathrm{IrO}}_3$. The dots in the three panels give the angle dependence in three different field regions. In panel (a), the low-field region in which the AFM order is preserved for any angle of the applied field. Panel (b) shows the highest field region in which the AFM order is absent for field at any angle. Panel (c) shows the intermediate field region in which the AFM order is suppressed above a field $H^{*}\left(\theta \right)$. The function $H^{*}\left(\theta \right)$ is shown in Fig. 2 of Ref. [5]. The solid curves in all three panels are best fits to the data with the functional form given in panel (b). The field dependence of the coefficients $A$ and $A_2$ are given in Fig. 3. The discrepancies of the fit in the intermediate field region are discussed in the text. The data as function of magnetic field at fixed angles have been shown in Refs. [4, 5].

Figure 2

Magnetic field evolution of the angle-dependent torque at low temperatures in ${\mathrm{RuCl}}_3$. The first panel (a) shows the field dependence at various fixed angles and the second panel (b) shows the angle dependence at various fixed fields. More comprehensive results in ${\mathrm{RuCl}}_3$, including for the discontinuity at near $\theta =\pi /2$, have been obtained recently, through measurements of the magnetotropic coefficient [16, 17]. and are consistent with the results shown here.

Figure 3

(a) The coefficients $A$ and $A_2$, determined by fitting the angle dependence of the $\tau /B$ in Fig. 1 and more such data at fixed temperature to $A\text{sign}\left(\text{cos}\theta \right)\text{sin}\theta +A_2\text{sin}2\theta$, as a function of magnetic field. (b) The low-field dependence of $A$ and $A_2$. $A$ is multiplied by 10 for viewing on the $A_2$ scale. $A$ plotted against $B^2$ (inset) displays the $B^3$ dependence of the anomalous component of the torque at low fields. The shaded region in the first figure shows the region in which AFM is found below $B<B^{*}\left(\theta \right)$; the latter varies from 3 T for field in the hexagonal planes to 18 T for field normal to them [4].