Field-angle dependence of thermal transport in Kitaev-Heisenberg systems

The Kitaev model, which describes magnetic interactions of quantum spins with $S=1/2$ on a honeycomb lattice, has recently attracted considerable attention not only in condensed matter physics but also in quantum information. In this model, the ground state is exactly shown to be a quantum spin liquid (QSL) with the fractionalization of spins into Majorana fermions and the half-integer quantization of the thermal Hall conductivity is predicted as a consequence of a Majorana edge mode. As candidate materials of the model, $\alpha \textrm{-} \mathrm{RuCl_3}$ and $\mathrm{A_2 IrO_3}$ $\mathrm{(A=Na,Li)}$ have been studied energetically. However, these materials do not show a QSL ground state but exhibit a magnetic order at low temperatures, which is due to additional interactions, such as Heisenberg and off-diagonal terms. Recently, in $\alpha \textrm{-} \mathrm{RuCl_3}$, the magnetic order was suppressed by applying a magnetic field in the honeycomb plane and the half-quantized plateau has been observed in the thermal Hall conductivity, which offers convincing evidence for a Majorana chiral edge mode. Stimulated by this experimental result, magnetic field effects on the Kitaev-related systems have intensively investigated. For example, classical phase diagrams of the Kitaev-related systems under the magnetic field have been obtained. In the phase diagrams, various magnetically ordered states, including noncollinear and noncoplanar spin configurations, appear by applying the magnetic field. Nevertheless, the thermal transport properties remain unclear in magnetic orders induced by the magnetic field.
In this study, we investigate magnetic-field effects on transport properties of magnetically ordered states in the Kitaev-Heisenberg systems with $S=1/2$ spins on a honeycomb lattice. By applying mean-field approximation and liner spin-wave theory, we calculate the magnon dispersion and the thermal Hall conductivity originating from the magnons by changing the direction and strength of an applied magnetic field. We reveal that the thermal Hall conductivity is enhanced in magnetically ordered states with noncoplanar spin configurations. Moreover, we also find that the thermal Hall effect originating from the topological nature of the magnon dispersion is absent when the field is applied at nodal angles where $h_x h_y h_z= 0$ in the Kitaev-Heisenberg system. We demonstrate that the Berry curvature generated by the magnons is zero and the feature does not depend on the strength of the Kitaev and Heisenberg interactions and the types of magnetic orders. This phenomenon is intrinsic to the Kitaev-Heisenberg systems and violated by introducing off-diagonal spin interactions, such as the $\Gamma$ term. The disappearance of the thermal Hall effect originates from the properties of the Hamiltonian, and the argument is applicable even in the case beyond the liner spin-wave theory. Thus, we expect that the strength of off-diagonal interactions can be deduced from the field-angle dependence of the thermal Hall conductivity in experiments.