Symmetry analysis of tensors in the honeycomb lattice of edge-sharing octahedra

We obtain the most general forms of rank-2 and rank-3 tensors allowed by the crystal symmetries of the honeycomb lattice of edge-sharing octahedra for crystals belonging to different crystallographic point groups, including the monoclinic point group $2/m$ and the trigonal (or rhombohedral) point group $\overline{3}$. Our results are relevant for two-dimensional materials, such as $\alpha -{\mathrm{RuCl}}_3, {\mathrm{CrI}}_3$, and the honeycomb iridates. We focus on the magnetic-field-dependent thermal conductivity tensor ${\kappa }_{ij}\left(\mathbf{H}\right)$, which describes a system's longitudinal and thermal Hall responses, for the cases when the magnetic field is applied along high-symmetry directions, perpendicular to the plane and in the plane. We highlight some unexpected results, such as the equality of fully longitudinal components to partially transverse components in rank-3 tensors for systems with threefold rotational symmetry, and make testable predictions for the thermal conductivity tensor.

Figure 1

Honeycomb lattice of edge-sharing octahedra and its possible point group generating symmetries. The dark blue circles are magnetic metal ions, and the violet and pink circles surrounding them are ligands above and below the honeycomb plane, respectively, that form octahedra. Each metal ion interacts with its three neighboring metal ions through superexchange mediated by their shared nonmagnetic ligands. This interaction is generally bond-dependent, so we can label the three different types of bonds as $x, y$, and $z$ bonds (shown in red, green, and blue, respectively). The three possible generating symmetries of the crystallographic point groups associated with this lattice are shown in orange. $C_2^{e_2}$ is the twofold (${180}^{\circ }$) rotational symmetry with respect to each $z$-bond axis, $C_3^{e_3}$ is the threefold (${120}^{\circ }$) rotational symmetry with respect to the out-of-plane axis through each site, and $\mathcal{I}$ is the inversion symmetry with respect to each bond center.

Figure 2

Honeycomb lattice for a monoclinic crystal (point groups 2 or $2/m$). This lattice lacks threefold (${120}^{\circ }$) rotational symmetry because $z$ bonds do not have the same length as $x$ and $y$ bonds, and it has twofold (${180}^{\circ }$) rotational symmetry because $x$ and $y$ bonds have the same length. The high-symmetry armchair axis (shown in orange) along $z$ bonds therefore has twofold rotational symmetry, whereas the other two armchair axes along $x$ and $y$ bonds lack this symmetry.

Figure 3

Example illustrating one of the unusual equalities for rank-3 tensors describing crystals with $C_3^{e_3}$ symmetry (i.e., belonging to the trigonal point groups 3, $\overline{3}$, 32, or $\overline{3}m$). In the absence of external magnetic or electric fields, (a) the change in the thermal Hall conductivity $\delta {\kappa }_{e_1{e}_2}$ that results from applying a small magnetic field $\delta \mathbf{H}$ along the zigzag direction $e_1$ will be the same as (b) the change in the longitudinal thermal conductivity $\delta {\kappa }_{e_2{e}_2}$ that results from applying a small magnetic field $\delta \mathbf{H}$ along the armchair direction $e_2$ [Eq. (28)]. ${\mathbf{J}}_Q$ is the heat current and $\mathbf{\nabla }T$ is the temperature gradient. Reversing the direction of any one of the three vectors simply reverses the sign of the change in the thermal conductivity. These results hold when the system is either not magnetized, or magnetized along the out-of-plane direction ($e_3$). Applying Eqs. (23) and (24) using the thermomagnetic susceptibility tensor ${\chi }_{ijk}^{\text{thermomag}}=(\partial {\kappa }_{ij}/\partial H_k){|}_{\mathbf{H}=\mathbf{0}}$ yields several other similar equalities, illustrated in Fig. 5.

Figure 4

Example illustrating the type of symmetry constraint imposed by the Grabner-Swanson equation [Eq. (34)] on systems in an external magnetic or electric field. For a square lattice with fourfold (${90}^{\circ }$) rotational symmetry with respect to the $z$ axis ($C_4^z$) in the presence of an external magnetic field $\mathbf{H}$ applied along the $x$ axis, even though the magnetic field breaks the system's $C_4^z$ symmetry, we can nevertheless use this symmetry to state that for a rank-2 field-dependent tensor $T_{ij}\left(\mathbf{H}\right)$, we must have $T_{ij}\left(\mathbf{H}\right)=T_{ij}^{\prime }\left(\tilde{\mathbf{H}}\right)$, where $T_{ij}^{\prime }\left(\tilde{\mathbf{H}}\right)$ is the tensor expressed in the transformed coordinates (passive transformation) and $\tilde{\mathbf{H}}$ is the transformed field (active transformation). For the element $T_{xz}\left(H\widehat{\mathbf{x}}\right)$, for example, this gives $T_{xz}\left(H\widehat{\mathbf{x}}\right)=T_{yz}\left(H\widehat{\mathbf{y}}\right)$.

Figure 5

Illustration of the two sets of unusual equalities [Eqs. (23) and (24)] for zero-field rank-3 tensors describing crystals with $C_3^{e_3}$ symmetry (i.e., belonging to the trigonal point groups 3, $\overline{3}$, 32, or $\overline{3}m$). The changes in the thermal conductivity that results from applying a small magnetic field $\delta \mathbf{H}$ in scenarios (a1)–(a4) all have the same magnitude, i.e., $|\delta {\kappa }_{e_1{e}_1}(\delta \mathbf{H}\parallel {\widehat{\mathbf{e}}}_1)|=|\delta {\kappa }_{e_1{e}_2}(\delta \mathbf{H}\parallel {\widehat{\mathbf{e}}}_2)|=|\delta {\kappa }_{e_2{e}_1}(\delta \mathbf{H}\parallel {\widehat{\mathbf{e}}}_2)|=|\delta {\kappa }_{e_2{e}_2}(\delta \mathbf{H}\parallel {\widehat{\mathbf{e}}}_1)|$, and similarly for scenarios (b1)–(b4). Reversing the direction of any one of these vectors simply reverses the sign of the change in the thermal conductivity. These results hold when the system is either not magnetized, or magnetized along the out-of-plane direction ($e_3$). Even though these illustrations describe the thermomagnetic susceptibility tensor ${\chi }_{ijk}^{\text{thermomag}}=(\partial {\kappa }_{ij}/\partial H_k){|}_{\mathbf{H}=\mathbf{0}}$ for concreteness, they more generally apply to any general rank-3 tensor describing a system in the absence of external magnetic or electric fields by simply replacing the heat current ${\mathbf{J}}_Q$, temperature gradient $\mathbf{\nabla }T$, and small magnetic field $\delta \mathbf{H}$ with any three vector quantities as long as they are not large external fields (however, small magnetic or electric fields that are used to probe the system, i.e., $\delta \mathbf{H}$ and $\delta \mathbf{E}$, are allowed).