Planar thermal Hall effects in the Kitaev spin liquid candidate ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$

We investigate both the longitudinal thermal conductivity $\left({\kappa }_{xx}\right)$ and the planar thermal Hall conductivity $\left({\kappa }_{xy}\right)$ in the Kitaev spin liquid candidate of the cobalt-based honeycomb antiferromagnet ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ in a magnetic field $\left(B\right)$ applied along the $a$- and $a^{*}$-axes. A finite ${\kappa }_{xy}$ is resolved for both field directions in the antiferromagnetic (AFM) phase below the Néel temperature of 27 K. The temperature dependence of ${\kappa }_{xy}/T$ shows the emergence of topological bosonic excitations. In addition, the field dependence of ${\kappa }_{xy}$ shows sign reversals at the critical fields in the AFM phase, suggesting the changes in the Chern number distribution of the topological magnons. Remarkably, a finite ${\kappa }_{xy}$ is observed in $B\parallel a^{*}$ between the first-order transition field in the AFM phase and the saturation field, which is prohibited in a disordered state by the two-fold rotation symmetry around the $a^{*}$ axis of the honeycomb lattice, showing the presence of a magnetically ordered state that breaks the two-fold rotation symmetry. Our results demonstrate the presence of topological magnons in this compound in the whole field range below the saturation field.

Figure 1

The temperature dependence of the longitudinal thermal conductivity (${\kappa }_{xx}$) under various magnetic fields applied along the $a$- (a) and $a^{*}$- (b) axes obtained in the VTI measurements. The inset in (a) illustrates the honeycomb lattice of ${\mathrm{Co}}^{2+}$ ions with the crystallographic axes.

Figure 2

The field-induced deviation of the longitudinal thermal conductivity [${\kappa }_{xx}\left(B\right)-{\kappa }_{xx}\left(0\right)$] normalized by the zero-field value under magnetic fields applied along the $a$- (a) and $a^{*}$- (b) axes. For clarity, the vertical axis of the data $T\le 15$ K is shifted. The data obtained in the field-up and field-down procedure is shown by filled and open symbols, respectively.

Figure 3

The field dependence of the thermal Hall conductivity divided by the temperature for $B\parallel a$ (a) and $B\parallel a^{*}$ (b) at different temperatures. The data are shifted by a constant amount for clarity. The error bars show the deviation of the data in the field-up and field-down measurements.

Figure 4

The temperature dependence of ${\kappa }_{xy}/T$ (a) and ${\kappa }_{xx}/T$ (b) at 7, 9, and 11 T for $B\parallel a$, and at 9 T for $B\parallel a^{*}$. The temperature dependence of ${\kappa }_{xx}/T$ at 10 T, the same data shown in Fig. 1, is also shown for comparison.

Figure 5

(a, b) A comparison of the field dependence of the longitudinal thermal conductivity (${\kappa }_{xx}$) (a) and the planar thermal Hall conductivity (${\kappa }_{xy}$) (b) for $B\parallel a$ (black) and $B\parallel a^{*}$ (blue) at 5 K. The critical fields are also marked by dashed lines. The arrows indicate the field used in the data of the same color in Fig. 4. (c, d) The suggested magnetic phases of ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ in $B\parallel a$ (c) and $B\parallel a^{*}$ (d), drawn from our thermal transport measurements.