Fig. 3: The theoretical calculation of the thermal Hall conductivity.

a \({{{{{\rm{Im}}}}}}\langle {H}_{-{{{{{\bf{q}}}}}}};{{{{{{\bf{J}}}}}}}_{{{{{{\bf{q}}}}}}}^{{{{{\rm{{E}}}}}}}\rangle\) appearing in Eq. (4). One can see the vortex-like structure in the momentum space. The color bar is given in an arbitrary unit. b The temperature dependence of the z-component of \({{{{{{\boldsymbol{\mu }}}}}}}^{E}\) defined in Eq. (4). The triangular dotted curve shows the high-temperature expansion result, and the blue and red points show the numerically obtained results with different momentum cut-offs. c The energy current correlation function \({\widetilde{C}}_{{xy}}\left(t\right)=({C}_{{xy}}\left(t\right)-{C}_{{yx}}\left(t\right))/2\) with \({C}_{\mu \nu }\left(t\right)=\langle {J}^{{{{{{\rm{E}}}}}},\mu }\left(t\right){J}^{{{{{{\rm{E}}}}}},\nu }\left(t\right)\rangle\) calculated by the stochastic LLG equation. The fitting by the superposition of the damped oscillators is also shown. d The obtained result of the thermal Hall conductivity as a function of temperature (solid blue squares) together with the experimental values for two samples. S1 and S2 indicate sample 1 and sample 2, respectively. All error bars are standard deviation from our calculations.