We investigate thermal Hall conductivity ${\kappa }_{xy}$ of a $J\text{−}K$ Kitaev-Heisenberg model with a Zeeman field in the (111) direction in light of the recent debate surrounding the possible reemergence of Ising topological order (ITO) and half-quantized ${\kappa }_{xy}/T$ upon field suppression of long-range magnetic order in Kitaev materials. We use the purification-based finite-temperature tensor network approach making no prior assumptions about the nature of the excitations: Majorana, visons, or spin waves. For purely Kitaev interactions and fields $h/K\gtrsim 0.02$ sufficient to degrade ITO, the peak ${\kappa }_{xy}/T$ monotonically decreases from half quantization associated with lower fields—a behavior reminiscent of vison fluctuation corrections. For higher fields $h/K\gtrsim 0.1,$ we find the results qualitatively consistent with a spin-wave treatment. In our $J\text{−}K$ model (with ferro-$K$ and antiferro-$J$), in the vicinity of field-suppressed magnetic order, we found ${\kappa }_{xy}/T$ to be significant, with peak magnitudes exceeding half quantization followed by a monotonic decrease with increasing $h.$ We thus conclude that the half-quantized thermal Hall effect in the vicinity of field-suppressed magnetic order in our model is a fine-tuning effect and is not associated with a Majorana Hall state with ITO.

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• Quantum spin liquid
• Thermal Hall effect
• Topological phases of matter
• Quantum spin models
• Kitaev-Heisenberg model
• Tensor network methods