We develop a real-space computational framework that integrates Chebyshev polynomial expansions with generalized Bogoliubov transformations to address anomalous quadratic interactions in magnon systems. The method enables efficient calculation of thermal transport coefficients through random trace evaluation and kernel damping techniques, significantly reducing computational costs compared to exact diagonalization methods. Applied to ${\mathrm{CrI}}_3$ ferromagnets, this approach reveals a temperature-driven sign reversal in thermal Hall effect that reflects the topology of magnon bands. The framework facilitates the implementation of real-space perturbations, for example, disorder, which slightly smears the sign-reversal behavior while markedly suppressing saturation magnitudes of thermal Hall conductivity. By incorporating thermal fluctuations near the Curie temperature, we can qualitatively reproduce experimental trends without introducing phenomenological parameters. The real-space implementation provides a universal platform for investigating topological thermal transport in magnetic systems with complex interactions.

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• Magnons
• Quantum transport
• Thermal conductivity
• Ferromagnets