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We compute the rotational anisotropy of the free energy of $\alpha$-RuCl$_3$in an external magnetic field. This quantity, known as the magnetotropicsusceptibility, $k$, relates to the second derivative of the free energy withrespect to the angle of rotation. We have used approximation-free,auxiliary-field quantum Monte Carlo simulations for a realistic model of$\alpha$-RuCl$_3$ and optimized the path integral to alleviate the negativesign problem. This allows us to reach temperatures down to $30~\rm{K}$ -- anenergy scale below the dominant Kitaev coupling. We demonstrate that themagnetotropic susceptibility in this model of $\alpha$-RuCl$_3$ displays uniquescaling, $k = Tf(B/T)$, with distinct scaling functions $f$ at high and lowtemperatures. In comparison, for the XXZ Heisenberg model, the scaling $k =Tf(B/T)$ breaks down at a temperature scale where the uniform spinsusceptibility deviates from the Curie law (i.e. at the energy scale of theexchange interactions) and never recovers at low temperatures. Our findingssuggest that correlations in $\alpha$-RuCl$_3$ lead to degrees of freedom thatrespond isotropically to a magnetic field. One possible interpretation for theapparent scale-invariance observed in experiments could be fractionalization ofthe spin degrees of freedom in the extended Kitaev model.

Scale-invariant magnetic anisotropy in $\alpha$-RuCl$_3$: A quantum Monte Carlo study