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

Abstract

We compute the rotational anisotropy of the free energy of $\alpha$-RuCl$_3$ in an external magnetic field. This quantity, known as the magnetotropic susceptibility, $k$, relates to the second derivative of the free energy with respect 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 negative sign problem. This allows us to reach temperatures down to $30~\mathrm{K}$, an energy scale below the dominant Kitaev coupling. We demonstrate that the magnetotropic spin susceptibility in this model of $\alpha$-RuCl$_3$ displays scaling behavior $k = T f(B/T)$ at high temperatures. Once the uniform susceptibility departs from the Curie law (i.e., at the energy scale of the exchange interactions), it appears to transition to an emergent scalinglike behavior, characterized by a different function $f$ at lower temperatures, stemming from the locality of torque fluctuations. We observe a remarkable numerical match between experiment and simulations and we also find qualitative agreement with the pure Kitaev model. In comparison, for the XXZ Heisenberg Hamiltonian, the scaling $k = T f(B/T)$ breaks down at a temperature scale where the uniform spin susceptibility deviates from the Curie law and never reemerges at low temperatures.