Revisiting spin Hamiltonian parameters in a Kitaev material via Bayesian optimization of magnetization curves

Determining the spin Hamiltonian of a magnetic compound is crucial for understanding its magnetic properties. A standard approach is to derive model parameters from $ab$ $initio$ calculations based on the crystal structure. However, the resulting Hamiltonian can depend sensitively on methodological details of the $ab$ $initio$ procedure. This issue is particularly evident in $\alpha$-RuCl$_3$, a candidate Kitaev material. Here, we present an alternative, data-driven approach to determine the spin Hamiltonian parameters of $\alpha$-RuCl$_3$ by Bayesian optimization of experimental magnetization curves along the $b$- and $c$-axis directions. We optimize five parameters, namely the Kitaev interaction $K$, off-diagonal interactions $\Gamma$ and $\Gamma'$, the Heisenberg interaction $J$, and the $c$-axis $g$-factor $g_c$. The parameter set that minimizes the cost function is $(K,\Gamma,\Gamma',J,g_c)=(-6.0,\,7.5,\,-0.3,\,-1.75,\,2.3)$, where the exchange couplings are in meV. We find that the cost function is insensitive to the absolute value of the Kitaev coupling $K$. Thus, the magnetization data alone do not determine its energy scale. The cost function also depends only weakly on $\Gamma'$ and $J$, while the optimization favors a large positive $\Gamma$. By computing the static spin structure factor, magnetic susceptibility, and specific heat, we show that these quantities favor the large-$\Gamma$ scenario over the small-$g_c$ scenario and that the parameter set that minimizes the cost function yields good agreement with experiment. The combination of Bayesian optimization and accurate low-energy solvers provides an effective approach for determining parameters of spin Hamiltonians. This methodology opens a systematic route to determining spin Hamiltonians in quantum magnets from experimental data.