Machine-learned phase diagrams of generalized Kitaev honeycomb magnets

We use a recently developed interpretable and unsupervised machine-learning method, the tensorial kernel support vector machine (TK-SVM), to investigate low-temperature classical phase diagram of generalized Heisenberg-Kitaev-$\Gamma$ ($J$-$K$-$\Gamma$) model on honeycomb lattice. Aside from reproducing phases reported by previous quantum studies, our finds hitherto missed nested zigzag-stripy order establishes robustness identified modulated $S_3 \times Z_3$ phase, which emerges through competition between Kitaev $\Gamma$ spin liquids, against Heisenberg interactions. The results imply that, in restricted parameter space spanned three primary exchange interactions -- $J$, $K$, $\Gamma$, representative material $\alpha$-${\rm RuCl}_3$ lies close boundaries several phases, including simple ferromagnet, unconventional magnets. A zigzag is stabilized finite $\Gamma^{\prime}$ and/or $J_3$ term, whereas four magnetic orders may compete particular if anti-ferromagnetic.