  The spin-orbital entangled states are of great interest as they hold exotic
phases and intriguing properties. Here we use first-principles calculations to
investigate the electronic and magnetic properties of RuI$_{3}$ and RuCl$_{3}$
in both bulk and monolayer cases. Our results show that RuI$_{3}$ bulk is a
paramagnetic metal, which is in agreement with recent experiments. We find that
the Ru$^{3+}$ ion of RuI$_{3}$ is in the spin-orbital entangled $j_{\rm
eff}=\frac{1}{2}$ state. More interestingly, a metal-insulator transition
occurs from RuI$_{3}$ bulk to monolayer, and this is mainly due to the band
narrowing with the decreasing lattice dimensionality and to the Ru-I
hybridization altered by the I $5p$ spin-orbit coupling. In contrast,
RuCl$_{3}$ bulk and monolayer both show Mott-insulating behavior, the Ru$^{3+}$
ion is in the formal $S=\frac{1}{2}$ and $L=1$ state with a large in-plane
orbital moment, and this result well explains the experimental large effective
magnetic moment of RuCl$_{3}$ and the strong in-plane magnetization. The
present work demonstrates the contrasting spin-orbital states and the varying
properties of RuI$_{3}$ and RuCl$_{3}$.
