Orbital-Spin Structure and Lattice Coupling in $R{\mathrm{T}\mathrm{i}\mathrm{O}}_3$ where $R=\mathrm{L}\mathrm{a}$, $\mathrm{P}\mathrm{r}$, $\mathrm{N}\mathrm{d}$, and $\mathrm{S}\mathrm{m}$

The origin of the G-type antiferromagnetism [AFM(G)] and puzzling properties of $R\mathrm{T}\mathrm{i}{\mathrm{O}}_{\mathrm{3}}$ with $R=\mathrm{L}\mathrm{a}$ are studied. We clarify that the crystal field from La caused by the $\mathrm{G}\mathrm{d}\mathrm{F}\mathrm{e}{\mathrm{O}}_{\mathrm{3}}$-type distortion lifts the $t_{2g}$ degeneracy at Ti $3d$ orbitals. The lift stabilizes the AFM(G) with spin-exchange constant in agreement with neutron-scattering results. The orbital-spin structures for $R=\mathrm{P}\mathrm{r}$, Nd, and Sm are also consistent with experiments. We propose that the $\mathrm{G}\mathrm{d}\mathrm{F}\mathrm{e}{\mathrm{O}}_{\mathrm{3}}$-type distortion has a universal mechanism of controlling orbital-spin structure competing with the Jahn-Teller (JT) mechanism.