Orbital fluctuations in the $R$VO$_3$ perovskites

The properties of Mott insulators with orbital degrees of freedom are described by spin-orbital superexchange models, which provide a theoretical framework for understanding their magnetic and optical properties. We introduce such a model derived for $(xy)^1(yz/zx)^1$ configuration of V$^{3+}$ ions in the $R$VO$_3$ perovskites, $R$=Lu,Yb,$\cdots$,La, and demonstrate that $\{yz,zx\}$ orbital fluctuations along the $c$ axis are responsible for the huge magnetic and optical anisotropies observed in the almost perfectly cubic compound LaVO$_3$. We argue that the GdFeO$_3$ distortion and the large difference in entropy of $C$-AF and $G$-AF phases is responsible for the second magnetic transition observed at $T_{N2}$ in YVO$_3$. Next we address the variation of orbital and magnetic transition temperature, $T_{\rm OO}$ and $T_{N1}$, in the $R$VO$_3$ perovskites, after extending the spin-orbital model by the crystal-field and the orbital interactions which arise from the GdFeO$_3$ and Jahn-Teller distortions of the VO$_6$ octahedra. We further find that the orthorhombic distortion which increases from LaVO$_3$ to LuVO$_3$ plays a crucial role by controlling the orbital fluctuations, and via the modified orbital correlations influences the onset of both magnetic and orbital order.