  Starting from the Mott insulator picture for cubic vanadates, we derive and
investigate the model of superexchange interactions between V$^{3+}$ ions, with
nearly degenerate $t_{2g}$ orbitals occupied by two electrons each. The
superexchange interactions are strongly frustrated and demonstrate a strong
interrelation between possible types of magnetic and orbital order. We
elucidate the prominent role played by fluctuations of $yz$ and $xz$ orbitals
which generate ferromagnetic superexchange interactions even in the absence of
Hund's exchange. In this limit we find orbital valence bond state which is
replaced either by $C$-type antiferromagnetic order with weak $G$-type orbital
order at increasing Hund's exchange, or instead by $G$-type antiferromagnetic
order when the lattice distortions stabilize $C$-type orbital order. Both
phases are observed in YVO$_3$ and we argue that a dimerized $C$-type
antiferromagnetic phase with stronger and weaker FM bonds alternating along the
c axis may be stabilized by large spin-orbital entropy at finite temperature.
This suggests a scenario which explains the origin of the exotic $C$-AF order
observed in YVO$_3$ in the regime of intermediate temperatures and allows one
to specify the necessary ingredients of a more complete future theory.

Strongly Correlated Electrons