  We investigate the phase diagrams of the spin-orbital $d^9$ Kugel-Khomskii
model for increasing system dimensionality: from the square lattice monolayer,
via the bilayer to the cubic lattice. In each case we find strong competition
between different types of spin and orbital order, with entangled spin-orbital
phases at the crossover from antiferromagnetic to ferromagnetic correlations in
the intermediate regime of Hund's exchange. These phases have various types of
exotic spin order and are stabilized by effective interactions of longer range
which follow from enhanced spin-orbital fluctuations. We find that orbital
order is in general more robust and spin order melts first under increasing
temperature, as observed in several experiments for spin-orbital systems.

Strongly Correlated Electrons;Quantum Physics