Abstract
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.
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