Abstract
We investigate the phase diagrams of the spin-orbital $d^9$ Kugel-Khomskii model for a bilayer and a monolayer square lattice using Bethe-Peierls-Weiss method. For a bilayer we obtain valence bond phases with interlayer singlets, with alternating planar singlets, and two entangled spin-orbital (ESO) phases, in addition to the antiferromagnetic and ferromagnetic order. Possibility of such entangled phases in a monolayer is under investigation at present.
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