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Orbital ordering in e_g orbital systems: Ground states and thermodynamics of the 120 degree model

Orbital degrees of freedom shape many of the properties of a wide class of Mott insulating, transition metal oxides with partially filled 3d-shells. Here we study orbital ordering transitions in systems where a single electron occupies the e_g orbital doublet and the spatially highly anisotropic orbital interactions can be captured by an orbital-only model, often called the 120 degree model. Our analysis of both the classical and quantum limits of this model in an extended parameter space shows that the 120 degree model is in close proximity to several T=0 phase transitions and various competing ordered phases. We characterize the orbital order of these nearby phases and their associated thermal phase transitions by extensive numerical simulations and perturbative arguments.