Effect of strong electron correlations and electron-lattice interaction on the ordering of electron orbitals

The interactions between electrons in degenerate orbitals and between electrons and the lattice degrees of freedom have been treated by the Monte Carlo method taking into account the anharmonicity of ionic vibrations in the framework of the phenomenological model. The existence regions of the ferromagnetic and antiferromagnetic orbital orders have been found. Two critical temperatures have been determined at which the long-range orbital order disappears and the distribution of electrons over the orbitals becomes uniform. The correlation between the orbital ordering types and the thermal expansion coefficient has been established. It has been found that an increase in anharmonicity leads to the stabilization of the antiferromagnetic orbital ordering.