Abstract: In this work, we propose a many-body model for cooperative Jahn-Teller/orbital ordering in crystals containing unpaired electrons in locally degenerate e-shells (KCuF3, CuO, CuCl2, ») that moves away from the usual Heisenberg-like expressions for solids and that is closely related to the usual local Jahn-Teller formulation for molecules and impurities. In particular, we show that the crystal problem can be expressed as a [Eg(?)+Eg(R)] - [egst(?)+egph(R)] combined Jahn-Teller and pseudo-Jahn-Teller problem where the involved distortion modes egst(?) and egph(R) are, respectively, a ferrodistortive strain mode and a phonon (ferro- or antiferrodistortive) mode that are strongly coupled among themselves. This model, fully consistent with experimental data, allows us to predict new phases that are separated by only a few meV from the ground state, and to propose ways to observe them. First-principles simulations fully support these findings. Comparison of our model with previous orbital ordering models shows subtle yet significant differences in the way orbital degrees of freedom work. We finally show that many orbital ordering models and density functional theory simulations yield solutions that are not consistent with Bloch's theorem and should be regarded with caution.

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