Spin-orbital model of stoichiometric LaMnO3 with tetragonal distortions

The spin-orbital superexchange model is derived for the cubic (perovskite) symmetry of ${\mathrm{LaMnO}}₃$, whereas real crystal structure is strongly deformed. We identify and explain three a priori important physical effects arising from tetragonal deformation: (i) the splitting of ${e}_g$ orbitals $\ensuremath{\propto}{E}_z$, (ii) the directional renormalization of $d\text{\ensuremath{-}}p$ hybridization ${t}_pd$, and (iii) the directional renormalization of charge excitation energies. Using the example of ${\mathrm{LaMnO}}₃$ crystal we evaluate their magnitude. It is found that the major effects of deformation are an enhanced amplitude of ${x}²\text{\ensuremath{-}}{y}²$ orbitals induced in the orbital order by ${E}_z\ensuremath{\simeq}300$ meV and anisotropic ${t}_pd\ensuremath{\simeq}2.0$ (2.35) eV along the $ab$ ($c$) cubic axis, in very good agreement with Harrison's law. We show that the improved tetragonal model analyzed within mean field approximation provides a surprisingly consistent picture of the ground state. Excellent agreement with the experimental data is obtained simultaneously for: (i) ${e}_g$ orbital mixing angle, (ii) spin exchange constants, and (iii) the temperatures of spin and orbital phase transition.