Spin-orbital model of stoichiometric LaMnO$_3$ with tetragonal distortions

The model developed for LaMnO$_3$ addresses the spin-orbital order by superexchange and Jahn-Teller orbital interactions in the cubic (perovskite) symmetry up to now whereas real crystal structure is strongly deformed. We identify and explain three \textit{a priori} important physical effects arising from tetragonal deformation: (i) the splitting of $e_g$ orbitals $\propto E_z$, (ii) the directional renormalization of $d-p$ hybridization $t_{pd}$, and (iii) the directional renormalization of charge excitation energies. Using the example of LaMnO$_3$ crystal we evaluate their magnitude. It is found that the major effects of deformation are enhanced amplitude of $x^2-y^2$ orbitals induced in the orbital order by $E_z\simeq 300$ meV and anisotropic $t_{pd}\simeq 2.0$ (2.35) eV along the $ab$ ($c$) cubic axis, in very good agreement with the Harrison's law. We show that the 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.