Orbital ordering in ${\mathrm{LaMnO}}_3:$ Electron-electron and electron-lattice interactions

The interactions between $e_g$ orbitals in neighboring sites are investigated in ${\mathrm{LaMnO}}_3$ by taking into account virtual exchange of electrons and phonons. The spin and orbital ordering temperatures and the spin-wave dispersion relation are calculated. We find that the orbital ordering is mainly caused by the electronic interactions and that the Jahn-Teller coupling is much smaller than that reported previously. We propose that the elastic constant shows a characteristic change at the Néel temperature by the spin and orbital couplings and the higher-order Jahn-Teller coupling.