Re-examination of orbital ordering in LaMnO$_3$: Electron-electron and electron-lattice interactions

The interactions between $e_g$ orbitals in neighboring sites are investigated in 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 N{\'e}el temperature by the spin and orbital coupling and the higher-order Jahn-Teller coupling.