Electron interaction effects on Jahn-Teller instability in LaMnO$_3$*

We investigate the origin of the orbital ordering in LaMnO$_3$ using the LDA+U method and Wannier functions analysis. We find that electron-electron interaction dramatically facilitates the distortion of the MnO$_6$ octahedra and effectively enhances electron-phonon interaction. In addition, electron-electron interaction plays a significant role in determining real lattice distortion and orbital ordering pattern, beyond the conventional Jahn-Teller picture. Our conclusion agrees with existing experimental data and could be directly verified by future measurements, e.g., soft X-ray scattering.