Toward a Theory of Orbiton Dispersion in LaMnO_3

At 750K, LaMnO_3 has a cooperative Jahn-Teller (JT) distortion, with Mn atoms in distorted oxygen octahedra. This lifts the degeneracy of the singly-occupied $e_g$ orbitals of the Mn$^{3+}$ ions, which then become orbitally ordered. We use a minimal model to describe the ordered phase at T=0. The on-site Coulomb repulsion $U$ is set to infinity. There are two electronic orbitals and three oxygen vibrational coordinates per unit cell. In addition to spin excitations and phonons, the model has electronic excitations consisting of mis-orienting orbitals on Mn ions. Neglecting coupling to the oxygen displacements, the gap to such excitations is $2\Delta=16g^2/K$ where $g$ is the electron-phonon coupling and $K$ is the oxygen spring constant. When static oxygen displacements are coupled, this excitation becomes a self-trapped exciton with energy $\Delta$, half the JT gap. Adding dynamic oxygen displacements in one-phonon approximation introduces dispersion to both the (previously Einstein-like) phonons and the orbital excitons (``orbitons''). One of the phonon branches has zero frequency at $\vec{k}=(0,0,\pi)$. This is the Goldstone mode of the JT broken symmetry.