Long Range Order in Orbital Model

We investigate the existence of N\'eel type long range order (LRO) in an orbital model which is highly anisotropic and frustrated. The model originated from magnetic materials such as LaMnO$_3$ where orbital degrees of freedom play important role. In the system described by the two-fold degenerate $e_g$ orbitals, due to the Kugel-Khomskii superexchange, the orbital degrees of freedom are represented by quantum pseudo-spin 1/2 operators. By applying the reflection-positivity method developed by Dyson, Lieb, and Simon, and adopting appropriate numerical variational method to obtain good estimations on the energy density and correlation functions, we are able to rigorously prove the existence of long range order in this orbital model on the square lattice.