Long-range order in the ground state of spin systems with orbital degeneracy

We apply the method of infrared bounds to the spin-orbital systems with isotropic and anisotropic interactions to examine the existence of long-range order (LRO) in two or more dimensions. For the isotropic case, we prove the existence of antiferro-spin and antiferro-orbital LRO in three or more dimensions and ferro-spin-orbital LRO in seven or more dimensions. For the anisotropic case, we prove the existence of antiferro-orbital LRO in two or more dimensions. Under some acceptable assumptions we conjecture on the existence of antiferro-spin LRO in three or more dimensions and extend the region where we can show antiferro-orbital LRO in two or more dimensions.