Orbital Liquid in Perovskite Transition-Metal Oxides

We study the effects of the degeneracy of the $e_g$ orbitals as well as the double exchange interaction with $t_{2g}$ spins in perovskite transition-metal oxides. In addition to the spin field ${\vec S}_i$, the isospin field ${\vec T}_i$ is introduced to describe the orbital degrees of freedom. The isospin is the quantum dynamical variable, and is represented by the boson with a constraint. The dispersion of this boson is flat along $(\pi/a,\pi/a,k_z)$ ($a$: lattice constant) and the other two equivalent directions. This enables the orbital disordered phase down to low temperatures. We interpret some of the anomalous experiments, i.e., optical absorption and d.c. resistivity, in the low temperature ferromagnetic phase of La$_{1-x}$Sr$_x$MnO$_3$ with $x > 0.2$ in terms of this orbital liquid picture.