  Ring-exchange (RE) interaction in Mott insulator with doubly degenerate $e_g$
orbitals is studied. Hamiltonian for the RE interaction is derived by the
perturbational calculation. A remarkably weak RE interaction destroys the
orbital order caused by the order-by-fluctuation mechanism in the
nearest-neighbor exchange-interaction model. A long range order of magnetic
octupole moment, i.e. complex orbital wave functions, appears. Competition
between octupole and quadrupole interactions strongly suppresses ordered
moments.

Strongly Correlated Electrons;Materials Science