Complex orbital state in manganites

The e(g)-orbital states with complex coefficients of the linear combination of x(2) -y(2) and 3z(2)-r(2) are studied for the ferromagnetic state in doped manganites. Especially the focus is put on the competition among uniform complex, staggered complex, and real orbital states. As the hole-doping x increases, the real, the canted complex, and the staggered complex orbital states appears successively. Uniform complex state analoguous to Nagaoka ferromagnet does not appear. These complex states can be expressed as a resonating state among the planer orbitals as the orbital liquid, accompanied by no Jahn-Teller distortion.