Orbital liquid in ferromagnetic manganites: The orbital Hubbard model for $e_g$ electrons

We have analyzed the symmetry properties and the ground state of an orbital Hubbard model with two orbital flavors, describing a partly filled spin-polarized $e_g$ band on a cubic lattice, as in ferromagnetic manganites. We demonstrate that the off-diagonal hopping responsible for transitions between $x^2-y^2$ and $3z^2-r^2$ orbitals, and the absence of SU(2) invariance in orbital space, have important implications. One finds that superexchange contributes in all orbital ordered states, the Nagaoka theorem does not apply, and the kinetic energy is much enhanced as compared with the spin case. Therefore, orbital ordered states are harder to stabilize in the Hartree-Fock approximation (HFA), and the onset of a uniform ferro-orbital polarization and antiferro-orbital instability are similar to each other, unlike in spin case. Next we formulate a cubic (gauge) invariant slave boson approach using the orbitals with complex coefficients. In the mean-field approximation it leads to the renormalization of the kinetic energy, and provides a reliable estimate for the ground state energy of the disordered state. Using this approach one finds that the HFA fails qualitatively in the regime of large Coulomb repulsion $U\to\infty$ -- the orbital order is unstable, and instead a strongly correlated orbital liquid with disordered orbitals is realized at any electron filling.