Orbital liquid in the \(e_g\) orbital Hubbard model in \(d=\infty\) dimensions

We demonstrate that the three-dimensional \(e_g\) orbital Hubbard model can be generalized to arbitrary dimension \(d\), and that the form of the result is determined uniquely by the requirements that (i) the two-fold degeneracy of the \(e_g\) orbital be retained, and (ii) the cubic lattice be turned into a hypercubic lattice. While the local Coulomb interaction \(U\) is invariant for each basis of orthogonal orbitals, the form of the kinetic energy depends on the orbital basis and takes the most symmetric form for the so-called complex-orbital basis. Characteristically, with respect to this basis, the model has two hopping channels, one that is orbital-flavor conserving, and a second one that is orbital-flavor non-conserving. We show that the noninteracting electronic structure consists of two nondegenerate bands of plane-wave real-orbital single-particle states for which the orbital depends on the wave vector. Due to the latter feature each band is unpolarized at any filling, and has a non-Gaussian density of states at \(d=\infty\). The \textit{orbital liquid} state is obtained by filling these two bands up to the same Fermi energy. We investigate the \(e_g\) orbital Hubbard model in the limit \(d\to\infty\), treating the on-site Coulomb interaction \(U\) within the Gutzwiller approximation, thus determining the correlation energy of the orbital liquid and the (disordered) para-orbital states. (...) We show that the orbital liquid is the ground state everywhere in the \((n,U)\) phase diagram except close to half-filling at sufficiently large \(U\), where ferro-orbital order with real orbitals occupied is favored. The latter feature is shown to be specific for \(d=\infty\), being of mathematical nature due to the exponential tails in the density of states.