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Orbital liquid in the eg orbital Hubbard model in d=∞ dimensions

Louis Felix Feiner

Andrzej M. Oleś

  • Institute for Theoretical Physics, Utrecht University, PO Box 80.000, 3508 TA Utrecht, Netherlands

  • Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany and Institute of Theoretical Physics, Jagiellonian University, Prof. Stanisława Łojasiewicza 11, PL-30348 Kraków, Poland

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Phys. Rev. Research 4, 043134 – Published 28 November, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.043134

Abstract

We demonstrate that the three-dimensional eg 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 twofold degeneracy of the eg 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 nonconserving. 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=∞. The orbital liquid state is obtained by filling these two bands up to the same Fermi energy. We investigate the eg orbital Hubbard model in the limit d→∞, treating the on-site Coulomb interaction U within the Gutzwiller approximation, thus determining the correlation energy of the orbital liquid and the (disordered) paraorbital states. In perfect analogy with the case of the spin Hubbard model, the Gutzwiller approximation is demonstrated to be exact at d=∞ for the orbital Hubbard model, because of the collapse of electron correlations to a single site. At half-filling (one electron per site on average, n=1) one finds a Brinkman-Rice type “metal-insulator” transition in the orbital liquid, which is analogous to the transition for a paramagnetic state in the spin model, but occurs at stronger Hubbard interaction U due to the enhanced kinetic energy provided by the nonconserving hopping channel. 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=∞, being of mathematical nature due to the exponential tails in the density of states.

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Phys. Rev. Research 4, 043134– Published 28 November, 2022
  • Received 11 June 2022
  • Revised 31 October 2022
  • Accepted 3 November 2022
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