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New Journal of Physics

Long-range orbital order in a degenerate-orbital Hubbard model: absence in low-dimensions

Canio Noce

Published Jul 20, 2007• 4 citations
Publisher

Abstract

We investigate the possibility of long range orbital order in low- dimensional systems. The study is performed within a two-orbital Hubbard model by means of the Bogoliubov's inequality. We show that, in one- and two-dimensions, and when a narrow conduction band is concerned, this model does not exhibit orbital order at any filling and any finite temperature.

DOI
10.1088/1367-2630/9/7/238

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We investigate the possibility of long range orbital order in low- dimensional systems. The study is performed within a two-orbital Hubbard model by means of the Bogoliubov's inequality. We show that, in one- and two-dimensions, and when a narrow conduction band is concerned, this model does not exhibit orbital order at any filling and any finite temperature.
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We studied ferromagnetism in the one-dimensional Hubbard model with doubly degenerate atomic orbitals by means of the density-matrix renormalization-group method and obtained the ground-state phase diagrams. It was found that ferromagnetism is stable from low to high (0<n<1.75)(0<n<1.75)(0<n<1.75) electron density when the interactions are sufficiently strong. The quasi-long-range order of triplet superconductivity coexists with the ferromagnetic order for a strong Hund coupling region, where the interorbital interaction ${U}^{\ensuremath{'}}\ensuremath{-}J$ is attractive. At quarter-filling (n=1),(n=1),(n=1), the insulating ferromagnetic state appears, accompanying orbital quasi-long-range order. For low densities (n<1),(n<1),(n<1), ferromagnetism occurs owing to the ferromagnetic exchange interaction caused by virtual hoppings of electrons, the same as in the quarter-filled system. This comes from separation of the charge and spin-orbital degrees of freedom in the strong-coupling limit. This ferromagnetism is fragile against variation of band structure. For high densities (n>1),(n>1),(n>1), the phase diagram of the ferromagnetic phase is similar to that obtained in infinite dimensions. In this case, the double exchange mechanism is operative for stabilizing the ferromagnetic order and this long-range order is robust against variation of the band dispersion. A partially polarized state appears in the density region $1.68\ensuremath{\lesssim}n\ensuremath{\lesssim}1.75$ and phase separation occurs for n just below the half-filling (n=2).(n=2).(n=2).
On the basis of Bogoliubov's 1/k2 theorem the relation between the long-range order in the ground state and the collective mode, namely, the Goldstone particle is investigated. It is pointed out that Bogoliubov's inequality rules out the long-range orders in the ground states of the isotropic Heisenberg model, the hafl-filled Hubbard model and the interacting Bose system for one dimension while it admits the long-range orders for two dimensions. The present proof is based on the fact that the lowest-excited state that can be regarded as the Goldstone particle has the energy E(k)∝|k| for small k. This energy spectrum is exactly given in the one-dimensional models and is shown to be proven in the ordered state on a reasonable assumption except for the ferromagnetic case.
A two-dimensional model of interacting electrons on a lattice, the Hubbard model, has been studied by means of a numerical simulation technique. Results for the half-filled-band sector are presented for lattices of up to 6\ifmmode\times\else\texttimes\fi{}6 spatial sites, as a function of temperature and electron-electron repulsion UUU. In the ground state, the system is found to exhibit antiferromagnetic long-range order for all values of UUU, although greatly reduced from the mean-field-theory predictions.