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The European Physical Journal Plus

Spin-orbital correlations in the two-orbital Hubbard model

Alfonso Romano, Delia Guerra, Filomena Forte, Canio Noce

Published May 31, 2023• 2 citations
Publisher

Abstract

Abstract We use exact diagonalization results to investigate how the ground state of a two-site, two-orbital Hubbard model evolves in terms of the crystal field energy and a magnetic spin symmetry breaking field. We show that, based on the behavior of a properly defined composite spin-orbital correlation function, the different regions of the related phase diagrams can be classified according to the extent to which spin and orbital degrees of freedom turn out to be correlated.

DOI
10.1140/epjp/s13360-023-04047-5

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Abstract We use exact diagonalization results to investigate how the ground state of a two-site, two-orbital Hubbard model evolves in terms of the crystal field energy and a magnetic spin symmetry breaking field. We show that, based on the behavior of a properly defined composite spin-orbital correlation function, the different regions of the related phase diagrams can be classified according to the extent to which spin and orbital degrees of freedom turn out to be correlated.
We study the magnetic phase diagram of a one-dimensional, orbitally degenerate Hubbard model using the Lanczos algorithm to project out the ground state. Using lattices larger than those previously explored with diagonalization approaches, we find a phase diagram which is in much better agreement with a strong coupling analysis. In addition to determining the phase diagram by calculating the spin sector of minimum energy, we also show results for the spin-spin correlations and participation ratio, which further characterize the magnetic order and the itineracy of the electrons. We then describe results for the case when the intersite hybridization is different for the two orbitals.
The ground-state phase diagrams of the three-orbital t2g{t}_{2\mathrm{g}}t2g​ Hubbard model are studied using a Hartree-Fock approximation. First, a complete set of multipolar order parameters for t2g{t}_{2\mathrm{g}}t2g​ models defined in terms of the effective total angular momentum jeff{j}_{\mathrm{eff}}jeff​ is theoretically derived. These order parameters can classify off-diagonal orders between jeff=1/2{j}_{\mathrm{eff}}=1/2jeff​=1/2 and jeff=3/2{j}_{\mathrm{eff}}=3/2jeff​=3/2 manifolds. Second, through extensive Hartree-Fock calculations, the ground-state phase diagrams in the space of (1) the on-site Coulomb repulsion UUU, (2) the spin-orbit coupling $\ensuremath{\lambda}$, and (3) the number of electrons nnn are mapped out. A variety of nontrivial quantum phases with jeff{j}_{\mathrm{eff}}jeff​-diagonal and jeff{j}_{\mathrm{eff}}jeff​--off-diagonal multipole orders are found. Finally, future studies using more numerically expensive methods, such as dynamical mean-field theory, are discussed.
We study a one-dimensional spin-orbital model with Heisenberg symmetry for spins and XY\mathrm{XY}XY symmetry for orbital degrees of freedom. We establish the ground-state phase diagram of this model. The gapless phases of the model consist of fully polarized ferromagnetic spin states together with the filled Jordan-Wigner Fermi sea for the orbitals. Two such phases, with ferromagnetic and antiferromagnetic XY\mathrm{XY}XY correlations for the orbitals, are found. All other phases are found to have a spin gap with degenerate ground states. Two types of correlated spin and orbital dimerizations are observed: (i) spin singlet with orbital singlet in alternating rungs and (ii) spin singlet with orbital triplet in the same rung. The phase boundaries, energy gaps, and order parameters are evaluated using the density matrix renormalization group method.
The ground-state properties of a simple model for transition metal oxide compounds in ferromagnetic phase have been studied. This model only contains the orbital degree of freedom. The two-orbital spinless Hubbard model in the limit of strong on-site coulomb interaction (U) has been used to describe the motion of election in e[subscript g] orbital. In the large-U and half-filling limits, the effective Hamilto-nian has been derived from the Hubbard model by the canonical transformation. After the transformation, the doubly occupied states have been projected out and the size of the Hillbert space has been reduced. For this effective Hamiltonian, the hopping amplitudes have been defined according to the managanite system which consists of three axes: x-, y- and z-axis. Since the ground-state properties of the effective Hamiltonian have been calculated only in one-dimension, then the results of this work have been separated in three cases; the electrons hopping in x, y, and z axes respectively. The ground-state wavefunction of this effective Hamiltionian has been calculated by infinite density matrix renormalization group algorithm. This algorithm leads to the ground state at the thermodynamic limit when the number of sties is large. It has been found that the ground-state energies are equivalent in all hopping axes. Furthermore, the nearest neighbor orbital correlations at the ground state for all axes are antiferro-orbital.
We study by dynamical mean-field theory the ground state of a quarter-filled Hubbard model of two bands with different bandwidths. At half-filling, this model is known to display an orbital selective Mott transition, with the narrower band undergoing Mott localization while the wider one being still itinerant. At quarter-filling, the physical behavior is different and to some extent reversed. The interaction generates an effective crystal field splitting, absent in the Hamiltonian, that tends to empty the narrower band in favor of the wider one, which also become more correlated than the former at odds with the orbital selective paradigm. Upon increasing the interaction, the depletion of the narrower band can continue till it empties completely and the system undergoes a topological Lifshitz transition into a half-filled single-band metal that eventually turns insulating. Alternatively, when the two bandwidths are not too different, a first order Mott transition intervenes before the Lifshitz's one. The properties of the Mott insulator are significantly affected by the interplay between spin and orbital degrees of freedom.