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arXiv:1809.09483 · Condensed Matter Physics

Two-orbital Hubbard model vs spin S=1 Heisenberg model: studies on clusters

R. Lemański, Matysiak

Published Sep 1, 2018• 3 citations
arXivHTMLPDF

Abstract

We perform exact numeric calculations for the two-orbital Hubbard model on the four-site cluster.In the limit of large on-site coupling the model becomes equivalent to the spin S " 1 Heisenberg model.By comparing energy spectra of these two models, we quantified the range of interaction parameters for which the Heisenberg model satisfactorily reproduces the two-orbital Hubbard model.Then we examined how the spectrum evolves when we are outside of this region, focusing especially on checking of how it is modified when various ways of interatomic hoppings of electrons between different orbitals are taken into account.We finally show how these modifications affect the dependence of specific heat on temperature.

DOI
10.5488/cmp.21.33301

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We perform exact numeric calculations for the two-orbital Hubbard model on the four-site cluster.In the limit of large on-site coupling the model becomes equivalent to the spin S " 1 Heisenberg model.By comparing energy spectra of these two models, we quantified the range of interaction parameters for which the Heisenberg model satisfactorily reproduces the two-orbital Hubbard model.Then we examined how the spectrum evolves when we are outside of this region, focusing especially on checking of how it is modified when various ways of interatomic hoppings of electrons between different orbitals are taken into account.We finally show how these modifications affect the dependence of specific heat on temperature.
Abstract We studied a multi-orbital Hubbard model at half-filling for two and three orbitals per site on a two-site cluster via full exact diagonalization, in a wide range for the onsite repulsion U, from weak to strong coupling, and multiple ratios of the Hund coupling J H to U. The hopping matrix elements among the orbitals were also varied extensively. At intermediate and large U, we mapped the results into a Heisenberg model. For two orbitals per site, the mapping is into a S = 1 Heisenberg model where by symmetry both nearest-neighbor (S i ⋅ S j ) and ( S i ⋅ S j ) 2 are allowed, with respective couplings J 1 and J 2. For the case of three orbitals per site, the mapping is into a S = 3/2 Heisenberg model with (S i ⋅ S j ), ( S i ⋅ S j ) 2 , and ( S i ⋅ S j ) 3 terms, and respective couplings J 1, J 2, and J 3. The strength of these coupling constants in the Heisenberg models depend on the U, J H, and hopping amplitudes of the underlying Hubbard model. Our study provides a first crude estimate to establish bounds on how large the ratios J 2/J 1 and J 3/J 1 can be. We show that those ratios appear rather limited and, as a qualitative guidance, we conclude that J 2/J 1 is less than 0.4 and J 3/J 1 is less than 0.2, establishing bounds on effective models for strongly correlated Hubbard systems. Moreover, the intermediate Hubbard U regime was found to be the most promising to enhance J 2/J 1 and J 3/J 1.
We study the effects of charge degrees of freedom on the spin excitation dynamics in quasi-one-dimensional magnetic materials. Using the density matrix renormalization group method, we calculate the dynamical spin structure factor of the Hubbard model at half electronic filling on a chain and on a ladder geometry, and compare the results with those obtained using the Heisenberg model, where charge degrees of freedom are considered frozen. For both chains and two-leg ladders, we find that the Hubbard model spectrum qualitatively resembles the Heisenberg spectrum---with low-energy peaks resembling spinonic excitations---already at intermediate on-site repulsion as small as $U/t\ensuremath{\sim}2--3$, although ratios of peak intensities at different momenta continue evolving with increasing U/tU/tU/t converging only slowly to the Heisenberg limit. We discuss the implications of these results for neutron scattering experiments and we propose criteria to establish the values of U/tU/tU/t of quasi-one-dimensional systems described by one-orbital Hubbard models from experimental information.
A simple way to determine the channel of scattering of the Hubbard chain electrons with additional quantum number, which means the orbital or band characteristic, is demonstrated on the integrable 1D modified Hubbard model. Exact solutions are obtained for each scattering channel of the electrons. The preferable channel is determined by means of numerical calculations of the ground-state energies and comparison its values. The dependence of the scattering matrices on the parameters of Coulomb, orbital (band), and spin interactions gives the possibility to present the properties of the system considered in different ranges (reasonable in physics sense) of parameters which correspond to the electron coupling. The ratio between magnetic susceptibility and heat capacity of the model is presented.
The interactions between holes in the Hubbard model, in the low density, intermediate- to strong-coupling limit, are investigated by systematically improving mean field calculations. The configuration-interaction basis set is constructed by applying to local unrestricted Hartree-Fock configurations all lattice translations and rotations. It is shown that this technique reproduces, correctly, the properties of the Heisenberg model, in the limit of large U. Upon doping, dressed spin polarons in neighboring sites have an increased kinetic energy and an enhanced hopping rate. Both effects are of the order of the hopping integral and lead to an effective attraction at intermediate couplings. The numerical results also show that when more than two holes are added to the system, they do not tend to cluster, but rather hole pairs remain far apart. Hole-hole correlations are also calculated and shown to be in qualitative agreement with exact calculations for $4\ifmmode\times\else\texttimes\fi{}4$ clusters. In particular our results indicate that for intermediate coupling the hole-hole correlation attains a maximum when the holes are in the same sublattice at a distance of 2\sqrt{2}2​ times the lattice spacing, in agreement with exact results and the $t\ensuremath{-}J$ model. The method is also used to derive known properties of the quasiparticle band structure of isolated spin polarons.