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Acta Physica Polonica A

Comparison of the Multi-Band Hubbard and Heisenberg Models on Small Finite Clusters

J. Matysiak, R. Lemański

Published Mar 1, 2018• 1 citation
Publisher

Abstract

We perform exact numeric calculations for a few versions of the two-orbital Hubbard model on the four site cluster.We show that the conventional spin 1 Heisenberg Hamiltonian can be obtained in the strong coupling limit, but only for some versions of the two-orbital Hubbard model.Then we propose a modified version of the Heisenberg Hamiltonian, that we call the two-orbital Heisenberg model, which is relevant in those cases, where the conventional spin 1 Heisenberg model fails.

DOI
10.12693/aphyspola.133.405

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We perform exact numeric calculations for a few versions of the two-orbital Hubbard model on the four site cluster.We show that the conventional spin 1 Heisenberg Hamiltonian can be obtained in the strong coupling limit, but only for some versions of the two-orbital Hubbard model.Then we propose a modified version of the Heisenberg Hamiltonian, that we call the two-orbital Heisenberg model, which is relevant in those cases, where the conventional spin 1 Heisenberg model fails.
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 present a study of the two-orbital degenerate Hubbard model in which the exact numerical solution on a regular tetrahedron is obtained via suitable implementation of the symmetries generated by the spin, the pairing and the orbital pseudospin operators. In particular, we show that a large variety of high-spin magnetic ground states can develop away from half filling, depending on the values of the electron density and the parameters of the model. As the tetrahedron is the simplest finite-size cluster where hopping processes connect all pairs of sites with constant probability, the study is extended by providing the exact analytical solution of the model on an infinite lattice in the unconstrained hopping limit.
This thesis addresses the question of a strongly interacting many-body problem: the Hubbard model on one- and two-dimensional lattices. The technique employed is the small-cluster method wherein the full many-body Hamiltonian for finite clusters with periodic boundary conditions (PBC's) is solved exactly. We first apply this technique to an extended one-dimensional chain and a two-dimensional square-lattice Hubbard model. We then construct a two-dimensional overlayer/substrate Hubbard model with first- and second-nearest-neighbor hopping, on-site and nearest-neighbor interactions, and overlayer-site orbital energy. The ground-state properties are investigated and discussed in a many-body picture.
The single-band Hubbard model (both with and without first-neighbor interactions) is studied by exact diagonalization on several one- and two-dimensional clusters, ranging from four to nine sites, including rings and arrangements of squares. Occupancies include the half-filled band (one electron per site) and configurations with one and two holes in the half-filled band. Ground-state properties are determined for a wide range of parameters. We discuss the phase diagram in the (U,V) plane and consider spin, charge, and pairing correlations. The binding of a pair of holes is investigated. Circumstances in which binding and pairing are enhanced by first-neighbor interactions are described.
Cluster perturbation theory is a technique for calculating the spectral weight of Hubbard models of strongly correlated electrons, which combines exact diagonalizations on small clusters with strong-coupling perturbation theory at leading order. It is exact in both the strong- and weak-coupling limits and provides a good approximation to the spectral function at any wave vector. Following the paper by S\'en\'echal et al. [Phys. Rev. Lett. 84,84,84, 522 (2000)], we provide a more complete description and derivation of the method. We illustrate some of its capabilities, in particular regarding the effect of doping, the calculation of ground state energy and double occupancy, the disappearance of the Fermi surface in the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model, and so on. The method is applicable to any model with on-site repulsion only.