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Acta Physica Polonica A

Extended Hubbard Model in the Dimer Representation. I. Dimer Hamiltonian in the Large U Limit

B. Grabiec, M. Matlak

Published Apr 1, 2002• 5 citations

Abstract

Ê I nst i t ut e of P hysi cs, Si l esian Uni versity Uni werstytec ka 4, 40-007 Ka to wi ce, Pol an d ( Recei ved December 14, 2001)W e consider t he extended H ubb ard mo del for the single cubic lattice and re w rite it in the form of interacting dimers , using the exact solutio n of the dimer problem .W e anal ytical l y deri ve the second quanti zation form of the dimer H amiltoni an elimina tin g from the consideratio ns uno ccupie d dimer energy levels in the large U limit (it is the only assumption ).T he resulting dimer H amiltonian w ritten with the use of the Hubbard op erators and spin operators contains three terms, visuali zin g explicitl y comp eting magnetic interactions (f erromagnetic, antif erromagnetic) as a generaliz atio n of the t { J mo del.T he presented, nonp erturbati ve metho d, can in principl e b e applied to the cluster of any size (e.g. one central atom and z its nearest neighb ours).T he use of the pro j ection technique can further b e appli ed in the case of a crystal to obtain the second quantization form of the extended H ubbard mo del for the sc lattice in the large U limit.

DOI: 10.12693/aphyspola.101.537

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Ê I nst i t ut e of P hysi cs, Si l esian Uni versity Uni werstytec ka 4, 40-007 Ka to wi ce, Pol an d ( Recei ved December 14, 2001)W e consider t he extended H ubb ard mo del for the single cubic lattice and re w rite it in the form of interacting dimers , using the exact solutio n of the dimer problem .W e anal ytical l y deri ve the second quanti zation form of the dimer H amiltoni an elimina tin g from the consideratio ns uno ccupie d dimer energy levels in the large U limit (it is the only assumption ).T he resulting dimer H amiltonian w ritten with the use of the Hubbard op erators and spin operators contains three terms, visuali zin g explicitl y comp eting magnetic interactions (f erromagnetic, antif erromagnetic) as a generaliz atio n of the t { J mo del.T he presented, nonp erturbati ve metho d, can in principl e b e applied to the cluster of any size (e.g. one central atom and z its nearest neighb ours).T he use of the pro j ection technique can further b e appli ed in the case of a crystal to obtain the second quantization form of the extended H ubbard mo del for the sc lattice in the large U limit.

We find the eigenvalues $E_α $ and eigenvectors

|E_α>

of the extended Hubbard dimer and we represent each part

E_α |E_α> < E_α |

of the dimer Hamiltonian

(α=1,2,...,16)

in the second quantizations with the use of the Hubbard and spin operators. This procedure gives a review of all competitive, intrinsic interactions, deeply hidden in the orginal form of the considered Hamiltonian. Among competitive interactions we can find ferromagnetic and antiferromagnetic interactions (also in the form appearing in the

t-J

model), hopping of the Cooper pairs between different dimer lattice sites (similar as in Kulik-Pedan, Penson-Kolb models), as well as, intersite Cooper pair interactions. We plot several coupling constants of these interactions vs Coulomb intrasite interaction

U

to show that the competition between them strongly depends on the model parameters. The thermodynamical activity of each particular interaction, belonging to a given energy level, depends, however, on the occupation of this level. From the presented review of interactions it is evident that the extended Hubbard model is capable to describe the properties of superconductors with magnetic ordering (including also high-

T_C

superconductors) where a strong competition between magnetism and superconductivity takes place.

The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].

The dimerized one-dimensional Hubbard model is studied in the framework of lattice density-functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable. The corresponding interaction-energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is defined by Levy's constrained search. Exact numerical results are obtained for $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23}),$ where ${\ensuremath{\gamma}}_{12}={\ensuremath{\gamma}}_{i,i+1}$ for odd i and ${\ensuremath{\gamma}}_{23}={\ensuremath{\gamma}}_{i,i+1}$ for even i are the nearest-neighbor density-matrix elements along the chain. The domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}}$ and the functional dependence of $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ are analyzed. A simple, explicit approximation to $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ is proposed, which is derived from scaling properties of W, exact dimer results, and known limits. Using this approximation, LDFT is applied to determine ground-state properties and charge-excitation gaps of finite and infinite dimerized chains as a function of the Coulomb-repulsion strength

U/t

and of the alternation $\ensuremath{\delta}t$ of the hopping integrals

{t}_{\mathrm{ij}}

${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.