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Physical review. B, Condensed matter

Cluster approach to the extended Hubbard model

W. N. Mei, Y. C. Lee

Published Jul 15, 1981• 4 citations
Publisher

Abstract

The eigenvalue problem of the two-site and three-site extended Hubbard Hamiltonian, which contains the intraatomic repulsion $U$, the nearest-neighbor repulsion $V$, the isotropic exchange $J$, the anisotropic exchange ${J}^{\ensuremath{'}}$, the hopping term $t$, the semihopping term $S$, and the double hopping term $D$, has been solved exactly and the grand partition function is obtained. The magnetic and thermodynamic quantities such as the magnetic susceptibility and the specific heat are calculated for arbitrary electron density. Several features believed to have wider validity have been deduced from this cluster model.

DOI
10.1103/physrevb.24.1111

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The eigenvalue problem of the two-site and three-site extended Hubbard Hamiltonian, which contains the intraatomic repulsion UUU, the nearest-neighbor repulsion VVV, the isotropic exchange JJJ, the anisotropic exchange J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′, the hopping term ttt, the semihopping term SSS, and the double hopping term DDD, has been solved exactly and the grand partition function is obtained. The magnetic and thermodynamic quantities such as the magnetic susceptibility and the specific heat are calculated for arbitrary electron density. Several features believed to have wider validity have been deduced from this cluster model.
With the aim of contributing to a better understanding of the finite-temperature properties of low-dimensional strongly correlated systems we study a one-dimensional half-filled-band extended Hubbard model with a term of exchange interaction between neighboring sites. We analyze the temperature dependence of the specific heat, internal energy, entropy, number of doubly occupied sites, magnetic susceptibility, spin-spin correlation functions, and also investigate a detailed diagram of magnetic correlations. We use the method of small-cluster exact diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined model exhibits a rich structure of magnetic properties due to the competition between the onsite Coulombian repulsion and the exchange interaction.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
Abstract The Hubbard model on a cube was revisited and extended by both nearest‐neighbor Coulomb correlation W and nearest‐neighbor Heisenberg exchange J. The complete eigensystem was computed exactly for all electron occupancies and all model parameters ranging from minus infinity to plus infinity. For two electrons on the cluster the eigensystem is given in analytical form. For six electrons and infinite on‐site correlation U we determinded the groundstate and the groundstate energy of the pure Hubbard model analytically. For fixed electron numbers we found a multitude of ground state level crossings depending on the various model parameters. Furthermore the groundstates of the pure Hubbard model in dependence on a magnetic field h coupled to the spins are shown for the complete U‐h plane. The critical magnetic field, where the zero spin groundstate breaks down is given for four and six electrons. Suprisingly we found parameter regions, where the ground state spin does not depend monotonously on J in the extended model. For the cubic cluster gas, i.e. an ensemble of clusters coupled to an electron bath, we calculated the density n (μ, T, h) and the thermodynamical density of states from the grand potential. The ground states and the various spin‐spin correlation functions are studied for both attractive and repulsive values of the three interaction constants. We determined the various anomalous degeneration lines, where n (μ, T = 0, h = 0) shows steps higher than one, since in this parameter regions exotic phenomena as phase separation are to expect in extended models. For the cases where these lines end in triple points, i.e. groundstates of three different occupation numbers are degenerated, we give the related parameter values. Regarding the influence of the nn‐exchange and the nn‐Coulomb correlation onto the anomalous degeneration we find both lifting and inducing of degeneracies depending on the parameter values.
The extended Hubbard model in the zero-bandwidth limit is studied.The eective Hamiltonian consists of (i) on-site U interaction, (ii) intersite densitydensity interaction W , and (iii) Ising-like magnetic exchange interaction J between the nearest-neighbors.We present rigorous (and analytical) results obtained within the transfer-matrix method for 1D chain in two particular cases: (a) W = 0 and n = 1; (b) U → +∞ and n = 1/2 (W = 0, J = 0).We obtain the exact formulae for the partition functions which enables to calculate thermodynamic properties such as entropy, specic heat (c), and double occupancy per site.In both cases the system exhibits an interesting temperature dependence of c involving a characteristic two-peak structure.There are no phase transitions at nite temperatures and the only transitions occur in the ground state.
For an infinite linear chain we consider the strong-coupling ($U\ensuremath{\rightarrow}\ensuremath{\infty}$) limit of an extended Hubbard Hamiltonian which takes into account the nearest-neighbor interatomic Coulomb repulsion. It is shown that the magnetic susceptibility obeys a Curie law, and, to this effect, a theorem is proved pertaining to the partition function. The specific heat is computed for the quarter-filled band to lowest order in the transfer integral. The density-density correlation function and the specific heat are also obtained in the atomic limit for an arbitrary density.