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Journal of Applied Physics

Linked-cluster expansion for the single-band Hubbard model

Kok-Kwei Pan, Yung‐Li Wang

Published Apr 15, 1991• 7 citations
Publisher

Abstract

Using the linked-cluster expansion, we have obtained the free energy, uniform susceptibility, and staggered susceptibility of the s-band Hubbard model to the sixth order in terms of βt, where t is the hopping integral to the nearest neighbors and β is the inverse of temperature. We show that a special technique based on the multiple-site Wick reduction theorem is more convenient than previous perturbation calculations. Our linked-cluster expansion result is for an arbitrary value of the on-site repulsion U. It has made corrections of previous results of several papers (up to fourth order) and achieved the sixth-order term.

DOI
10.1063/1.348287

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Using the linked-cluster expansion, we have obtained the free energy, uniform susceptibility, and staggered susceptibility of the s-band Hubbard model to the sixth order in terms of βt, where t is the hopping integral to the nearest neighbors and β is the inverse of temperature. We show that a special technique based on the multiple-site Wick reduction theorem is more convenient than previous perturbation calculations. Our linked-cluster expansion result is for an arbitrary value of the on-site repulsion U. It has made corrections of previous results of several papers (up to fourth order) and achieved the sixth-order term.
Using the fourth-order perturbation expansion, we have calculated the uniform and the staggered magnetic susceptibility of the sss-band Hubbard model. The Curie and the N\'eel temperatures are derived and the magnetic phase diagrams are constructed. Our calculation improves on the earlier work of Kubo.
The Green’s-function diagrammatic high-density expansion technique for Hubbard’s operators is constructed and applied to the extended Hubbard model. The electron Green’s functions and the band structure of the ground state for arbitrary electron density are calculated in the zeroth-order in 1/z. The phase diagram, staggered magnetic susceptibility and the Néel temperature for the half-filled band case are derived in the first order of the high-density expansion. Connections of the high-density and linked cluster expansions are also discussed.
The Wick reduction theorem is used in a linked-cluster-expansion calculation to facilitate the evaluation of the multi-integrals of the ordered products encountered in the expansion. The method resolves a major problem in the general linked-cluster expansion for quantum spin systems. We apply it to the evaluation of the perpendicular susceptibility of the Ising model. An eighth-order linked-cluster series is obtained for a general lattice. The temperature behavior of the perpendicular susceptibility of the three-dimensional Ising model (fcc lattice) is discussed.
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 ${J}^{\ensuremath{'}}$, 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.