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Physical review. B, Solid state

Partially filled linear Hubbard model near the atomic limit

Daniel Klein, W. A. Seitz

Published Oct 15, 1974• 92 citations
Publisher

Abstract

The linear Hubbard model is considered when the intrasite Coulomb repulsion is very large compared to the intersite charge-transfer interaction. The solution for a general density of electrons per site is carried out by a perturbative treatment which relates this Hubbard model to a generalized Heisenberg spin Hamiltonian describing effective exchange interactions between (partially) delocalized electrons. Accurate numerical results for the linear Heisenberg model are employed to obtain good estimates of the static thermodynamic properties for Hubbard models near the atomic limit. Some modifications of the simple Hubbard model are also briefly discussed.

DOI
10.1103/physrevb.10.3217

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The linear Hubbard model is considered when the intrasite Coulomb repulsion is very large compared to the intersite charge-transfer interaction. The solution for a general density of electrons per site is carried out by a perturbative treatment which relates this Hubbard model to a generalized Heisenberg spin Hamiltonian describing effective exchange interactions between (partially) delocalized electrons. Accurate numerical results for the linear Heisenberg model are employed to obtain good estimates of the static thermodynamic properties for Hubbard models near the atomic limit. Some modifications of the simple Hubbard model are also briefly discussed.
A linked-diagram perturbation expansion of the linear half-filled-band Hubbard model is developed and carried out through seventh order. The zero-order Hamiltonian is taken to be the intrasite Coulomb repulsion terms, and the perturbation is taken to be the intersite interaction. The degenerate perturbation theory applied here results in an effective Hamiltonian defined on a simple spin space, and is found to consist of terms involving nearest-neighbor, next-nearest-neighbor, and biquadratic spin operators.
Based on exact diagonalization results for small clusters we discuss the effect of intersite Coulomb repulsion in Mott-Hubbard or charge transfers insulators. Whereas the exchange constant JJJ for direct exchange is enhanced by intersite Coulomb interaction, that for superexchange is suppressed. The enhancement of JJJ in the single-band models holds up to the critical value for the charge density wave (CDW) instability, thus opening the way for large values of JJJ. Single-band Hubbard models with sufficiently strong intersite repulsion to be near a CDW instability thus may provide physical realizations of $t\ensuremath{-}J$-like models with the "unphysical" parameter ratio $\frac{J}{t}\ensuremath{\approx}1$.
The exact eigenvalues for the ground state and the excitation spectra were obtained for a system with two electrons on two sites, taking into account the transfer energy, the Coulomb repulsion on the same site, and the intersite correlation energy. The vibrational modes couple linearly to the electronic density. We discuss the interplay of these interactions resulting in different phases of distorted and undistorted eigenstates with corresponding localized and delocalized electronic wave functions and accompanying charge transfer. Several simple approximation schemes generalizable to larger systems are evaluated. The comparison with the exact results shows that their validity depends strongly on the system parameters. The according limits of applicability are discussed.