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Physical review. B, Solid state

Perturbation Expansion of the Linear Hubbard Model

Daniel Klein, W. A. Seitz

Published Sep 1, 1973• 80 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.8.2236

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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.
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.
We have incorporated the projection operators to the canonical transformation to derive an analytical infinite perturbation-expansion series. This canonical perturbation expansion (CPE) is valid if the unperturbed Hamiltonian H0{H}_{0}H0​ and the perturbation H1{H}_{1}H1​ can be expressed as ${H}_{0}=\ensuremath{\Sigma}{j}^{}{P}_{j}H{P}_{j}$ and ${H}_{1}=\ensuremath{\Sigma}{j\ensuremath{\ne}k}^{}{P}_{j}H{P}_{k}$, where Pj{P}_{j}Pj​ is the projection operator corresponding to a group of closely spaced effective one-electron orbital energies ${E}_{j\ensuremath{\mu}}$ with $\ensuremath{\mu}=1, 2, \dots{}, {d}_{j}$, and if $|{E}_{j\ensuremath{\mu}}\ensuremath{-}{E}_{j\ensuremath{\nu}}|\ensuremath{\ll}|{E}_{j\ensuremath{\omega}}\ensuremath{-}{E}_{k\ensuremath{\delta}}|$ with $j\ensuremath{\ne}k$. We have shown that the CPE is equivalent to the time-dependent perturbation theory. An extremely simple effective Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is obtained when the CPE is applied to the sss-band Hubbard model at the atomic limit. An explicit form of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ to the eighth order is given, and the magnetic interaction in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ is of the form of Heisenberg exchange ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}$, including far neighbors. We then use this form to compute the antiferromagnetic groundstate energy to the seventh order. Our result is compared with other works.
We investigate the Hubbard model on a two-dimensional square lattice by the perturbation expansion to the fourth order in the on-site Coulomb repulsion U. Numerically calculating all diagrams up to the fourth order in self-energy, we examine the convergence of perturbation series in the lattice system. We indicate that the coefficient of each order term rapidly decreases as in the impurity Anderson model for T > 0.1t in the half-filled case, but it holds in the doped case even at lower temperatures. Thus, we can expect that the convergence of perturbation expansion in U is very good in a wide parameter region also in the lattice system, except for T < 0.1t in the half-filled case. We next calculate the density of states in the fourth-order perturbation. In the half-filled case, the shape in a moderate correlation regime is quite different from the three peak structure in the second-order perturbation. Remarkable upper and lower Hubbard bands locate at w = +(-)U/2, and a pseudogap appears at the Fermi level w=0. This is considered as the precursor of the Mott-Hubbard antiferromagnetic structure. In the doped case, quasiparticles with very heavy mass are formed at the Fermi level. Thus, we conclude that the fourth-order perturbation theory overall well explain the asymptotic behaviors in a strong correlation regime.
We derive an effective spin Hamiltonian for the one-dimensional half-filled Alternating Hubbard model in the limit of strong on-site repulsion. We show that the effective Hamiltonian is a spin S=1/2S=1/2S=1/2 Heisenberg chain with asymmetric next-nearest-neighbor (nnn) exchange.
The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong Hubbard repulsion.The expansion in powers of the hopping constant is expressed in terms of site cumulants of electron creation and annihilation operators.For Green's function an equation of the Larkin type is derived and solved in a one-loop approximation for the case of two dimensions and nearest-neighbor hopping.With decreasing the electron concentration in addition to the four bands observed at half-filling, a narrow band arises near the Fermi level.The dispersion of the new band, its bandwidth and the variation with filling are close to those of the spin-polaron band in the t-J model.