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Physical Review

Electron Correlation in Narrow Energy Bands. I. The Two-Pole Approximation in a Narrow S Band

Laura M. Roth

Published Aug 10, 1969• 254 citations
Publisher

Abstract

An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's $T$-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.

DOI
10.1103/physrev.184.451

Related papers

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Magnetic Order in Hubbard Narrow-Energy-Band Model

D. M. Esterling

We must understand the physics of strongly interacting electrons if we are to have a fundamental understanding of magnetism. This paper treats the simplest model with this property — the Hubbard model. The moments of the spectral weight function (SWF), derived by Harris and Lange, are calculated in the atomic limit. These moments are used to generate an approximate SWF (including lifetime effects). The paramagnetic and ferromagnetic ground state energies are calculated and compared. There is no restriction on the electron density.

AIP conference proceedings•Jan 1, 1972•0 citations
An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's TTT-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.
In a previous paper, a new Green's-function decoupling scheme was applied to the Hubbard Hamiltonian, and an improved version of Hubbard's first approximation was obtained. That result did not reduce to the correct low-density limit as obtained by Kanamori. In the present article, the theory is improved for the special case of a single reversed spin in an otherwise fully aligned band, and the improved theory is correct in the low-density limit. Numerical results are presented for the simple cubic lattice. If we define an effective exchange-interaction parameter Ueff{U}_{\mathrm{eff}}Ueff​ as the k=0k=0k=0 reversed-spin self-energy for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, divided by the number $n\ensuremath{\uparrow}$ of up-spin electrons per site, we find that the present result departs rather rapidly from the Kanamori result as $n\ensuremath{\uparrow}$ is increased, and it is concluded that the Kanamori result overestimates the increase in Ueff{U}_{\mathrm{eff}}Ueff​ with $n\ensuremath{\uparrow}$, at least in the present case. For intermediate values of $n\ensuremath{\uparrow}$, the two-pole approximation of the previous article and the present calculation give very similar results for this quantity.
The Hubbard hamiltonian is analysed by means of Gutzwiller's variational scheme for both the nondegenerate and the degenerate band models. The single-particle excitations in the correlated ground state, which is determined by a balance between the electron hopping energy and the intra-atomic interaction energy, are investigated for arbitrary electron density and arbitrary correlation strength. It is found that the single-particle excitation energies forming a quasi-energy band. The centre of gravity of the quasi band is shifted from that of the bare band, and the bulk part of the density of states function for the quasi particles is narrowed by the correlation effect. Both the shift and the narrowing are spin-dependent. In the limit of strong correlation, our results agree with those obtained by Harris and Lange (1967) and Roth (1969).
To investigate the effects of electron correlation in narrow energy bands, a general formula of static spin susceptibility for interacting electron systems is derived. It is expressed in terms of the one-particle Green's function and its derivative with respect to the average number of the opposite spin electrons, the existence of the latter being the characteristic of electron correlation. The formula is applied to the case of Hubbard's improved approximation (Hubbard III), and gross feature of both the correlation and temperature dependences of the susceptibility is investigated in the insulating and metallic regions. The correlation in the metallic region is reduced for the Hubbard III case so that there occurs no ferromagnetism in a half-filled band.
A solution to the Hubbard model is presented which invokes the single assumption that the low-energy peak in the spectral weight function can be well approximated by a function which duplicates its zeroth, first, and second moments to a given order in the bandwidth. The solution takes advantage of the rigorous moment relations of Harris and Lange. A simple, exact, and unambiguous technique to evaluate the correlation functions, required for the moment relations, in the atomic limit is developed. These functions are used to calculate the moments which in turn generate a spectral weight function with no restriction on the electron density. The ground-state energy for the paramagnetic and ferromagnetic configurations are calculated and compared. The model is found to be ferromagnetic for moderate densities.
The ground-state wave function for the electrons in a narrow sss band is investigated for arbitrary density of electrons and arbitrary strength of interaction. An approximation is proposed which limits all the calculations to counting certain types of configurations and attaching the proper weights. The expectation values of the one-particle and two-particle density matrix are computed for the ferromagnetic and for the non-ferromagnetic case. The ground-state energy is obtained under the assumption that only the intra-atomic Coulomb interaction is of importance. Ferromagnetism is found to occur if the density of states is large at the band edges rather than in the center, and if the intra-atomic Coulomb repulsion is sufficiently strong. The relation of this approximation to certain exact results for one-dimensional models is discussed.
We must understand the physics of strongly interacting electrons if we are to have a fundamental understanding of magnetism. This paper treats the simplest model with this property — the Hubbard model. The moments of the spectral weight function (SWF), derived by Harris and Lange, are calculated in the atomic limit. These moments are used to generate an approximate SWF (including lifetime effects). The paramagnetic and ferromagnetic ground state energies are calculated and compared. There is no restriction on the electron density.