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Journal of the Physical Society of Japan

Correlation Effects on the Spin Susceptibility of Narrow-Band Electrons

M. Kanehisa, Hiroshi Kamimura

Published Jan 1, 1973• 7 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.34.44

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International Journal of Modern Physics B•Nov 10, 1992•18 citations
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.
Correlation and magnetic effects in narrow bands are studied using a new type of self-consistent cluster treatment of Hubbard's Hamiltonian. Here the dynamics consist of electron hopping between lattice sites and Coulomb repulsion between electrons only on the same site. The environment of the cluster is regarded as a particle reservoir, one such for each spin. Hopping between the cluster and reservoirs is described by fermion source terms. It is required that the thermodynamic average of the particle currents of each spin within the cluster equal the corresponding particle exchange between cluster and reservoir. Electron motion in the system is examined for varying ratios of the strength of the hopping to the Coulomb repulsion. This is done as a function of electron density, spin, temperature, and external magnetic field. A Mott transition is found under dynamical conditions close to those predicted by Hubbard when there is one particle per site. For 〈n〉 ≠ 1, no metal insulator transition exists although discontinuous changes in the cluster-reservoir coupling occur. The magnetic susceptibility of the cluster reveals antiferromagnetic behavior for 〈n〉 = 1, ferromagnetic behavior for 〈n〉 > 1, and partial ferromagnetic character for n < 1. This appears to agree with Nagoaka's results.
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.
Some correlation effects are studied in a narrow-band model with electron–hole asymmetry. A peculiar feature of the model is that it takes into account the hopping integral associated with electron–electron interaction (correlated hopping). The method of Green’s functions is used to find the quasiparticle energy spectrum, as well as expressions for the upper and lower Hubbard subband widths. It is shown that the specific features of the model lead in a number of cases to consequences differing significantly from those following from the Hubbard model. The obtained results mainly illustrate the peculiarities of the model in the low-temperature region and are used for interpreting the physical properties of systems with narrow energy bands.
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.
On the basis of a regular perturbation theory in the form of diagrammatic technique with X-operators, a ferromagnetic state in the Hubbard model with infinite on-site Coulomb interaction U is studied for a wide range of electron concentration n. Treating the kinetic energy of electrons as perturbation, we get the exact graphic representations for three Green’s functions [Formula: see text] and [Formula: see text] for electrons with spin ↑ and ↓ (spontaneous magnetic moment is in ↑ direction) and [Formula: see text] for spin waves. All of them are expressed through the unique system of four-point and three-point vertex parts for effective electron-magnon and electron-electron interactions. These vertex parts are calculated in two approximations: a low-density approximation for n≪1 or nh=1–n≫1, and the generalized random-phase approximation (GRPA), which was suggested earlier by us for the description of paramagnetic phase in this model. An important result in both cases is the understanding of essential difference of electron states with spin ↑ and ↓. The state for ↑ has coherent character in all region of electron concentration (n>nc) where a ferromagnetic state exists, while the state for spin ↓ has mixed characters including both coherent and incoherent contribution. For the saturated ferromagnetism (n>ns) when nh≪1, [Formula: see text] is an entirely incoherent (nonquasiparticle) state. Appearance of incoherent state is probably a general property of strongly correlated systems distinguishing its behavior from the Fermi liquid. We show also that the Hubbard-I approximation has no region of applicability for the electron with spin ↓ in a ferromagnet and we found a new factor describing the correlation-narrowing of ↓ spin electron band. Green’s function [Formula: see text] calculated in two different limits n≪1 and nh≪1, are jointed together to allow us to calculate ns which lies in the intermediate concentration regime. For simple cubic lattice we found ns=0.87. In the limit nh≪1 our results reduce to the earlier known ones, including Nagaoka’s result for the spin wave energy. We point out the way to construct a rigorous theory for ferromagnetic state in the intermediate range of electron concentration. In the conclusion several important problems are discussed in connection to the continuation of the present work.