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physica status solidi (b)

Nearest‐Neighbour Poisson Approximation to a Functional Approach for the Hubbard Model

Ulrich Behn

Published Mar 1, 1981• 5 citations
Publisher

Abstract

Abstract In the framework of the Poisson approximation to a functional scheme for the Hubbard model the effect of the spatial correlation of the discrete stochastic potential Uvi(t) simulating the spin‐↓ subsystem is considered. For a half‐filled band in the paramagnetic case at zero temperature the density of states is calculated in a nearest‐neighbour approximation. For strong Coulomb repulsion the subbands occur sharper and the minimum in between is deeper compared with the single‐site approximation. The influence of the spatial correlation on the possible appearance of a gap within a dynamic approximation is discussed.

DOI
10.1002/pssb.2221040121

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Abstract In the framework of the Poisson approximation to a functional scheme for the Hubbard model the effect of the spatial correlation of the discrete stochastic potential Uvi(t) simulating the spin‐↓ subsystem is considered. For a half‐filled band in the paramagnetic case at zero temperature the density of states is calculated in a nearest‐neighbour approximation. For strong Coulomb repulsion the subbands occur sharper and the minimum in between is deeper compared with the single‐site approximation. The influence of the spatial correlation on the possible appearance of a gap within a dynamic approximation is discussed.
Abstract In a previous paper the Green function for the Hubbard model is represented by a functional average of the Green function for an electron moving in a local time dependent potential Uvi(t) taking only the values 0 and U. Here this functional average is calculated with an ad hoc assumed Poisson distribution governing the jumps of Uvi(t). In a single‐site approximation the Poisson average can be calculated exactly and leads to a self‐consistent ‚dynamic’︁ scheme interpolating between band and atomic limit. For a half‐filled band in the paramagnetic case at zero temperature for strong Coulomb repulsion U ≈ 2 two subbands are obtained separated by a minimum of the density of states.
Abstract The chemical potential μ and the electrical dc conductivity σ in the one‐dimensional Hubbard model in the strong‐coupling region U≫t (U intra‐atomic Coulomb energy, –t nearest‐neighbour hopping integral) are calculated as functions of temperature and electron density n. In the non‐half‐filled band case n ≠ 1 the Bari‐Kaplan approximation lim (σ/t2) is shown to yield an un magnified image physical singularity in the dc conductivity at T = 0 which is connected with the discontinuity of the zero‐temperature chemical potential at n = 1. An improved calculation for n ≠ 1 going beyond the Bari‐Kaplan approximation (i.e. taking into account the band terms in μ and σ/t2) yields a finite zero‐temperature conductivity which is proportional to the first power in t. For small deviations from the half‐filled band case the conductivity as a function of increasing temperature subsequently shows a metal‐like behaviour, a semiconductor‐like behaviour, and a repeated decrease at high temperatures.
Abstract In the single‐site and the nearest‐neighbour approximation, respectively, the density of electronic states at the Fermi level for normal systems at T = 0 is considered as a functional of the unperturbed density of states. For electron–hole symmetric systems, e.g. the half‐filled‐band Hubbard model, the density of states at the Fermi level in a normal single‐site approximation is either equal to the unperturbed value or vanishes. The expected gradual change in the density of states can be obtained in the nearest‐neighbour approximation.
Abstract Equations for the electron Green's function of the two‐dimensional Hubbard model, derived using the strong coupling diagram technique, are self‐consistently solved for different electron concentrations n and tight‐binding dispersions. Comparison of spectral functions calculated for the ratio of Hubbard repulsion to the nearest neighbor hopping with Monte Carlo data shows not only qualitative, but in some cases quantitative agreement in position of maxima. General spectral shapes, their evolution with momentum and filling in the wide range are also similar. At half‐filling and for the next nearest neighbor hopping constant the Mott transition occurs at , where is the initial bandwidth. This value is close to those obtained in the cases of the semi‐elliptical density of states and for . In the case and the Mott gap reaches maximum width at , and it is larger than that at for half‐filling. In all considered cases positions of spectral maxima are close to those in the Hubbard‐I approximation.
Correlation effects in the one-dimensional Hubbard model are studied by application of the L-resolvent method. The calculations are well beyond one-particle mean field approximation. All correlations involving two neighboring sites are included exactly and higher order correlations are approximated. The effects of varying temperature, band filling, Coulomb repulsion and external magnetic field are investigated. Two-site correlations for non-half-filled band fillings are calculated for the first time. Some exact relations between correlation functions are derived by electron-hole symmetry. Results indicate that short-range spin correlations always favor antiferromagnetic order. Large Coulomb repulsion induces a gap in the excitation spectrum and a metal-insulator transition is predicted as the Coulomb repulsion parameter is changed.
We study the density of states for a structurally random Hubbard model with a half-filled band. Calculations are carried out with the hopping integral t(R) obtained from a hydrogenic 1S wave function and with an on-site Coulomb repulsion [Formula: see text]. The equation of motion for the retarded single-particle Green's function is treated in the Hubbard III decoupling approximation. The configurational averaging for the Green's function is simplified using the approximation of Kikuchi. Coefficients in the locator–interactor expansion are calculated analytically up to seventh order in t. Several approximants for this formal series are investigated, including Padé approximant, continued fraction approximation, and quadratic Padé approximant. The last two approximants yield densities of state in satisfactory agreement with the result of Aoki and Kamimura in the high concentration region and our results are expected to represent an improvement for intermediate densities. However, for very low densities the Hubbard III scheme will underestimate the effect of spin correlations and our previous treatment in the U = ∞ limit would be more appropriate for the lower Hubbard band.