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physica status solidi (b)

Poisson Approximation to a Functional Approach for the Hubbard Model

Ulrich Behn

Published Jul 1, 1980• 12 citations
Publisher

Abstract

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.

DOI
10.1002/pssb.2221000113

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Physical review. B, Condensed matter•Feb 15, 1999•1 citation
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 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 Describing spin ↑ ‐electrons the particle number operators ni↓ are replaced under the functional integral directly by c‐numbers vi↓(t) using a δ‐functional instead of the Hubbard‐Stratonovich transformation. The Green function of the Hubbard model is thus exactly represented as an averaged Green function of electrons moving in a stochastic potential Uvi↓(t). The probability distribution for the stochastic process v ↓ containing the many body aspect and the information about the ↓‐electrons is determined. The vi↓(t) are only allowed to jump between 0 and 1 reflecting the property of Fermion particle number operators. This exact scheme can be interpreted as a dynamically generalized alloy analogy.
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.
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.
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].
The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.