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physica status solidi (b)

On the Connection between Electric and Magnetic Properties of the Hubbard Model I. Electrical Conductivity

Wolfgang Nolting

Published Aug 1, 1975• 7 citations
Publisher

Abstract

Abstract An approximate expression for the electrical conductivity σ is derived by means of the method of spectral moments. Therefore, σ is written in terms of a spectral density, for which a suitable ansatz is chosen. Free parameters of this ansatz are fixed by equating moments. The so obtained approximate solution of the conductivity tensor fulfils the well‐known sum rules and contains nearly all exactly calculable limiting cases. Furthermore, Moneckes statement that the conductivity exhibits free acceleration behaviour even in the case of strong electronic correlations, is shown to be incorrect.

DOI
10.1002/pssb.2220700209

Related papers

Electrical conductivity in the strong‐coupling non‐half‐filled band Hubbard model

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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.

physica status solidi (b)•Mar 1, 1978•4 citations

An exact sum-rule for the Hubbard model: an historical/pedagogical approach

S. Di Matteo, Yann Claveau

Abstract The aim of the present article is to derive an exact integral equation for the Green function of the Hubbard model through an equation-of-motion procedure, like in the original Hubbard papers. Though our exact integral equation does not allow to solve the Hubbard model, it represents a strong constraint on its approximate solutions. An analogous sum rule has been already obtained in the literature, through the use of a spectral moment technique. We think however that our equation-of-motion procedure can be more easily related to the historical procedure of the original Hubbard papers. We also discuss examples of possible applications of the sum rule and propose and analyse a solution, fulfilling it, that can be used for a pedagogical introduction to the Mott–Hubbard metal-insulator transition.

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Magnetic Order in Hubbard Narrow-Energy-Band Model

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AIP conference proceedings•Jan 1, 1972•0 citations

Hubbard model of strong correlations

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A state-of-the-art review is given of the investigations of the Hubbard model, which is the main model used in the theory of strongly correlated electron systems. It is shown how the main correlation effects — the metal–insulator phase transition, the appearance of localised magnetic moments, and the breakdown of the Fermi-liquid behaviour — appear in the case of strong electron correlations when U ~ W, where U is the Coulomb repulsion parameter and W is the band width. Different approaches to the problem, not based on perturbation theory in terms of the parameter U/W or W/U, are presented. A new method in the theory of strongly correlated systems, in which the limit of infinite number of dimensions of space, d = ∞, is the main feature of the review. The physical results obtained by this approach are compared with those deduced by traditional methods. An overview of the behaviour of strongly correlated systems is given for wide ranges of the model parameters U and W, and of the electron density n.

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The Moment Sum Rule and Its Consequences for Ferromagnetism in the Hubbard Model

Michael Potthoff, T. Herrmann, Thomas Wegner +1 more

The sum rule for the moments of the spectral density is discussed for the single-band Hubbard model. It is shown that respecting the sum rule up to the order m = 3 is conceptually important for a qualitatively correct description of the quasi-particle band structure in the strong-correlation regime. Different analytical approximations for the self-energy are analyzed with respect to their compatibility with the moment sum rule. To estimate the practical usefulness of the sum rule, correlation functions and dynamical quantities are determined. The results obtained within the various approximation schemes of different complexity are compared with each other and also with essentially exact results available for infinite-dimensional lattices. It turns out that the m = 3 moment is rather unimportant for the paramagnetic phase on the hyper-cubic lattice. Contrary, it decisively influences the magnetic phase boundary as well as the critical temperature for the ferromagnetic phase on an f.c.c.-type lattice.

arXiv:cond-mat/9809212•physica status solidi (b)•Nov 1, 1998•51 citations
Abstract An approximate expression for the electrical conductivity σ is derived by means of the method of spectral moments. Therefore, σ is written in terms of a spectral density, for which a suitable ansatz is chosen. Free parameters of this ansatz are fixed by equating moments. The so obtained approximate solution of the conductivity tensor fulfils the well‐known sum rules and contains nearly all exactly calculable limiting cases. Furthermore, Moneckes statement that the conductivity exhibits free acceleration behaviour even in the case of strong electronic correlations, is shown to be incorrect.
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 The aim of the present article is to derive an exact integral equation for the Green function of the Hubbard model through an equation-of-motion procedure, like in the original Hubbard papers. Though our exact integral equation does not allow to solve the Hubbard model, it represents a strong constraint on its approximate solutions. An analogous sum rule has been already obtained in the literature, through the use of a spectral moment technique. We think however that our equation-of-motion procedure can be more easily related to the historical procedure of the original Hubbard papers. We also discuss examples of possible applications of the sum rule and propose and analyse a solution, fulfilling it, that can be used for a pedagogical introduction to the Mott–Hubbard metal-insulator transition.
We study the frequency-dependent electrical conductivity of a structurally disordered single-band Hubbard model near half-filling and in the strong-correlation limit. The electronic transfer integral is assumed to have a quasi-exponential form as used in the impurity-band problem of doped semiconductors. The short-time expansion of the current correlation function is calculated diagrammatically to the second order in the inverse temperature β and up to the seventh order in electronic hopping. Based on this expansion an approximate current correlation function is constructed. By means of the Kubo formula, the configurational average of the electrical conductivity is then calculated to order β2 for various atomic densities. The validity of the high-temperature expansion is examined. We also study the effect of spin polarization on the absorption line shape and the dc resistivity at different temperatures.
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.
A state-of-the-art review is given of the investigations of the Hubbard model, which is the main model used in the theory of strongly correlated electron systems. It is shown how the main correlation effects — the metal–insulator phase transition, the appearance of localised magnetic moments, and the breakdown of the Fermi-liquid behaviour — appear in the case of strong electron correlations when U ~ W, where U is the Coulomb repulsion parameter and W is the band width. Different approaches to the problem, not based on perturbation theory in terms of the parameter U/W or W/U, are presented. A new method in the theory of strongly correlated systems, in which the limit of infinite number of dimensions of space, d = ∞, is the main feature of the review. The physical results obtained by this approach are compared with those deduced by traditional methods. An overview of the behaviour of strongly correlated systems is given for wide ranges of the model parameters U and W, and of the electron density n.
The sum rule for the moments of the spectral density is discussed for the single-band Hubbard model. It is shown that respecting the sum rule up to the order m = 3 is conceptually important for a qualitatively correct description of the quasi-particle band structure in the strong-correlation regime. Different analytical approximations for the self-energy are analyzed with respect to their compatibility with the moment sum rule. To estimate the practical usefulness of the sum rule, correlation functions and dynamical quantities are determined. The results obtained within the various approximation schemes of different complexity are compared with each other and also with essentially exact results available for infinite-dimensional lattices. It turns out that the m = 3 moment is rather unimportant for the paramagnetic phase on the hyper-cubic lattice. Contrary, it decisively influences the magnetic phase boundary as well as the critical temperature for the ferromagnetic phase on an f.c.c.-type lattice.