Physical review. B, Solid state

Narrow-Band Expansions in the Hubbard Model: A Comment

Robert A. Bari

Published Sep 15, 1970• 8 citations

Abstract

The connection between a solution of the Hubbard Hamiltonian obtained by Esterling and Lange and the low-density solution is discussed. It is shown that their solution agrees to first order in the bandwidth with the exact solution for the two-electron $N$-site case. However, even this correct first-order expansion fails to adequately account for the low-frequency behavior of the Green's function and related quantities.

DOI: 10.1103/physrevb.2.2260

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Physical review. B, Solid state

Narrow-Band Expansions in the Hubbard Model: A Comment

Robert A. Bari

Published Sep 15, 1970• 8 citations

Publisher

Abstract

N

-site case. However, even this correct first-order expansion fails to adequately account for the low-frequency behavior of the Green's function and related quantities.

DOI: 10.1103/physrevb.2.2260

Related papers

Moment-Generated Solution to the Hubbard Narrow-Energy-Band Model

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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.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.

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Comment on 'narrow band solution to the Hubbard model'

John A. Blackman, D. M. Esterling

W. D. Langer's solution to the Hubbard model is shown to be incorrect to lowest order in the bandwidth. (see ibid. vol. 4, l56, 1971).

Journal of Physics C Solid State PhysicsAug 16, 19712 citations

Functional-derivative study of the Hubbard model. III. Fully renormalized Green's function

Tadashi Arai, Morrel H. Cohen

The functional-derivative method of calculating the Green's function developed earlier for the Hubbard model is generalized and used to obtain a fully renormalized solution. Higher-order functional derivatives operating on the basic Green's functions, $G$ and $\ensuremath{\Gamma}$, are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms $\ensuremath{\Phi}$ generated from functional derivatives of equal-time Green's functions of the type $\frac{{\ensuremath{\delta}}^{n}〈N〉}{\ensuremath{\delta}{\ensuremath{\epsilon}}^{n}}$, etc., with $n\ensuremath{\ge}2$. It is found that the $\ensuremath{\Phi}'\mathrm{s}$ are, in fact, renormalization factors involved in the self-energy $\ensuremath{\Sigma}$ and that the structure of the $\ensuremath{\Phi}'\mathrm{s}$ resembles that of $\ensuremath{\Sigma}$ and contains the same renormalization factors $\ensuremath{\Phi}$. The renormalization factors $\ensuremath{\Phi}$ are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the $\ensuremath{\Phi}'\mathrm{s}$, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum $\ensuremath{\omega}$ can now be evaluated for all occupations $n$ . The Schwinger relation is the only basic relation used in generating this fully self-consistent Green's function, and the Baym-Kadanoff continuity condition is automatically satisfied.The functional-derivative method of calculating the Green's function developed earlier for the Hubbard model is generalized and used to obtain a fully renormalized solution. Higher-order functional derivatives operating on the basic Green's functions, $G$ and $\ensuremath{\Gamma}$, are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms $\ensuremath{\Phi}$ generated from functional derivatives of equal-time Green's functions of the type $\frac{{\ensuremath{\delta}}^{n}〈N〉}{\ensuremath{\delta}{\ensuremath{\epsilon}}^{n}}$, etc., with $n\ensuremath{\ge}2$. It is found that the $\ensuremath{\Phi}'\mathrm{s}$ are, in fact, renormalization factors involved in the self-energy $\ensuremath{\Sigma}$ and that the structure of the $\ensuremath{\Phi}'\mathrm{s}$ resembles that of $\ensuremath{\Sigma}$ and contains the same renormalization factors $\ensuremath{\Phi}$. The renormalization factors $\ensuremath{\Phi}$ are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the $\ensuremath{\Phi}'\mathrm{s}$, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum $\ensuremath{\omega}$ can now be evaluated for all occupations $n$. The Schwinger relation is the only basic relation used in generating this fully self-consistent Green's function, and the Baym-Kadanoff continuity condition is automatically satisfied.

Physical review. B, Condensed matterApr 15, 19807 citations

N

-site case. However, even this correct first-order expansion fails to adequately account for the low-frequency behavior of the Green's function and related quantities.

G

and calculated iteratively in a perturbation scheme which takes the Hubbard I solution

{G}_{0}

as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting

{G}_{0}

into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\ensuremath{\epsilon}$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.

G

and $\ensuremath{\Gamma}$, are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms $\ensuremath{\Phi}$ generated from functional derivatives of equal-time Green's functions of the type $\frac{{\ensuremath{\delta}}^{n}〈N〉}{\ensuremath{\delta}{\ensuremath{\epsilon}}^{n}}$, etc., with $n\ensuremath{\ge}2$. It is found that the $\ensuremath{\Phi}'\mathrm{s}$ are, in fact, renormalization factors involved in the self-energy $\ensuremath{\Sigma}$ and that the structure of the $\ensuremath{\Phi}'\mathrm{s}$ resembles that of $\ensuremath{\Sigma}$ and contains the same renormalization factors $\ensuremath{\Phi}$. The renormalization factors $\ensuremath{\Phi}$ are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the $\ensuremath{\Phi}'\mathrm{s}$, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum $\ensuremath{\omega}$ can now be evaluated for all occupations

n

. The Schwinger relation is the only basic relation used in generating this fully self-consistent Green's function, and the Baym-Kadanoff continuity condition is automatically satisfied.