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Physical review. B, Condensed matter

Determination of the radius of convergence of the perturbation expansion using Padé approximants: Application to the Hückel limit of the Hubbard model for finite cyclic polyenes

M. Takahashi, Jakub Čı́žek, Josef Paldus

Published Jan 15, 1986• 7 citations
Publisher

Abstract

The radius of convergence of a perturbation expansion is estimated using Pad\'e approximants. Location of the branch point which is nearest to the origin is evaluated directly from the pole of the denominator of an [M,M] Pad\'e approximant and indirectly from the logarithmic derivative of this approximant. Both methods are applied to the perturbation expansion for the ground-state energy of the finite cyclic Hubbard systems. The method using the logarithmic derivative yields numerical results which are in good agreement with those obtained by other methods. Kato's formula is also applied.

DOI
10.1103/physrevb.33.1203

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The radius of convergence of a perturbation expansion is estimated using Pad'e approximants. Location of the branch point which is nearest to the origin is evaluated directly from the pole of the denominator of an [M,M] Pad'e approximant and indirectly from the logarithmic derivative of this approximant. Both methods are applied to the perturbation expansion for the ground-state energy of the finite cyclic Hubbard systems. The method using the logarithmic derivative yields numerical results which are in good agreement with those obtained by other methods. Kato's formula is also applied.
Abstract The perturbation expansion coefficients for the ground‐state energy of the half‐filled one‐dimensional Hubbard model with N = 4ν + 2, (ν = 1, 2,…) sites and satisfying cyclic boundary conditions were calculated in the Hückel limit (U/β → 0), where β designates the one‐electron hopping or resonance integral, and U, the two‐electron on‐site Coulomb integral. This was achieved by solving—order by order—the Lieb–Wu equations, a system of transcendental equations that determines the set of quasi‐momenta {ki} and spin variables {τα} for this model. The exact values for these quantities were found for the N = 6 member ring up to the 20th order in terms of the coupling constant B = U/2β, as well as numerically for 10 ⩽ N ⩽ 50, and the N = 6 Lieb–Wu system was reduced to a system of sextic algebraic equations. These results are compared with those of the infinite system derived analytically by Misurkin and Ovchinnikov [Teor. Mat. Fiz. 11, 127 (1972)]. It is further shown how the results of this article can be used for the interpolation by the root of a polynomial. It is demonstrated that a polynomial of relatively small degree provides a very good approximation for the energy in the intermediate coupling region. © 1995 John Wiley & Sons, Inc.
We report results for the ground-state energy of the Hubbard model in the form of perturbation expansions in t/U, i.e., expansions about the strong-coupling or ``atomic'' limit. Expansions to order (t/U)40{)}^{40})40 are obtained for finite cells with number of sites \ensuremath{\le}10, for the half-filled case, for both the linear chain and two-dimensional (2D) square lattice. Analysis of the series by Pad\'e approximant methods yields, in the 1D case, results consistent with the known exact results. For the largest cell, N=10, on the square lattice there is an indication of a singularity on the real t axis, which may indicate a phase transition at U/t\ensuremath{\simeq}4.
We investigate the ground state of the Hubbard model in one and two dimensions. Long perturbation expansions in t/U, i.e., about the atomic limit, are computed for the ground-state energy of finite lattices with N\ensuremath{\le}10 for arbitrary electron density. We also extend the series for the one-dimensional infinite lattice by expanding the Bethe-ansatz equations. Pad\'e-approximant analysis of the series shows a distribution of singularities that varies dramatically with the electron density n. We also investigate the effect of inclusion of next-nearest-neighbor hopping. This appears to have little effect at half-filling but is significant away from half-filling.
A general method is presented for obtaining an accurate intermediate-coupling theory from weak- and strong-coupling perturbation theory. The method uses two-point Pad\'e approximants to extrapolate (low-order) expansions about the weak- and the strong-coupling limits into the intermediate-coupling regime. The method is used to evaluate the ground-state energy and effective mass of the polaron with gratifying success. In addition, the weak-coupling perturbation theory of the polaron dispersion relation, ground-state energy, and effective mass are extended to fourth, sixth, and fourth order, respectively. A scheme based on two-point Pad\'e approximants is used to obtain an optimal polaron dispersion relation.