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International Journal of Quantum Chemistry

Perturbation expansion of the ground‐state energy for the one‐dimensional cyclic Hubbard system in the Hückel limit

M. Takahashi, Paul Bracken, Jakub Čı́žek, Josef Paldus

Published Mar 5, 1995• 10 citations

Abstract

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.

DOI: 10.1002/qua.560530502

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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.

Abstract The applicability of the finite‐order many‐body perturbation theory to the electron correlation problem in extended one‐dimensional systems is examined. The cyclic polyenes CNHN, N = 4ν + 2, ν = 1, 2, …, with the DNh geometry as described by both the Pariser–Parr–Pople and Hubbard Hamiltonians, are employed to model the metallic‐like one‐dimensional systems. The second‐order perturbation theory contributions to the correlation energy are obtained with three different partitionings of the Hamiltonian (Hückel, M⊘ller–Plesset, and Epstein–Nesbet). The third‐ and fourth‐order contributions are also calculated in special cases. A comparison with other methods is given wherever available. For the Hubbard Hamiltonian the asymptotic behavior of the perturbation theory expansion is examined numerically. It is shown that the finite‐order perturbation expansion can provide reliable results for the correlation energy of one‐dimensional systems even in the correlation region which corresponds to the spectroscopically determined physical value of the coupling constant.

The one-dimensional Hubbard model is studied for general electron densities and large U. On the basis of the coupled integral equations of Lieb and Wu exact asymptotic expansions are derived to fifth order in ${U}^{\mathrm{\ensuremath{-}}1}$ for the ground-state energy, the kinetic energy, and the number of doubly occupied sites. A closed-form expression for the distribution functions of charge and spin degrees of freedom is proposed which yields the same large-U expansion as that derived directly from the Lieb-Wu equations. Our analytical results are in excellent agreement with previous numerical studies.

The ground-state energy E and momentum distribution n k of the electrons are expanded from the atomic limit for the half-filled Hubbard model. The coefficients of expansion are represented by the spin correlation functions of the spin 1 / 2 Heisenberg model. Using the spin-wave theory, approximate values of coefficients are calculated for the square lattice and the simple cubic lattice. In one dimension, the theory shows a good agreement with the exact solution. An effective spin Hamiltonian for the half-filled Hubbard model with arbitrary hopping integrals is obtained up to the fifth order. It is shown that the fourth term contains four spin interactions.

Using a straightforward extension of the analysis of Lieb and Wu, we derive a simple analytic form for the ground state energy of a one-dimensional Hubbard ring in the atomic limit. This result is valid for an \textit{arbitrary} number of lattice sites

L

and electrons

N \leq L

. Furthermore, our analysis, including an application of the theory of stochastic matrices, provides insight into the degeneracy and spin properties of the ground states in the atomic limit. We give numerical results which illustrate how the atomic limit is approached.