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Journal of Mathematical Physics

Reconstruction of secular polynomials for Hubbard model from energy perturbation series for weak and strong coupling cases

Paul Bracken, Jiřı́ Čı́žek

Published Nov 1, 1997• 3 citations
Publisher

Abstract

An interpolation technique which is based on the strongly and weakly correlated cases of the one-dimensional Hubbard model is proposed. The input information consists of the perturbation expansions which are obtained from the Lieb–Wu equations in both limits. The Hubbard model is used to describe cyclic polyene rings, and for the case of N=6 sites, this would correspond to benzene. The technique has been applied to several symmetries of the model. It has been shown that the exact secular problem can be reconstructed for these symmetries, and the results for N=6 agree exactly with the results which have been obtained in a different way, that is, by using a full configuration interaction calculation.

DOI
10.1063/1.532179

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arXiv:cond-mat/9704005•Physical review. B, Condensed matter•Jun 15, 1997•86 citations
An interpolation technique which is based on the strongly and weakly correlated cases of the one-dimensional Hubbard model is proposed. The input information consists of the perturbation expansions which are obtained from the Lieb–Wu equations in both limits. The Hubbard model is used to describe cyclic polyene rings, and for the case of N=6 sites, this would correspond to benzene. The technique has been applied to several symmetries of the model. It has been shown that the exact secular problem can be reconstructed for these symmetries, and the results for N=6 agree exactly with the results which have been obtained in a different way, that is, by using a full configuration interaction calculation.
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 optimized-inner-projection (OIP) method is implemented and applied to determine the lower bounds to the ground-state correlation energy for the Pariser-Parr-Pople and Hubbard models of cyclic polyenes CN{\mathrm{C}}_{\mathit{N}}CN​HN{\mathrm{H}}_{\mathit{N}}HN​, N=4\ensuremath{\nu}+2, \ensuremath{\nu}=3--5, in the whole range of the coupling constant. In actual calculations, the orthogonally spin-adapted form of the OIP equations is used, and full advantage is taken of the high symmetry of the cyclic polyene model, so that systems with N\ensuremath{\ge}14 can be explored. Comparison is made with the results of both full and limited configuration-interaction calculations and with the correlation energies obtained with certain approximate coupled-cluster schemes. Contrary to the six- and ten-membered rings, the OIP lower bounds to the ground-state correlation energy of cyclic polyenes with larger number of sites are very poor and cannot serve as a meaningful source of information. Since the eigenvalues of the intermediate Hamiltonian, in terms of which the OIP technique is defined, are highly degenerate even in the weakly correlated region, and in view of the random behavior of the OIP bracketing function, the OIP method becomes computationally very demanding for N\ensuremath{\ge}14 rings. Moreover, simple iterative schemes usually diverge and sophisticated root searching procedures have to be applied. The usefulness of the OIP technique for the determination of lower bounds to energy eigenvalues in larger systems is thus questioned.
We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.
We develop an analytical expression for the self-energy of the infinite-dimensional Hubbard model that is correct in a number of different limits. The approach represents a generalization of the iterative perturbation theory to arbitrary fillings. In the weak-coupling regime perturbation theory to second order in the interaction U is recovered. The theory is exact in the atomic limit. The high-energy behavior of the self-energy up to order 1/E2{\mathrm{E}}^{2}E2 and thereby the first four moments of the spectral density are reproduced correctly. Referring to a standard strong-coupling moment method, we analyze the limit U->\ensuremath{\infty}. Different modifications of the approach are discussed and tested by comparing with the results of an exact diagonalization study.