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Physical review. B, Condensed matter

Exact calculation of the two-electron interaction in the ground state of the Hubbard model

Liang Chen, Changjiang Mei

Published May 1, 1989• 18 citations
Publisher

Abstract

The single-band Hubbard model has been studied in both the one-dimensional ring and two-dimensional torus. By the exact calculation of the pair correlation function F(i,j), defined as the possibility of finding an electron at site j when there has been an electron at site i, it is shown that for two electrons, the interaction is always repulsive in the ground state for any positive value of the on-site Coulomb interaction U.

DOI
10.1103/physrevb.39.9006

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The single-band Hubbard model has been studied in both the one-dimensional ring and two-dimensional torus. By the exact calculation of the pair correlation function F(i,j), defined as the possibility of finding an electron at site j when there has been an electron at site i, it is shown that for two electrons, the interaction is always repulsive in the ground state for any positive value of the on-site Coulomb interaction U.
We study the one-band Hubbard model in a three dimensional simple cubic lattice, with periodic boundary conditions, by means of a variational analytic approach. Ground state energies and pairing correlations depend implicitly on the interaction strength (U/41). It is shown that for two electrons, the interaction is always repulsive in the ground state for any positive value of the on-site Coulomb interaction U. Implications of this result for superconductivity are also discussed. Nigerian Journal of Physics Vol. 16(2) 2004: 167-171
The major characteristic of the single-band Hubbard model (HM) is to redistribute electrons at a uniform lattice separation distance within the molecular lattice. Hence, it is only linearly dependent on lattice separations distance. Thus the single-band Hubbard model does not consider the lattice gradient encountered by interacting electrons as they hop from one lattice point to another. The linear dependence of the single-band HM only on lattice separations would certainly not provide a thorough understanding of the interplay between interacting electrons. Consequently, we have in this study developed a gradient Hamiltonian model to solve the associated defects pose by the limitations of the single band Hubbard model. Thus, we utilized the single-band HM and the gradient Hamiltonian model to study the behaviour of two interacting electrons on a two dimensional (2D) 9X9 square lattice. It is revealed in this study that the results of the ground-state energies produced by the gradient Hamiltonian model are more favourable when compared to those of the single-band Hubbard model. We have also shown in this work, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength.
We use the C4vC_{4v}C4v​ symmetry group of the 4-site Hubbard model to construct a ground state variational wave function of two- and four interacting electrons. In the limit U→0U\rightarrow 0U→0, ground state energies of the two- and four interacting electrons system is of the order −4t-4t−4t. The variational wave function of the four interacting electrons obtained using the B1B_1B1​ irreducible representation is valid for on-site Coulomb repulsion, while the one obtained using the A1A_1A1​ representation is valid for negative values of the Coulomb interaction. The system exhibits antiferromagnetic correlations.
The single-band Hubbard model (both with and without first-neighbor interactions) is studied by exact diagonalization on several one- and two-dimensional clusters, ranging from four to nine sites, including rings and arrangements of squares. Occupancies include the half-filled band (one electron per site) and configurations with one and two holes in the half-filled band. Ground-state properties are determined for a wide range of parameters. We discuss the phase diagram in the (U,V) plane and consider spin, charge, and pairing correlations. The binding of a pair of holes is investigated. Circumstances in which binding and pairing are enhanced by first-neighbor interactions are described.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.