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Acta Physica Polonica A

A Pairwise Approximation in the Hubbard Model

Jacob Szeftel

Published Feb 1, 1994• 2 citations
Publisher

Abstract

The Hubbard Hamiltonian is projected onto a representation consisting of electron pairs characterised by the momentum of their centre of mass.Within this approximation the electron gas can be viewed as a collection of subsets, each of which contains a constant number of electron pairs, all having the same centre of mass momentum.As these subsets are decoupled, the Hubbard Hamiltonian is diagonalised to give two types of many-body eigenstates: correlated and uncorrelated.The uncorrelated pairs build up an ideal Fermi gas.Excellent agreement is found for the uncorrelated energy calculated at zero temperature in one dimension between this model and the Bethe ansatz, for arbitrary electron concentration and magnitude of the electron interaction.The correlated states turn out to be of the BCS type.

DOI
10.12693/aphyspola.85.329

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The Hubbard Hamiltonian is projected onto a representation consisting of electron pairs characterised by the momentum of their centre of mass.Within this approximation the electron gas can be viewed as a collection of subsets, each of which contains a constant number of electron pairs, all having the same centre of mass momentum.As these subsets are decoupled, the Hubbard Hamiltonian is diagonalised to give two types of many-body eigenstates: correlated and uncorrelated.The uncorrelated pairs build up an ideal Fermi gas.Excellent agreement is found for the uncorrelated energy calculated at zero temperature in one dimension between this model and the Bethe ansatz, for arbitrary electron concentration and magnitude of the electron interaction.The correlated states turn out to be of the BCS type.
We extend previous studies of the (BCS) canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard model and a simpler reduced Hamiltonian. Using direct diagonalization, exact one- and two-dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one-dimensional findings, the BCS approach is an excellent approximation for the Hubbard ground state energy but not the energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wave functions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. Even for weak coupling pronounced differences are found between the BCS and Hubbard on-site correlators. Significantly, in the extreme strong coupling limit the Hubbard model exhibits no dimensional crossover, or density induced transitions, to long-range order in two dimensions. However, for the reduced Hamiltonian the BCS approach is an excellent approximation for all observables in both one and two dimensions.
A novel variational approach is outlined for treating electron correlation in many-particle systems and is applied to the one-band Hubbard model. The authors' treatment is based on the concept of fermion pairs and their representation in terms of ideal-boson operators. In this representation approximate eigen-states of the Hamiltonian are obtained in the form of an antisymmetrised product of pair orbitals. For the ground state of the Hubbard model these orbitals are then determined by the use of the variational principle. Because of the complexity of the resulting equation a simplified ansatz is used for the orbitals, which leads to the same energy expression as is obtained by the traditional Bogolubov method. The trial states used in both approaches are, however, quite different. They find reasonable qualitative agreement with well established results for the Hubbard model in one dimension even in those cases where the Bogolubov or related approaches fail. They are also able to show that in the atomic limit the Bogolubov energy bound is the best upper bound that can be obtained by using a single determinant of pair orbitals. Therefore, the only way to improve the ground-state energy in the atomic limit is by means of configuration interaction.
One- and two-dimensional attractive Hubbard models in the presence of magnetic field h are studied in a wide range of on-site interaction strength U, temperature T and electron concentration n by means of the generalized self-consistent field (GSCF) approach with renormalized chemical potential. Pairing, the smooth crossover from the BCS regime into the Bose condensation state and the inhomogeneous superconductivity with the non-uniform order parameter are investigated. The excitation spectrum, the momentum distribution function and thermodynamic properties, including the concentration of double occupied sites, the kinetic energy, and the chemical potential, are calculated. The numerical calculations are found in a good agreement with the Bethe-ansatz results and quantum Monte-Carlo calculations in a wide range of system parameters U, n and h. The GSCF theory correctly displays the separation of the energy gap from the order parameter (crossover) as n or |U| decrease. The concentration of double occupied sites and the bound electron pairs n pair are calculated and we found a simple relationship between the order parameter and n pair , valid for the arbitrary U and all n in any dimension. The developed GSCF approach in magnetic field suggests a smooth transition into the inhomogeneous superconducting state.
We start from a Hamiltonian describing non-interacting fermions and add bosons to the model, with a Jaynes-Cummings-like interaction between the bosons and fermions. Because of the specific form of the interaction the model can be solved exactly. In the ground state, part of the electrons form bound pairs with opposite momentum and spin. The model also shows a gap in the kinetic energy of the fermions, but not in the spectrum of the full Hamiltonian. This gap is not of a mean-field nature, but is due to the Pauli exclusion principle.
Ground-state properties of the attractive Hubbard model in one dimension are studied by means of both the exact Bethe-ansatz formalism and the self-consistent field (SCF) approach with renormalized chemical potential for general band fillings n and a wide range of coupling strength U/t.U/t.U/t. The energy, the concentration of double occupied sites, the kinetic energy and the chemical potential of the ground state are in a good numerical agreement with the exact results over a wide range of parameters U/tU/tU/t and n. The concentration of local pairs or double occupied sites in the Bethe-ansatz solution serves as a suitable parameter measuring the electron pairing correlations. The SCF theory provides a simple analytical relationship between the concentration of double occupied sites, the band filling and the BCS order parameter, valid for arbitrary U/tU/tU/t and n. The calculated energy gap, the BCS order parameter, the phase diagram and the compressibility are also discussed. The SCF theory in one dimension distinguishes the order parameter from the excitation gap and suggests a smooth crossover away from half-filling (at $n\ensuremath{\ne}1)$ from the BCS pairing to the Bose condensation regime under the variation of U/tU/tU/t and n.