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Modern Physics Letters B

GROUND-STATE ENERGY OF THE ONE-DIMENSIONAL HUBBARD MODEL IN A SIMPLE SELF-CONSISTENT VERSION OF THE LADDER APPROXIMATION

Florin D. Buzatu

Published Aug 10, 1995• 7 citations
Publisher

Abstract

The ground-state energy of the one-dimensional Hubbard model is calculated within the ladder approximation; from the comparison with the exact results in the repulsive case, it follows that the approximation is good at low densities or small couplings. The ladder approximation can be improved by imposing a self-consistency condition; using a simple assumption, the results become close to the exact ones in a large range of the model parameters.

DOI
10.1142/s0217984995001145

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International Journal of Modern Physics B•Dec 10, 2003•6 citations
The ground-state energy of the one-dimensional Hubbard model is calculated within the ladder approximation; from the comparison with the exact results in the repulsive case, it follows that the approximation is good at low densities or small couplings. The ladder approximation can be improved by imposing a self-consistency condition; using a simple assumption, the results become close to the exact ones in a large range of the model parameters.
The ground-state energy of a one-dimensional fermion system interacting via a repulsive $\ensuremath{\delta}$-function potential is calculated rigorously within the "ladder" approximation as a function of the coupling constant and the results compared with those given by the exact solution of Yang. Comparison with other many-body approximate theories has also been made. The density-density response function has also been calculated in an approximation in which the particle-hole interaction in the polarization bubble is taken into account to all orders. Results of this dynamic theory are compared with those of other approximations.
The ground-state energy and energy gap to the first excited state are calculated for the attractive Hubbard model in one dimension using both the Bethe ansatz equations and the variational BCS wave function. Comparisons are provided as a function of coupling strength and electron density. While the ground-state energies are always in very good agreement, the BCS energy gap is sometimes incorrect by an order of magnitude, particularly at half-filling. Finite-size effects are also briefly discussed for cases where an exact solution in the thermodynamic limit is not possible. In general, the BCS result for the energy gap is poor compared to the exact result.
The ground-state properties in one-dimensional Hubbard model with on-site attraction and repulsion of electrons in the presence of magnetic field h are calculated by means of the exact Bethe-ansatz formalism and the generalized self-consistent field (GSCF) approach for general electron concentrations n and arbitrary interaction strength. The ground-state properties, including the energy, the average spin (magnetization) and the kinetic energy are compared over a wide range of parameter space. The GSCF theory is in qualitative and in some cases in good quantitative agreement with the exact results. The GSCF theory at U≤0 (or U≥0) differentiates the spin (or charge) energy gap from the BCS (or antiferromagnetic) order parameter and suggests a smooth crossover from the phase with the itinerant BCS-like behavior to the Bose condensation regime of the local pairs.
The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.
The shortcomings and advantages of the generalized self-consistent field (GSCF) approach, which includes the electron–hole order parameter [Formula: see text] and the average spin s within the one-dimensional repulsive Hubbard model, are analyzed by comparison of the ground state properties with the corresponding Bethe ansatz results in an entire parameter space of interaction strength U/t ≥ 0, magnetic field h ≥ 0 and electron concentration 0 ≤ n ≤ 1. The GSCF spectral characteristics are derived and criteria are found for the stability of incommensurate magnetic phase with the wave number 0 < q < π. The GSCF theory displays a simple relationship for the double occupancy D(+) in terms of n, s and [Formula: see text], where D(+) underestimates electron correlations at weak and intermediate ranges and overestimates correlations at large interaction strengths. Beyond some critical U/t and n≠1 the GSCF D(+) for spatially homogeneous state vanishes, while the exact D for all n at h=0 decreases gradually as U/t increases. At n ≠ 1 the GSCF chemical potential μ(+) overestimates electron correlations everywhere and variation μ(+)versusn displays electron instability toward the phase separation in the vicinity of n=1. Exactly at n=1 the GSCF μ(+)versusU/t always underestimates electron correlation and as U/t → ∞ at h=0 we have μ(+) → 0, while the Bethe ansatz result gives μ → 2t. In the limiting cases U/t → 0 and U/t → ∞ for all h ≥ 0 and 0 ≤ n ≤ 1 the GSCF ground state energy is exact, which is necessary for formulation of converging perturbation procedure about mean field solution in the entire parameter space.