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Physical review. B, Condensed matter

Large-Uexpansions for the Hubbard model atT=0

J. Oitmaa, Jon Henderson

Published Oct 1, 1991• 9 citations
Publisher

Abstract

We report results for the ground-state energy of the Hubbard model in the form of perturbation expansions in t/U, i.e., expansions about the strong-coupling or ``atomic'' limit. Expansions to order (t/U${)}^{40}$ are obtained for finite cells with number of sites \ensuremath{\le}10, for the half-filled case, for both the linear chain and two-dimensional (2D) square lattice. Analysis of the series by Pad\'e approximant methods yields, in the 1D case, results consistent with the known exact results. For the largest cell, N=10, on the square lattice there is an indication of a singularity on the real t axis, which may indicate a phase transition at U/t\ensuremath{\simeq}4.

DOI
10.1103/physrevb.44.7433

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We report results for the ground-state energy of the Hubbard model in the form of perturbation expansions in t/U, i.e., expansions about the strong-coupling or ``atomic'' limit. Expansions to order (t/U)40{)}^{40})40 are obtained for finite cells with number of sites \ensuremath{\le}10, for the half-filled case, for both the linear chain and two-dimensional (2D) square lattice. Analysis of the series by Pad'e approximant methods yields, in the 1D case, results consistent with the known exact results. For the largest cell, N=10, on the square lattice there is an indication of a singularity on the real t axis, which may indicate a phase transition at U/t\ensuremath{\simeq}4.
We investigate the ground state of the Hubbard model in one and two dimensions. Long perturbation expansions in t/U, i.e., about the atomic limit, are computed for the ground-state energy of finite lattices with N\ensuremath{\le}10 for arbitrary electron density. We also extend the series for the one-dimensional infinite lattice by expanding the Bethe-ansatz equations. Pad\'e-approximant analysis of the series shows a distribution of singularities that varies dramatically with the electron density n. We also investigate the effect of inclusion of next-nearest-neighbor hopping. This appears to have little effect at half-filling but is significant away from half-filling.
We investigate the Hubbard model on a two-dimensional square lattice by the perturbation expansion to the fourth order in the on-site Coulomb repulsion U. Numerically calculating all diagrams up to the fourth order in self-energy, we examine the convergence of perturbation series in the lattice system. We indicate that the coefficient of each order term rapidly decreases as in the impurity Anderson model for T > 0.1t in the half-filled case, but it holds in the doped case even at lower temperatures. Thus, we can expect that the convergence of perturbation expansion in U is very good in a wide parameter region also in the lattice system, except for T < 0.1t in the half-filled case. We next calculate the density of states in the fourth-order perturbation. In the half-filled case, the shape in a moderate correlation regime is quite different from the three peak structure in the second-order perturbation. Remarkable upper and lower Hubbard bands locate at w = +(-)U/2, and a pseudogap appears at the Fermi level w=0. This is considered as the precursor of the Mott-Hubbard antiferromagnetic structure. In the doped case, quasiparticles with very heavy mass are formed at the Fermi level. Thus, we conclude that the fourth-order perturbation theory overall well explain the asymptotic behaviors in a strong correlation regime.
The ground-state energy of the Hubbard model on a Bethe lattice with infinite connectivity at half filling is calculated for the insulating phase. Using Kohn's transformation to derive an effective Hamiltonian for the strong-coupling limit, the resulting class of diagrams is determined. We develop an algorithm for an algebraic evaluation of the contributions of high-order terms and check it by applying it to the Falicov-Kimball model that is exactly solvable. For the Hubbard model, the ground-state energy is exactly calculated up to order t12/U11{t}^{12}/{U}^{11}t12/U11. The results of the strong-coupling expansion deviate from numerical calculations as quantum Monte Carlo (or density-matrix renormalization group) by less than 0.130.13%0.13 (0.320.32%0.32 respectively) for U>4.76U>4.76U>4.76.
We develop series expansions for the ground state properties of the Hubbard model by introducing an Ising anisotropy into the Hamiltonian. For the two-dimensional square lattice half-filled Hubbard model, the ground state energy, local moment, sublattice magnetization, uniform magnetic susceptibility, and spin stiffness are calculated as a function of U/t, where U is the Coulomb constant and t is the hopping parameter. Magnetic susceptibility data indicate a crossover around U\ensuremath{\approxeq}4 between spin density wave antiferromagnetism and Heisenberg antiferromagnetism. Comparisons with Monte Carlo simulations, random phase approximation result, and mean-field solutions are also made.
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.