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Journal of the Physical Society of Japan

Three-Body Correlations on the Antiferromagnetic State - An Extension of Local Approximation -

Jun-ichi Igarashi, Tatsuya Nagao

Published Apr 15, 2000• 0 citations
Publisher

Abstract

We propose an approximate version of the three-body scattering theory as an extension of local approximation. We solve the three-body problem by limiting the number of three-body states, instead of solving the Faddeev equation, in the calculation of the self-energy. Applying this simple extension to the antiferromagnetic state at half-filling in the two-dimensional Hubbard model, we demonstrate that it gives a reasonable spectral function for the single-particle Green's function in comparison with the results of the exact diagonalization for finite clusters and the quantum Monte Carlo method. We also show that it gives the ground state energy and the sublattice magnetization comparable to previous variational Monte Carlo results.

DOI
10.1143/jpsj.69.1219

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We propose an approximate version of the three-body scattering theory as an extension of local approximation. We solve the three-body problem by limiting the number of three-body states, instead of solving the Faddeev equation, in the calculation of the self-energy. Applying this simple extension to the antiferromagnetic state at half-filling in the two-dimensional Hubbard model, we demonstrate that it gives a reasonable spectral function for the single-particle Green's function in comparison with the results of the exact diagonalization for finite clusters and the quantum Monte Carlo method. We also show that it gives the ground state energy and the sublattice magnetization comparable to previous variational Monte Carlo results.
As a continuation of a previous paper [J. Phys. Soc. Jpn. 56 (1987) 1490], the variational Monte-Carlo method is extended to include the antiferromagnetic long-range order. The theory is based on the Gutzwiller-type correlation factor and its effect is exactly taken into account by the Monte-Carlo procedure. An application is made to the half-filled-band case of one-dimensional lattice (50 sites), two-dimensional square lattice (up to 20×20 sites) and three-dimensional simple cubic lattice (6×6×6 sites). The result is qualitatively different from previous studies relying on the random-phase-type "Gutzwiller approximation." The variational energy for two and three dimensions is favorably compared with Hirsch's quantum Monte-Carlo data.
The exchange-correlation potential formalism previously introduced and applied to the one-dimensional Hubbard model has been extended to spin systems and applied to the case of the one-dimensional antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg model. Within the spin exchange-correlation potential formulation, a sum rule for spin systems is derived. The exchange-correlation potential for the Heisenberg model is extrapolated from exact diagonalization results of small antiferromagnetic Heisenberg clusters. This procedure is also employed to revisit and computationally improve the previous investigation of the exchange-correlation potential of the half-filled Hubbard model, which was based on the exchange-correlation potential of the dimer. Numerical comparisons with exact benchmark calculations for both the Heisenberg and the Hubbard models indicate that, starting from the exchange-correlation potential of a finite cluster, the extrapolation procedure yields a one-particle spectral function with favorable accuracy at a relatively low computational cost. In addition, a comparison between the ground-state energies for the one-dimensional Hubbard and Heisenberg models displays how the well-known similarity in behavior of the two models at large interactions manifests itself within the exchange-correlation potential formalism.
Antiferromagnetic states of the half-filled Hubbard model on the d=1, 2, 3, and \ensuremath{\infty} dimensional lattices are systematically studied by using the Gutzwiller-type analytical approximation based on the slave-boson functional-integral approach. In all the cases investigated, antiferromagnetic states have lower energies than paramagnetic states even for infinitesimally small interactions, U. Calculated results of the ground-state energy E and the sublattice moment m for d=2 and 3 are in very good agreement with those of the Monte Carlo simulations. Even for d=1 the antiferromagnetic state in our Gutzwiller-type approximation has energy much closer to the exact one than the paramagnetic states in Gutzwiller's approximation and in Metzner-Vollhardt's approximation-free method. On the basis of the 1/d plot of E and m, a possibility of the 1/d expansion starting from the limit of d=\ensuremath{\infty} is discussed. It is also shown that the Brinkman-Rice metal-insulator transition does not take place at finite U for all d.
Electron correlation is studied in the special case of a single reversed spin in an otherwise fully aligned band in the one-dimensional Hubbard model for the purpose of examining our previous theory which exactly takes account of the three-body correlation. The importance of the three-body correlation is demonstrated through a comparison of our results with those of exact theories. The ground-state energy and the single particle spectra are in good agreement with McGuire-Lieb-Wu's exact solution and also with exact numerical solutions for finite size systems over a wide range of the intraatomic Coulomb integral and the electron number. The properties of the ground state are studied in detail through the calculation of pair correlation functions.
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.