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Progress of Theoretical Physics

The Antiferromagnetic Ground State of a Half-Filled Hubbard Model

K. Kubo, M. Uchinami

Published Nov 1, 1975• 12 citations
Publisher

Abstract

The effect of electron correlations on the antiferromagnetic ground state of a half-filled Hubbard model is examined by means of two approximate ways; Gutzwiller's variational method and the perturbational method. Both results by the two ways indicate that the antiferromagnetism is not destroyed by correlation effect in the limit of weak interaction.

DOI
10.1143/ptp.54.1289

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By using the classic approximation method of Singwi, Tosi, Land, and Sj\"olander (STLS), we calculate the ground-state energy of the one- and two-dimensional Hubbard model in the half-filled antiferromagnetic state. In the one-dimensional case, by comparison with the exact result of Lieb and Wu, we find that the STLS correlation energy is generally better than that obtained from the random-phase approximation (RPA) and competes in quality with the best values obtained from variational calculations based on the antiferromagnetic Gutzwiller wave function (AFGWF). In particular, the STLS approximation yields a maximum in the absolute value of the correlation energy as a function of U, which is also found in the exact treatment, but not in the RPA. However, the numerical values of the STLS correlation energy are not as accurate as in the case of the uniform electron gas. In two dimensions, where the exact energy is not known, we find a similar relationship between the results of STLS, RPA, and AFGWF calculations, but the absolute value of the correlation energy is smaller. For large values of U, in both dimensions, the STLS method (like the RPA) runs into difficulties, due to the insufficient treatment of transverse spin fluctuations.

Physical review. B, Condensed matter•Nov 1, 1989•22 citations
The effect of electron correlations on the antiferromagnetic ground state of a half-filled Hubbard model is examined by means of two approximate ways; Gutzwiller's variational method and the perturbational method. Both results by the two ways indicate that the antiferromagnetism is not destroyed by correlation effect in the limit of weak interaction.
We explore the phase diagram of the Hubbard model by generalizing the Gutzwiller variational wave function to take into account nearest-neighbor spin correlations. The energies of various variational states are calculated and compared. At half filling the lowest energy state has long-range antiferromagnetic correlations, in agreement with previous results. Away from half filling, when the antiferromagnet no longer has the lowest energy, the Gutzwiller approximation yields a lower energy than that found for small spin polarons.
The method of Gutzwiller in the problem of electron correlation in the Hubbard model is extended to the antiferromagnetic state. It is shown that the transition from the paramagnetic to the antiferromagnetic state occurs at a certain value of U (intraatomic Coulomb energy) for a fixed value of the electron concentration. In contrast to the result in the Hartree-Fock approximation, this value of U does not vanish even in the case when the magnetic zone boundary coincides with the Fermi surface. For extremely large values of U, the critical electron concentration is shown to approach to 1/2. Special care is taken in the case of the half-filled band, and our ground state energy is compared with that in the Hartree-Fock approximation. It is shown that for large value of U the Hartree-Fock energy is lower, but that for intermediate value of U our energy becomes lower for the special band structure.
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.
The role of the fluctuation around the mean field of the antiferromagnetic state has been examined by applying the path integral method to the calculation of the ground state energy of the one-dimensional half-filled Hubbard model with the repulsion, U . In the case of the weak interaction, our result is compared with that of the exactly solvable one the dominant part of which consists of the power series of U . It is found that the present method with the lowest order contribution due to the fluctuation gives the sufficient energy for small U while the higher order of the fluctuation is needed for large U .
By using the classic approximation method of Singwi, Tosi, Land, and Sj\"olander (STLS), we calculate the ground-state energy of the one- and two-dimensional Hubbard model in the half-filled antiferromagnetic state. In the one-dimensional case, by comparison with the exact result of Lieb and Wu, we find that the STLS correlation energy is generally better than that obtained from the random-phase approximation (RPA) and competes in quality with the best values obtained from variational calculations based on the antiferromagnetic Gutzwiller wave function (AFGWF). In particular, the STLS approximation yields a maximum in the absolute value of the correlation energy as a function of U, which is also found in the exact treatment, but not in the RPA. However, the numerical values of the STLS correlation energy are not as accurate as in the case of the uniform electron gas. In two dimensions, where the exact energy is not known, we find a similar relationship between the results of STLS, RPA, and AFGWF calculations, but the absolute value of the correlation energy is smaller. For large values of U, in both dimensions, the STLS method (like the RPA) runs into difficulties, due to the insufficient treatment of transverse spin fluctuations.