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Journal of the Physical Society of Japan

Effect of the Antiferromagnetic Fluctuation on a One-Dimensional Electron System

Yoshikazu Suzumura, T. Aoyama, Ken'ichi Takano

Published Apr 15, 1992• 1 citation
Publisher

Abstract

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 .

DOI
10.1143/jpsj.61.1232

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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 .
The Hubbard model was originally introduced as the simplest system which may exhibit an insulating (Mott) state. This state is the result of strong electron–electron interactions. In this chapter we consider the Hubbard model at half-filling. The main goal here is the study of the magnetic properties of its ground state. Apart from an exact solution in one dimension, no exact results are available for this problem. This leads to the use of several approximations. The most popular one, and the oldest, is the mean-field theory (MFT). In the MFT one has the bias that the ground state does have some sort of magnetic order (i.e. ferromagnetic, Néel antiferromagnetic, etc.). The problem is then usually solved by means of a variational ansatz. However, one is usually interested in more than just the ground-state energy, which, after all, is not directly measurable and depends very sensitively on the properties at short distances. Most often we wish to evaluate the long-distance, low-frequency, properties of the correlation and response functions of this theory. Moreover, in some cases, such as in one dimension, the fluctuations overwhelm the MFT predictions. In this chapter we will consider the standard MFT (i.e. Hartree–Fock), which is expected to become accurate at weak coupling. We will consider both ferromagnetic and antiferromagnetic states. We will also rederive these results using path integrals. As a byproduct, we will also have a theory of the fluctuations: the non-linear sigma model.
We have calculated the ground-state energy and the fluctuation contribution to the free-energy for the Hubbard model at half filling. We have used a gauge that leads to the Hartree-Fock solution in the saddle-point approximation. The mean-field ground-state energy calculated here agrees with the results obtained by Dagotto et al. [Phys. Rev. B 45, 10 741 (1992)] with use of the exact-diagonalization techniques on a 4\ifmmode\times\else\texttimes\fi{}4 site cluster. In the large-U limit, the ground-state energy per site is obtained as Eg{\mathit{E}}_{\mathit{g}}Eg​/N=-U/4-J+O(J2{\mathit{J}}^{2}J2/U). The fluctuation contribution to the free-energy was found to have a stronger temperature and U dependence than the mean-field contribution, and to play a dominant role for strongly correlated systems.
The lowest energies of the one-dimensional half-filled Hubbard model as a function of the total spin are computed from the exact solution of one-dimensional interacting fermions. The lowest energies are used to calculate the magnetization curve at zero temperature. In the case of repulsive interaction our work is an extension of Griffiths' theory for the Heisenberg model to the itinerant electron model of antiferromagnet. In the case of attractive interaction it is shown that the magnetic susceptibility in zero field vanishes.
Correlation effects in the one-dimensional Hubbard model are studied by application of the L-resolvent method. The calculations are well beyond one-particle mean field approximation. All correlations involving two neighboring sites are included exactly and higher order correlations are approximated. The effects of varying temperature, band filling, Coulomb repulsion and external magnetic field are investigated. Two-site correlations for non-half-filled band fillings are calculated for the first time. Some exact relations between correlation functions are derived by electron-hole symmetry. Results indicate that short-range spin correlations always favor antiferromagnetic order. Large Coulomb repulsion induces a gap in the excitation spectrum and a metal-insulator transition is predicted as the Coulomb repulsion parameter is changed.
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.
The lowest energies of the one-dimensional half-filled Hubbard model as a function of the total spin are computed from the exact solution of one-dimensional interacting fermions. The lowest energies are used to calculate the magnetization curve at zero temperature. In the case of repulsive interaction our work is an extension of Griffith's theory for the Heisenberg model to the itinerant electron model of antiferromagnet. In the case of attractive interaction it is shown that the magnetic susceptibility in zero field vanishes.