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International Journal of Modern Physics B

FERMION CLUSTER APPROACH TO THE HUBBARD MODEL

A. Ziegler

Published Jan 1, 1993• 0 citations
Publisher

Abstract

The deficiencies of the fermion cluster approximation by Caron and Pratt out-lined in a previous paper are corrected and the approximation is then applied to the one-site Hubbard model. Because the approximation in its standard form is unable to describe energy bands the boundary condition integration technique of C. Gros for free clusters is generalized to embedded clusters by averaging over the solutions to different self-consistency equations.

DOI
10.1142/s0217979293001268

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The deficiencies of the fermion cluster approximation by Caron and Pratt out-lined in a previous paper are corrected and the approximation is then applied to the one-site Hubbard model. Because the approximation in its standard form is unable to describe energy bands the boundary condition integration technique of C. Gros for free clusters is generalized to embedded clusters by averaging over the solutions to different self-consistency equations.
The self-energy of a translational invariant system of interacting fermions may be expanded in diagrams contributing to the self-energy of finite clusters with open boundary conditions. The exact solution of small clusters might therefore be used to construct a systematic approximation to the self-energy of the infinite system. This approximation incorporates both the local and the itinerant degrees of freedom on an equal footing. We develop this method for the one-band Hubbard Hamiltonian and apply it to the three-band Hamiltonian of the CuO superconductors. Already the lowest nontrivial approximation yields interesting results for the spectral density useful for the interpretation of photoemission experiments. We find (i) transfer of spectral weight from the upper to the lower Hubbard band upon doping, (ii) the formation of an isolated band of Zhang-Rice singlets separated from the band of triplet states by a many-body gap, and (iii) creation of density of states above the top of the oxygen band upon doping.
Abstract Band models with intrasite (Hubbard) and intersite Coulomb interactions are treated in a cluster approximation taking into account the interactions and the transfer within clusters of sites rigorously. Between different clusters the interaction is decoupled in a mean field approximation and the transfer term is treated as a perturbation. The ground‐state energy of the one‐dimensional Hubbard model is calculated up to fourth order in the perturbation and is compared with other approximations and the exact result. In the one‐dimensional spinless fermion model the perturbation is treated without further approximations by solving Dyson's equation for the one‐particle Green's function. The order‐disorder transition in this model is discussed.
The previous chapter is devoted to the formulation of DMFT and applications that are exact in the limit of infinite dimensions; however, in finite dimensions this is only a single-site approximation. The present chapter is devoted to clusters and other methods to treat correlation between sites. The equations are derived in a unified way, applicable to a single site or a cluster. For a cluster with more than one site there are various ways to choose the boundary conditions and the embedding procedure, and special care must be taken to satisfy causality and translation invariance. A few selected applications illustrate different techniques and results for Hubbard-type models. The essential features of DMFT are brought out in the previous chapter in the simplest form: the single-site approximation in which correlation between sites can be neglected. In this case the self-energy due to interactions can be calculated using an auxiliary system of a single site embedded in an effective medium that represents the surrounding crystal. The resulting many-body problem is equivalent to an Anderson impurity model that must be solved self-consistently. The results are exact for infinite dimensions d → ∞; however, in any finite dimension, i.e., real systems, this is just a mean-field approximation analogous to the Weiss mean field in magnetic systems or the CPA for disordered alloys. There are, however, many important properties and phenomena that require us to go beyond mean-field approximations. For example, basic questions concerning the nature of a metal–insulator transition can be established only by quantitative assessment of the correlations between different sites. Is a single site sufficient (the central idea in the arguments by Mott discussed in Sec. 3.2) or is correlation between sites (for example, an ordered antiferromagnetic state) essential for interactions to lead to an insulating state? A complete theory must establish the range of correlation needed to understand the mechanisms and to make quantitative calculations. A striking example that requires a k -dependent self-energy is the opening of a “pseudogap” in only part of the Brillouin zone in a high-temperature superconductor, as illustrated in Fig. 2.16. This chapter is devoted to methods that go beyond the single-site approximation to take into account interactions and correlations between electrons on sites.
The one-dimensional Hubbard model with nearest-neighbor hopping, first- and second-nearest-neighbor particle interactions, and one particle per site is reexamined by means of a small-crystal approach---an exact solution of a four-site cluster with periodic boundary conditions. Lines of symmetry crossing as well as singular lines and points are found. Analysis of correlation functions indicates that although there are smooth, strong changes as the parameters approach the values of the charge-to-spin-density-wave transition (a suggestion of an incipient second-order transition), the actual change involves a crossover between unrelated symmetries (a clear indication of a first-order transition). The relatively simple calculation and its results agree well with previous Monte Carlo numerical simulations and settles the unresolved question of the order of the density-wave transition.
A periodic cluster perturbation theory is used to investigate a fermion system. The theory is based on the dynamical cluster approximation with relaxing of self-consistency. Within the periodic cluster perturbation theory the self-energy is approximated by the self-energy of an isolated cluster with periodic boundary conditions. The small parameter of the theory is of order of 1∕Lc1∕{L}_{c}1∕Lc​, where Lc{L}_{c}Lc​ is the linear size of the cluster. It is found that within the periodic cluster perturbation theory, quantities averaged over the cluster converge rapidly with corrections O(1∕Lc2)O(1∕{L}_{c}^{2})O(1∕Lc2​), whereas within the cluster perturbation theory, they converge with corrections O(1∕Lc)O(1∕{L}_{c})O(1∕Lc​).