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Physica B Condensed Matter

Application of a novel cluster-effective-field theory to the three-dimensional Hubbard model

Hitoshi Asakawa, Masuo Suzuki

Published Jul 1, 1995• 0 citations
Publisher
DOI
10.1016/0921-4526(95)00019-6

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Almost 50 years after the invention of the Hubbard model [1–3] and despite modern petaflop supercomputers, a precise analysis of the criticality of this most basic model for electronic correlations has not been achieved so far, at least not in three dimensions. Dynamical mean field theory (DMFT) [4–6] was a big step forward to quantitatively calculate the three dimensional Hubbard model since the major contribution of electronic correlations, i.e., the local ones, are well captured within this theory. Local correlations give rise to quasiparticle renormalization, the Mott-Hubbard transition, magnetism, and even more subtle issues such as kinks in a purely electronic models [7]. However, non-local spatial correlations are also naturally generated by a purely local Hubbard interaction, and, as it is well known, they become of essential importance in the vicinity of secondorder phase transitions. As these correlations are neglected in DMFT, this scheme can provide only for a conventional mean-field (MF) description of the critical properties. To overcome this shortcoming cluster extensions to DMFT such as the dynamical cluster approximation (DCA) and cluster-DMFT have been proposed, see Ref. 8 for a review. In these approaches non-local correlations beyond DMFT are taken into account, however only within the range of the cluster size; and due to computational limitations the actual size of the cluster is severely restricted to about 3-5 sites for each direction. Hence, short-range correlations are included by these approaches, whereas long-range ones are not. Nonetheless, Kent et al. [9] were able to extrapolate the cluster size of so-called Betts clusters to infinity, albeit assuming from the very beginning the critical exponents to be those of the Heisenberg model. This way they extrapolated the critical temperature of the paramagnetic-toantiferromagnetic phase transition which was found to be in agreement with earlier quantum Monte Carlo (QMC) simulations of Staudt et al. [10] on finite lattices.

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Almost 50 years after the invention of the Hubbard model [1–3] and despite modern petaflop supercomputers, a precise analysis of the criticality of this most basic model for electronic correlations has not been achieved so far, at least not in three dimensions. Dynamical mean field theory (DMFT) [4–6] was a big step forward to quantitatively calculate the three dimensional Hubbard model since the major contribution of electronic correlations, i.e., the local ones, are well captured within this theory. Local correlations give rise to quasiparticle renormalization, the Mott-Hubbard transition, magnetism, and even more subtle issues such as kinks in a purely electronic models [7]. However, non-local spatial correlations are also naturally generated by a purely local Hubbard interaction, and, as it is well known, they become of essential importance in the vicinity of secondorder phase transitions. As these correlations are neglected in DMFT, this scheme can provide only for a conventional mean-field (MF) description of the critical properties. To overcome this shortcoming cluster extensions to DMFT such as the dynamical cluster approximation (DCA) and cluster-DMFT have been proposed, see Ref. 8 for a review. In these approaches non-local correlations beyond DMFT are taken into account, however only within the range of the cluster size; and due to computational limitations the actual size of the cluster is severely restricted to about 3-5 sites for each direction. Hence, short-range correlations are included by these approaches, whereas long-range ones are not. Nonetheless, Kent et al. [9] were able to extrapolate the cluster size of so-called Betts clusters to infinity, albeit assuming from the very beginning the critical exponents to be those of the Heisenberg model. This way they extrapolated the critical temperature of the paramagnetic-toantiferromagnetic phase transition which was found to be in agreement with earlier quantum Monte Carlo (QMC) simulations of Staudt et al. [10] on finite lattices.
We introduce a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.
We discuss recent applications of exact numerical continued fraction expansion (CFE) techniques to calculate dynamical correlation functions of various strong-coupling models related to the high-temperature cuprate superconductors. For the two-dimensional square-lattice spin-1/2 Heisenberg antiferromagnet, we present exact results for the zero-temperature dynamical structure factor on finite-sized lattices and compare them to approximate results from a Schwinger boson mean-field theory, recently proposed by Arovas and Auerbach. We find that the mean-field theory represents a very good approximation to the exact spin excitation spectra and to the static spin correlations. We then investigation the dynamical spin-spin structure factor and the single-particle spectral function for finite model clusters with dopant induced hole-type charge carriers, in the strong-coupling limit of both the single-band Hubbard model (t-J-model) and the three-band Hubbard model (Kondo-Heisenberg model). Our results are consistent with the physical picture recently proposed by Zhang and Rice which implies an approximate mapping of the low-energy states in the three-band model onto an effective single-band theory. 33 refs., 4 figs.