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arXiv:cond-mat/9904400 · Physical review. B, Condensed matter

Coupled-cluster approach to electron correlations in the two-dimensional Hubbard model

Yoshihiro Asai, Hideki Katagiri

Published Nov 15, 1999• 6 citations
arXivHTMLPDF

Abstract

We have studied electron correlations in the doped two-dimensional (2D) Hubbard model by using the coupled-cluster method (CCM) to investigate whether or not the method can be applied to correct the independent particle approximations actually used in ab initio band calculations. The double excitation version of the CCM, implemented using the approximate coupled pair (ACP) method, accounts for most of the correlation energies of the 2D Hubbard model in the weak $(U/t\ensuremath{\simeq}1)$ and the intermediate $U/t$ regions $(U/t\ensuremath{\simeq}4).$ The error in this case is always less than $1%.$ The ACP approximation gets less accurate for large $U/t$ $(U/t\ensuremath{\simeq}8)$ and near half-filling. Further incorporation of electron correlation effects is necessary in this region. The accuracy depends neither on the system size, nor the gap between the lowest unoccupied level and the highest occupied level in the finite size effect. Hence, the CCM may be favorably applied to ab initio band calculations on metals as well as semiconductors and insulators.

DOI
10.1103/physrevb.60.r13946

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We have studied electron correlations in the doped two-dimensional (2D) Hubbard model by using the coupled-cluster method (CCM) to investigate whether or not the method can be applied to correct the independent particle approximations actually used in ab initio band calculations. The double excitation version of the CCM, implemented using the approximate coupled pair (ACP) method, accounts for most of the correlation energies of the 2D Hubbard model in the weak (U/t\ensuremath≃1)(U/t\ensuremath{\simeq}1)(U/t\ensuremath≃1) and the intermediate U/tU/tU/t regions (U/t\ensuremath≃4).(U/t\ensuremath{\simeq}4).(U/t\ensuremath≃4). The error in this case is always less than 11%.1 The ACP approximation gets less accurate for large U/tU/tU/t (U/t\ensuremath≃8)(U/t\ensuremath{\simeq}8)(U/t\ensuremath≃8) and near half-filling. Further incorporation of electron correlation effects is necessary in this region. The accuracy depends neither on the system size, nor the gap between the lowest unoccupied level and the highest occupied level in the finite size effect. Hence, the CCM may be favorably applied to ab initio band calculations on metals as well as semiconductors and insulators.
Abstract We recently applied the coupled‐cluster method (CCM) with considerable success to several novel quantum spin‐lattice systems in the infinite bulk limit. In this article, we extend our CCM analysis to the electronic Hubbard models. In particular, based on a systematic approximation scheme within the CCM, we investigate the zero‐temperature properties of Hubbard models at half‐filling and with a single hole on a general bipartite lattice. Our aim was to provide the CCM framework for a systematic calculation of the properties of Hubbard models. © 1995 John Wiley & Sons, Inc.
We discuss the theory and implementation of the finite temperature coupled cluster singles and doubles (FT-CCSD) method including the equations necessary for an efficient implementation of response properties. Numerical aspects of the method including the truncation of the orbital space and integration of the amplitude equations are tested on some simple systems, and we provide some guidelines for applying the method in practice. The method is then applied to the 1D Hubbard model, the uniform electron gas (UEG) at warm, dense conditions, and some simple materials. The performance of model systems at high temperatures is encouraging: for the one-dimensional Hubbard model, FT-CCSD provides a qualitatively accurate description of finite-temperature correlation effects even at U = 8, and it allows for the computation of systematically improvable exchange-correlation energies of the warm, dense UEG over a wide range of conditions. We highlight the obstacles that remain in using the method for realistic ab initio calculations on materials.
The coupled cluster method (CCM) of microscopic quantum many-body theory has become an ab initio method of first choice in quantum chemistry and many fields of nuclear, subnuclear and condensed matter physics, when results of high accuracy are required. In recent years it has begun to be applied with equal success to strongly correlated systems of electrons or quantum spins defined on a regular spatial lattice. One regularly finds that the CCM is able to describe accurately the various zero-temperature phases and the quantum phase transitions between them, even when frustration is present and other methods such as quantum Monte Carlo often fail. We illustrate the use and powerfulness of the method here by applying it to a square-lattice spin-half Heisenberg model where frustration is introduced by competing nearest neighbour bonds. The model exhibits the physically interesting phenomenon of competition between magnetic order and dimerization. Results obtained for the model with the CCM are compared with those found from spin-wave theory and from extrapolating the results of exact diagonalizations of small lattices. We show that the CCM is essentially unique among available methods in being able both to describe accurately all phases of this complex model and to provide accurate predictions of the various phase boundaries and the order of the corresponding transitions.
The coupled-cluster method (CCM) is applied to anisotropic-quantum-antiferromagnetic models in one and two dimensions (1D and 2D). Several hierarchical approximation schemes of the CCM are defined, which are specialized for the spin-lattice models. Good results are obtained for both ground-state and excited-state energies. The spin-spin correlation functions and the staggered magnetizations are also calculated. We have found that the CCM gives a qualitatively correct description of the entire Heisenberg-Ising phase, including the critical point where a phase transition occurs in 1D. Some interesting differences between 1D and 2D models near the critical point are discussed.
The many-electron correlation problem for metalliclike systems with Born---von K\'arm\'an boundary conditions, as modeled by the cyclic polyene Pariser-Parr-Pople and Hubbard Hamiltonians, is examined over the entire range of the coupling constant using the coupled-pair (CP) many-electron theory based on the exponential-cluster ansatz for the exact wave function. It is shown that the standard CP theory breaks down not only in the highly correlated, but even in the intermediately correlated, regions, and the nature of its singular behavior in these regions is examined. This breakdown is linked with the increasingly important role played by the connected quadruply excited clusters, which invalidate the basic assumption of the CP theory, as the highly correlated limit and/or the extended character of the model are approached. The contribution from the quadruply excited clusters is then taken into account using the new version of the approximate coupled-pair theory corrected for connected quadruply excited clusters, called ACPQ. This approximation is almost identical with the standard approximate coupled-pair (ACP) -theory approach, in which only factorizable (with respect to hole pairs) nonlinear terms are retained, and differs from it only by a numerical factor associated with one of the nonlinear diagrams. It is shown that the ACPQ not only removes the singularities and associated convergency problems of the standard CP approaches, but, in fact, provides excellent quantitative results over the entire range of the coupling constant, yielding the exact correlation energy in the strongly correlated limit.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.