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Physica Scripta

Variational Studies of Strongly Correlated Electron Systems

P. Fazekas

Published Jan 1, 1989• 14 citations
Publisher

Abstract

Generalized Gutzwiller-type trial ground states for the Hubbard model are considered, which include extra variational parameters to control the nearest-neighbour charge-charge, charge-spin, and spin-spin correlations. The correlation between doubly occupied and empty sites is found to be particularly important. The results for the large- U regime show qualitative improvement when compared to the predictions of the simple Gutzwiller Ansatz: We get the correct order of magnitude for the energy density, and the band state occupation number is found to be a monotonically decreasing function, with the leading term corresponding to that of the t/U -expansion. We find that to get the small- U limit right, the pair-correlated Ansatz has to be used in conjunction with a cluster Gutzwiller approximation, and introduce a four-site cluster approximation which gives promising results. We argue that small-cluster Gutzwiller approximations get accurate in the limit of large lattice dimensionality.

DOI
10.1088/0031-8949/1989/t29/023

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Generalized Gutzwiller-type trial ground states for the Hubbard model are considered, which include extra variational parameters to control the nearest-neighbour charge-charge, charge-spin, and spin-spin correlations. The correlation between doubly occupied and empty sites is found to be particularly important. The results for the large- U regime show qualitative improvement when compared to the predictions of the simple Gutzwiller Ansatz: We get the correct order of magnitude for the energy density, and the band state occupation number is found to be a monotonically decreasing function, with the leading term corresponding to that of the t/U -expansion. We find that to get the small- U limit right, the pair-correlated Ansatz has to be used in conjunction with a cluster Gutzwiller approximation, and introduce a four-site cluster approximation which gives promising results. We argue that small-cluster Gutzwiller approximations get accurate in the limit of large lattice dimensionality.
The well-known Gutzwiller wave function is generalized by including new variational parameters to control nearest-neighbour charge-charge, charge-spin, and spin-spin correlations. The non-magnetic state of the one-dimensional, half-filled Hubbard model is studied. Within the Gutzwiller approximation, the expression for the ground state energy can be worked out analytically. The correlation between empty and doubly occupied sites is found to play the most essential role. Minimization in the large-U limit shows that the Brinkman-Rice transition has been pushed to U → ∞, and the leading term of the ground state energy density is of order −t2/ U . In contrast to results obtained with the Gutzwiller wave function, we find that the band occupation number nk is monotonically decreasing both above and below kF. The dominant k–dependence is given by ~(t/U) cos k, in agreement with t/U–expansion results. nk has also a weak step-function component, with the discontinuity at kF vanishing as (t/U)2 in the limit U/t ≫ 1.
We introduce a new lattice version of the correlated-basis-function approach for the study of strongly interacting electrons. As a first application we have chosen the one-dimensional Hubbard model. For this system, we have implemented different types of correlations: namely, (a) Gutzwiller; (b) Jastrow; (c) nearest-neighbor spin-spin; (d) excitonic (e-d); and (f) pairing (BCS). The resulting energies, momentum distribution, and other quantities are compared with exact results (when available) and are discussed, both from the methodological and physical point of view. The most important correlations that are needed to obtain good ground-state energies are found to be Gutzwiller (reduced double occupancy) plus e-d correlations (between empty and doubly occupied sites), in agreement with other studies.
A new variational scheme based on a modified Bethe–Peierls method is used to study the ground state properties of the one-dimensional t–J model. Expectation values are evaluated by cutting out a four-site cluster from a correlated Fermi sea, the ground state of which is described by a variational trial wave function. We study a generalized Gutzwiller state where nearest-neighbour hole–hole correlations are controlled variationally. From the electron concentration dependence of the ground state energy, we determine the true thermodynamic boundary where segregation into an electron-rich, and purely hole phase sets in. We also determine the spinodal line and pair susceptibilities. The variational method is applied also to an extended t–J–V model, where V is the coupling constant of the charge interaction term.
We investigate the ground state properties of the one- and the two-dimensional Hubbard models using a new variational Monte Carlo method. By generalizing the Gutzwiller variational wave function (GF) to take into account an off-diagonal intersite correlation, we examine the metal-insulator transition mechanism at the half filling. As Kaplan et al. first pointed out, the binding mechanism between doubly occupied sites and empty sites plays a crucial role for this transition; we confirm this by examining the real space correlation function between these two kinds of sites. Results for the spin and the charge structure factors are presented as well as the momentum distribution function and are discussed in connection with the pairing effect. We also find that in less-than half filling, the improvements of the structure factors at 2 k F are realized as compared with GF results.
The electronic ground states for two- and three-dimensional Anderson-lattice clusters with large Hubbard repulsion are determined variationally within the Gutzwiller ansatz by use of a stochastic algorithm for exact evaluation of matrix elements of Gutzwiller states. The algorithm is similar to one normally used for finite-temperature simulation and is described in some detail. Variational states for a variety of Hamiltonian parameters and particle densities were considered. Of particular interest is the regime of small interband hopping, which is antiferromagnetic at densities near two particles per unit cell. We found that for some Hamiltonian-parameter regimes, doping the antiferromagnetic cluster with enough particles or holes can destroy the magnetic ordering and produce a state that resembles a heavy-Fermi-liquid state, a metal with a strongly-frequency-dependent self-energy. For a 4\ifmmode\times\else\texttimes\fi{}4 Anderson lattice, we present results of low-temperature quantum Monte Carlo simulations to evaluate the validity of the variational Gutzwiller ansatz.
A new variational method is developed to calculate the ground state energy of Fermi systems with strong short-range correlations. A trial wave function of Gutzwiller's type contains additional variational parameters corresponding to configurations of pairs of nearest-neighbor sites. To evaluate the ground state energy, generalized Kikuchi's pseudo-ensemble method is used. The Hubbard model at half band-filling is investigated. The ground state energy of the paramagnetic phase is calculated for a chain, square and simple cubic lattices. It is shown that the short-range order lowers drastically the ground state energy of the Hubbard model at intermediate interaction strength. The paramagnetic phase of the Kondo-Hubbard model (S=1/2S=1/2S=1/2 and S=5/2S=5/2S=5/2) at half band-filling is investigated. The ground state energy, correlation functions and effective mass are calculated for chain, square and simple cubic lattices. A quantum phase transition was found for simple cubic lattice.