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Physical review. B, Condensed matter

Variational Monte Carlo evaluations of Gutzwiller states for the Anderson-lattice model

P. G. McQueen, C. S. Wang

Published Nov 1, 1991• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.10021

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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 approach is presented for variational Monte Carlo simulations of Hubbard-Anderson models using Gutzwiller projection operators to restrict charge fluctuations in noninteracting wave functions. The Gutzwiller matrix elements are mapped onto a statistical model, which is evaluated using a stochastic algorithm that is normally applied to finite-temperature simulation. The method has been successfully tested for one-dimensional (1D) Hubbard and 1D Anderson models, and results are presented for the 2D Hubbard model. We find a region close to the half-filled limit where the Gutzwiller states of different magnetic ordering are very close in energy.
We investigate the properties of the frustrated underdoped Hubbard model on the square lattice using two complementary approaches, i.e., the dynamical cluster extension of the dynamical mean-field theory and the variational Monte Carlo simulations of Gutzwiller-Jastrow wave functions with backflow corrections. We compare and discuss data for the energy and the double occupancies, as obtained from both approaches. At small dopings, we observe a rapid crossover from a weakly correlated metal at low interaction strength UUU to a non-Fermi-liquid correlated state with strong local spin correlations. Furthermore, we investigate the stability of the correlated state against phase separation. We observe phase separation only for large values of UUU or very large frustration. No phase separation is present for the parameter range relevant for the cuprates.
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 authors consider the variational Monte Carlo method of studying fermion systems in condensed matter physics. This method is a powerful tool for studying the properties of interacting electrons which are responsible for phenomena like the new high-Tc superconductors. Using a parallel algorithm for updating the determinants of large complex matrices on transputer arrays they study the energy and spin correlations for a 12*12 and a 22*22 lattice. Using the Hubbard model and the Gutzwiller projected wavefunction they have evaluated these quantities as functions of band filling and coupling constant. They find that although the Gutzwiller state is stable for densities near half filling it is not at low densities. For the stable states they find antiferromagnetic correlations. Statistically the thermodynamic limit is already attained with a 12*12 lattice.
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.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.