A variety of self-consistent-field techniques have recently been used to calculate correlation functions of the two-dimensional Hubbard model. Such techniques extend Hartree–Fock theory by including a large class of fluctuation processes. The computational requirements of the self-consistent-field approach decrease in lower dimensional systems. For this reason, a thorough test of techniques may be performed by examining the Anderson impurity model, the zero-dimensional limit of the Hubbard model. Using the fluctuation exchange and pseudopotential parquet approximations applied previously in two dimensions, we calculate static and dynamic correlation functions for the Anderson model and compare with quantum Monte Carlo results.
A variety of self-consistent-field techniques have recently been used to calculate correlation functions of the two-dimensional Hubbard model. Such techniques extend Hartree–Fock theory by including a large class of fluctuation processes. The computational requirements of the self-consistent-field approach decrease in lower dimensional systems. For this reason, a thorough test of techniques may be performed by examining the Anderson impurity model, the zero-dimensional limit of the Hubbard model. Using the fluctuation exchange and pseudopotential parquet approximations applied previously in two dimensions, we calculate static and dynamic correlation functions for the Anderson model and compare with quantum Monte Carlo results.
Abstract. A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by 〈Si · Sj〉, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorderinteraction phase diagram such physics may be important. PACS numbers: 71.27.+a,71.23.-k,71.10.-wCritiquing HF for the Anderson-Hubbard Model 2 1.
We describe an iterative technique for solution of the de Dominicis--Martin parquet equations for lattice electrons and state the relationship of this approach to conserving extensions of Hartree-Fock theory. We propose a physically motivated and computationally feasible pseudopotential approximation, which allows the solution of the parquet equations for the two-dimensional Hubbard model. We present calculations of static and dynamic Hubbard-model correlation functions based on the pseudopotential parquet and a simpler conserving approximation. For a wide parameter range the pseudopotential parquet results are in nearly quantitative agreement with finite-lattice quantum Monte Carlo results.
We present self-consistent Green-function calculations for the properties of the symmetric Anderson impurity model at zero temperature. A stable method for calculating the spectral density directly on the real frequency axis in the spin-fluctuation limit is described. Results using an approximation to the self-energy, previously termed the fluctuation-exchange approximation, which includes the effects of spin and density fluctuations (within the random-phase approximation) and multiple electron-electron scattering are compared with second-order perturbation-theory calculations. The static spin and charge susceptibilities, calculated in a manner consistent with conservation laws, are compared with the exact results from the Bethe ansatz. We find that the fluctuation-exchange approximation gives a good description of the charge susceptibility over a wide range of interaction strengths and is a great improvement over the self-consistent second-order theory. However, the T-linear coefficient of the specific heat is greatly overestimated by the fluctuation-exchange approximation and this leads to the underestimation of the spin susceptibility. The self-consistent second-order perturbation theory, which underestimates the specific-heat coefficient, predicts a magnetic ground state for sufficiently high interaction strengths.
A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by < S_i . S_j >, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorder-interaction phase diagram such physics may be important.
We have obtained the exact ground state wave functions of the Anderson-Hubbard model for different electron fillings on a 4x4 lattice with periodic boundary conditions - for 1/2 filling such ground states have roughly 166 million states. When compared to the uncorrelated ground states (Hubbard interaction set to zero) we have found strong evidence of the very effective screening of the charge homogeneities due to the Hubbard interaction. We have successfully modelled these local charge densities using a non-interacting model with a static screening of the impurity potentials. In addition, we have compared such wave functions to self-consistent real-space unrestricted Hartree-Fock solutions and have found that these approximate ground state wave functions are remarkably successful at reproducing the local charge densities, and may indicate the role of dipolar backflow in producing a novel metallic state in two dimensions.
Abstract This study extended the fixed-node QMC method to a lattice model for hopping fermions using Green’s function techniques. In general, the modifications of the procedures for continuous systems to adapt them to a lattice system were minor, and there appeared to be no surprises. Applications were made to the two-dimensional Hubbard model with different on-site interactions, lattice dimensions, boundary conditions, node specifications, and trial wavefunctions for importance sampling. Results were compared with those of Hartree-Fock and variational QMC calculations. For the HF calculations with zero interaction, the exact solutions are known, and the Green’s function QMC results agreed with those within statistical error. Trial wavefunctions specifying the nodes were obtained directly as those of HF determinantal solutions for zero interaction, or as variationally optimized determinantal functions with correlation terms. The procedure for fixed nodes was to eliminate walkers crossing the boundaries of a nodal region, a procedure not quite correct, but probably a reasonable approximation. The results in all cases were consistent with the HF and variational QMC calculations, giving lower energies than either for nonzero interaction energies. The existence of a domain-wall phase suggested by the HF and variational QMC results was confirmed by the fixed-node Green’s function results. It was noted that the fixed-node method would be applicable to systems other than the Hubbard model.