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Molecular Physics

An efficient random-sampling method for calculating double occupancy of Gutzwiller wave function in single-band 1D and 2D lattices

Feng Zhang, Zhuo Ye, Yongxin Yao, Cai‐Zhuang Wang, Kai‐Ming Ho

Published Aug 28, 2020• 3 citations
Publisher

Abstract

We report a random-sampling method for computing the expectation value of physical quantities based on the Gutzwiler variational wave function. As the first application, we calculated the double occupancy, which is a critical quantity for understanding the correlation effects in many-body systems, for single-band 1D and 2D lattices. We demonstrated that the random-sampling scheme is more efficient than an existing Metropolis Monte-Carlo algorithm. For the 1D Hubbard model with only nearest-neighbour hopping, our results are almost identical to the exact analytic solution. We have also studied systems to which analytic solutions are not available, including the 1D lattices with next-nearest-neighbour hopping and 2D lattices. In addition, constraints on real-space configurations can be easily implemented in the current scheme to further improve the Gutzwiller wave function. As an example, we calculated the double occupancy for 1D Hubbard model by applying the constraint that all double-occupied sites are paired with an empty site. With enhanced correlation between double-occupied and empty sites, the constraint results in much improved ground-state energy for 1D Hubbard model with strong on-site repulsion.

DOI
10.1080/00268976.2020.1812745

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We report a random-sampling method for computing the expectation value of physical quantities based on the Gutzwiler variational wave function. As the first application, we calculated the double occupancy, which is a critical quantity for understanding the correlation effects in many-body systems, for single-band 1D and 2D lattices. We demonstrated that the random-sampling scheme is more efficient than an existing Metropolis Monte-Carlo algorithm. For the 1D Hubbard model with only nearest-neighbour hopping, our results are almost identical to the exact analytic solution. We have also studied systems to which analytic solutions are not available, including the 1D lattices with next-nearest-neighbour hopping and 2D lattices. In addition, constraints on real-space configurations can be easily implemented in the current scheme to further improve the Gutzwiller wave function. As an example, we calculated the double occupancy for 1D Hubbard model by applying the constraint that all double-occupied sites are paired with an empty site. With enhanced correlation between double-occupied and empty sites, the constraint results in much improved ground-state energy for 1D Hubbard model with strong on-site repulsion.
Abstract We present a random-sampling (RS) method for evaluating expectation values of physical quantities using the variational approach. We demonstrate that the RS method is computationally more efficient than the variational Monte Carlo method using the Gutzwiller wavefunctions applied on single-band Hubbard models as an example. Non-local constraints can also been easily implemented in the current scheme that capture the essential physics in the limit of strong on-site repulsion. In addition, we extend the RS method to study the antiferromagnetic states with multiple variational parameters for 1D and 2D Hubbard models.
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.
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.
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.
A variational Monte-Carlo method, which is a combination of the variational theory and the Monte-Carlo technique is proposed to study the Hubbard model with the Gutzwiller's wave function. An application is made to one-, two-, and three-dimensional models having sites up to 216 (6×6×6) and arbitrary values of U / t (on-site repulsion U and hopping integral t ). Contrary to resorting to the Gutzwiller approximation, it provides a virtually exact way to carry out the Gutzwiller-type variational theory and opens up a possibility of further development.
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.