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arXiv:cond-mat/9903070 · Physical review. B, Condensed matter

Optimization of Gutzwiller wave functions in quantum Monte Carlo

Erik Koch, O. Gunnarsson, Richard M. Martin

Published Jun 15, 1999• 12 citations
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Abstract

Gutzwiller functions are popular variational wave functions for correlated electrons in Hubbard models. Following the variational principle, we are interested in the Gutzwiller parameters that minimize, e.g., the expectation value of the energy. Rewriting the expectation value as a rational function in the Gutzwiller parameters, we find a very efficient way for performing that minimization. The method can be used to optimize general Gutzwiller-type wave functions both in variational and in fixed-node diffusion Monte Carlo.

DOI
10.1103/physrevb.59.15632

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Gutzwiller functions are popular variational wave functions for correlated electrons in Hubbard models. Following the variational principle, we are interested in the Gutzwiller parameters that minimize, e.g., the expectation value of the energy. Rewriting the expectation value as a rational function in the Gutzwiller parameters, we find a very efficient way for performing that minimization. The method can be used to optimize general Gutzwiller-type wave functions both in variational and in fixed-node diffusion Monte Carlo.
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.
We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.
This work covers three lectures on Gutzwiller wave functions for correlated electron systems. In the first lecture, some basic ideas on model Hamiltonians and their motivation are outlined, Gutzwiller's variational approach to Landau Fermi-liquid theory is sketched, and, as a first application, the theory is applied to the one-band Hubbard model where the Brinkman–Rice metal-to-insulator transition is found. The second lecture is devoted to the perturbative calculation of expectation values. In this rather technical part the variational analog of Wick's theorem, Feynman diagrams, and the linked-cluster theorem are introduced. The important differences between the variational approach and the standard Green function technique are the absence of a time-dependence and the flexibility to select the expansion parameter of the variational perturbation theory. When this flexibility is used appropriately, not a single self-energy diagram must be calculated in the limit of large dimensions, i.e. in the limit of large lattice coordination number the expectation values can be evaluated exactly for general Hubbard-type models for all interaction strengths and band fillings. The third lecture covers the application of the theory to ferromagnetism of nickel. After a discussion of the nickel problem, multi-band Hubbard models and Gutzwiller wave functions are introduced and parameterized. The favourable comparison between the results from Gutzwiller theory and a large number of experiments shows that correlations play an important role in itinerant ferromagnets.
We use a variational Monte Carlo method (VMC) and a quantum Monte Carlo method to study the two-dimensional half-filled Hubbard model. We use the Gutzwiller wave function as a trial wave function in VMC and measure energy, correlation functions, and momentum distribution for lattices up to 12\ifmmode\times\else\texttimes\fi{}12 sites. We then use a Monte Carlo algorithm to study the Hubbard model with the same boundary conditions as in VMC at very low temperatures. By comparing the results from these two methods, we assess the accuracy of the Gutzwiller wave function as a representation of the ground state of the Hubbard model. Major differences are found, implying that the Gutzwiller wave function is not a useful approximation to the ground state of the Hubbard model in two dimensions.
The Gutzwiller wave function for the three-band Hubbard model on a two-dimensional CuO2{\mathrm{CuO}}_{2}CuO2​ plane is studied by using the variational Monte Carlo (VMC) method in the limit Ud{\mathit{U}}_{\mathit{d}}Ud​\ensuremath{\rightarrow}\ensuremath{\infty}, where Ud{\mathit{U}}_{\mathit{d}}Ud​ is the Coulomb repulsion between holes on a Cu site. The VMC results for the energy and the fraction of d holes are compared with those of the Rice-Ueda type Gutzwiller approximation (GA). The difference between the VMC and the GA results are most pronounced at half-filling, and away from half-filling the two results agree well in both the hole-doped and the electron-doped cases. The doping dependence of the momentum distribution function is also studied.