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arXiv:cond-mat/9307032 · Physical review. B, Condensed matter

Gutzwiller wave function in the three-band Hubbard model: A variational Monte Carlo study

Akira Oguri, Tetsuro Asahata, Sadamichi Maekawa

Published Mar 1, 1994• 18 citations
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Abstract

The Gutzwiller wave function for the three-band Hubbard model on a two-dimensional ${\mathrm{CuO}}_{2}$ plane is studied by using the variational Monte Carlo (VMC) method in the limit ${\mathit{U}}_{\mathit{d}}$\ensuremath{\rightarrow}\ensuremath{\infty}, where ${\mathit{U}}_{\mathit{d}}$ 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.

DOI
10.1103/physrevb.49.6880

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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.
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 Ud{\mathit{U}}_{\mathit{d}}Ud​=\ensuremath{\infty} three-band Hubbard model on a two-dimensional CuO2{\mathrm{CuO}}_{2}CuO2​ network is studied based on a Gutzwiller wave function by use of the variational Monte Carlo (VMC) method, where Ud{\mathit{U}}_{\mathit{d}}Ud​ is the Coulomb repulsion between holes on the Cu site (hole picture). The VMC results are compared with those of the Rice-Ueda-type Gutzwiller approximation (GA) in a half-filled band, where the average number of holes per Cu site is n=1. In the GA the Brinkman-Rice (BR) transition occurs at a finite value of ${\mathrm{\ensuremath{\Delta}}}_{0}$=(${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{d}}$)/2, and then the number of d holes becomes nd{\mathit{n}}_{\mathit{d}}nd​=n=1. Whether the BR transition really occurs or not within the Gutzwiller wave function is discussed in detail. In finite systems the absence of the BR transition is shown analytically. The thermodynamic limit is examined assuming the asymptotic behavior of nd{\mathit{n}}_{\mathit{d}}nd​ is given by a scaling function. The results of the finite-size scaling analysis suggest the absence of the BR transition as that in the single-band Hubbard model.
Choosing the Gutzwiller-type variational wave function and using the variational Monte Carlo (VMC) method, we have calculated the ground-state energies of two-dimensional (2D) Hubbard squares at temperature T=0 and have obtained the phase diagram in the small doping region (\ensuremath{\delta}=0--0.15). We do not find the spiral phase until at least U\ensuremath{\approxeq}20 in the half-filled and dilute-doping cases. This is different from the results obtained by C. Jayaprakash et al., who calculated the phase diagram of the 2D t-J model at temperature T=0 with the slave-fermion mean-field theory and found the spiral phase at much lower values of U/t.
As a continuation of a previous paper [J. Phys. Soc. Jpn. 56 (1987) 1490], the variational Monte-Carlo method is extended to include the antiferromagnetic long-range order. The theory is based on the Gutzwiller-type correlation factor and its effect is exactly taken into account by the Monte-Carlo procedure. An application is made to the half-filled-band case of one-dimensional lattice (50 sites), two-dimensional square lattice (up to 20×20 sites) and three-dimensional simple cubic lattice (6×6×6 sites). The result is qualitatively different from previous studies relying on the random-phase-type "Gutzwiller approximation." The variational energy for two and three dimensions is favorably compared with Hirsch's quantum Monte-Carlo data.
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 propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.