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Physical review. B, Condensed matter

Brinkman-Rice transition in the three-band Hubbard model

Akira Oguri, Tetsuro Asahata

Published Dec 1, 1992• 9 citations
Publisher

Abstract

The ${\mathit{U}}_{\mathit{d}}$=\ensuremath{\infty} three-band Hubbard model on a two-dimensional ${\mathrm{CuO}}_{2}$ network is studied based on a Gutzwiller wave function by use of the variational Monte Carlo (VMC) method, where ${\mathit{U}}_{\mathit{d}}$ 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 ${\mathit{n}}_{\mathit{d}}$=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 ${\mathit{n}}_{\mathit{d}}$ 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.

DOI
10.1103/physrevb.46.14073

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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 \ensuremathΔ0{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremathΔ0​=(\ensuremathεp{\mathrm{\ensuremath{\varepsilon}}}_{\mathit{p}}\ensuremathεp​-\ensuremathεd{\mathrm{\ensuremath{\varepsilon}}}_{\mathit{d}}\ensuremathεd​)/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.
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.
Two very different methods — exact diagonalization on finite chains and a variational method — are used to study the possibility of a metal-insulator transition in the symmetric half-filled periodic Anderson-Hubbard model. With this aim we calculate the density of doubly occupied d sites (gn d ) as a function of various parameters. In the absence of on-site Coulomb interaction (U f ) between f electrons, the two methods yield similar results. The double occupancy of d levels remains always finite just as in the one-dimensional Hubbard model. Exact diagonalization on finite chains gives the same result for finite U f , while the Gutzwiller method leads to a Brinkman-Rice transition at a critical value (U ), which depends on U f and V.
Normal states of the attractive Hubbard model, mainly in two dimensions, are studied in the context of a transition from a Fermi liquid to an insulating or gapped state. A series of variational Monte Carlo calculations with better statistics was carried out to estimate accurately the expectation values of several many-body wave functions. Although a relatively clear crossover is observed, even for the plain Gutzwiller wave function, the states in both regimes are metallic. Meanwhile, a substantial metal-insulator transition takes place at ∣U∣∼W (band width) for an improved wave function, into which an intersite correlation is introduced by taking account of virtual states in the second-order perturbation in the infinite ∣U∣ limit. The critical value compares favorably with recent results of the dynamical-mean-field approximation. In contrast, a conventional Jastrow-type wave function scarcely improves the normal state. In addition, the issue of the Brinkman-Rice metal-insulator transition is reconsidered with much larger systems.
The main flaw of the well-known Brinkman-Rice description, obtained through the Gutzwiller approximation, of the paramagnetic Mott transition in the Hubbard model is in neglecting high-energy virtual processes that generate, for instance, the antiferromagnetic exchange $J\ensuremath{\sim}{t}^{2}/U$. Here, we propose a way to capture those processes by combining the Brinkman-Rice approach with a variational Schrieffer-Wolff transformation, and apply this method to study the single-band metal-to-insulator transition in a Bethe lattice with infinite coordination number, where the Gutzwiller approximation becomes exact. We indeed find for the Mott transition a description very close to the real one provided by the dynamical mean-field theory, an encouraging result in view of possible applications to more involved models.
Two very different methods – exact diagonalization on finite chains and a variational method – are used to study the possibility of a metal-insulator transition in the symmetric half-filled periodic Anderson-Hubbard model. With this aim we calculate the density of doubly occupied d sites (νd) as a function of various parameters. In the absence of on-site Coulomb interaction (Uf ) between f electrons, the two methods yield similar results. The double occupancy of d levels remains always finite just as in the one-dimensional Hubbard model. Exact diagonalization on finite chains gives the same result for finite Uf , while the Gutzwiller method leads to a Brinkman-Rice transition at a critical value (U c d ), which depends on Uf and V .
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.