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Physical review. B, Condensed matter

Variational Monte Carlo study of the spiral state in the Hubbard model

Yongjun Liu, Jinming Dong, Chang-De Gong, Tingyang Chen

Published Jul 1, 1993• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.1306

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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.
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 investigate the Mott transition in a two-dimensional two-orbital Hubbard model on a square lattice at half filling by using the variational Monte Carlo method. As a variational trial wave function, we consider Gutzwiller's on-site correlations and nearest-neighbor doublon-holon correlations. We calculate the ground-state energy and the other physical quantities, all of which clearly signal a first-order Mott transition. We thus obtain the phase diagram at zero temperature, where we find the coexistence region of the metal and the Mott insulator. We discuss how Hund's coupling affects the nature of the Mott transition in this system.
Two methods are used, Hartree-Fock and a variational wave function, to investigate spiral states in the three-band Hubbard model. Our variational wave function is a Gutzwiller-projected Hartree-Fock spiral state. We use quantum Monte Carlo methods to calculate the variational energy exactly. In order to avoid boundary defects in the spiral spin structure, we work in a frame where the spin-quantization axis is locally rotated. Both methods show that upon doping the antiferromagnetic structure present at half-band-filling is unstable against a (1,1) spiral state. For moderate (large) values of the Cu on-site Hubbard repulsion, there is a transition to a (1,0) spiral (ferromagnetic) state. The Hartree-Fock calculations show that most of the (1,1) spiral phase is unstable. On the other hand, the (1,0) spiral state is stable.
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.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
We use a variational Monte Carlo technique to study the ground state of the two-dimensional Hubbard and t-J models on a square lattice. We use a trial wave function that allows a continuous description of the paramagnetic, antiferromagnetic, and superconducting phases, as well as the coexistence of these phases, with no a priori constraint on double occupancy. The phase diagram of both models is given for intermediate coupling (U=10t). We show that the apparent discrepancies between the two models, that appeared in previous variational Monte Carlo studies, are not present when this sufficiently good variational wave function is used. The two models are in good qualitative agreement. We find that except at half-filling a pure antiferromagnetic phase does not exist but is always in coexistence with a d-wave superconducting phase, and is followed by a pure d-wave superconducting phase at doping up to \ensuremath{\delta}\ensuremath{\sim}0.3. The staggered magnetization and the superconducting gap are measured, and a comparison with other analytical or numerical results is made.