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Journal of Physics Condensed Matter

Variation studies of the 2D Hubbard model: less than half filled

Robert B. Jones, W Yeung

Published Apr 2, 1990• 1 citation
Publisher

Abstract

The authors consider the variational Monte Carlo method of studying fermion systems in condensed matter physics. This method is a powerful tool for studying the properties of interacting electrons which are responsible for phenomena like the new high-Tc superconductors. Using a parallel algorithm for updating the determinants of large complex matrices on transputer arrays they study the energy and spin correlations for a 12*12 and a 22*22 lattice. Using the Hubbard model and the Gutzwiller projected wavefunction they have evaluated these quantities as functions of band filling and coupling constant. They find that although the Gutzwiller state is stable for densities near half filling it is not at low densities. For the stable states they find antiferromagnetic correlations. Statistically the thermodynamic limit is already attained with a 12*12 lattice.

DOI
10.1088/0953-8984/2/13/007

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The authors consider the variational Monte Carlo method of studying fermion systems in condensed matter physics. This method is a powerful tool for studying the properties of interacting electrons which are responsible for phenomena like the new high-Tc superconductors. Using a parallel algorithm for updating the determinants of large complex matrices on transputer arrays they study the energy and spin correlations for a 1212 and a 2222 lattice. Using the Hubbard model and the Gutzwiller projected wavefunction they have evaluated these quantities as functions of band filling and coupling constant. They find that although the Gutzwiller state is stable for densities near half filling it is not at low densities. For the stable states they find antiferromagnetic correlations. Statistically the thermodynamic limit is already attained with a 12*12 lattice.
A new approach is presented for variational Monte Carlo simulations of Hubbard-Anderson models using Gutzwiller projection operators to restrict charge fluctuations in noninteracting wave functions. The Gutzwiller matrix elements are mapped onto a statistical model, which is evaluated using a stochastic algorithm that is normally applied to finite-temperature simulation. The method has been successfully tested for one-dimensional (1D) Hubbard and 1D Anderson models, and results are presented for the 2D Hubbard model. We find a region close to the half-filled limit where the Gutzwiller states of different magnetic ordering are very close in energy.
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 ground state of the Hubbard model on one-dimensional and two-dimensional square lattices is studied with the effective Hamiltonian derived for the strong correlation regime and Gutzwiller-type wave function. The non-half-filled band as well as the half-filled band is discussed. The variational Monte-Carlo method enables us to evaluate the energy for the singlet liquid (resonating-valence-bond state) and for Néel state. The result shows that the stability of the singlet liquid depends on the dimensionality of the system.
New variational wave functions are introduced to describe the Hubbard model in strong correlation. Intersite correlation (or Jastrow) factors are taken into account beyond the Gutzwiller wave function (GF). A study is made on the one-dimensional lattice and the two-dimensional square lattice at half-filling as well as less-than-half-filling. Much improvement can be achieved thereby in the description of physical quantities such as the momentum distribution and spin and charge density correlation functions. In addition, the ground state energy is lowered appreciably compared with the GF. In this connection, the Mott-Hubbard-type metal-insulator transition is also re-examined.
We present recent theoretical results on superconductivity in correlated-electron systems, especially in the two-dimensional Hubbard model and the three-band d-p model. The mechanism of superconductivity in high-temperature superconductors has been extensively studied on the basis of various electronic models and also electron-phonon models. In this study, we investigate the properties of superconductivity in correlated-electron systems by using numerical methods such as the variational Monte Carlo method and the quantum Monte Carlomethod. The Hubbard model is one of basic models for strongly correlated electron systems, and is regarded as the model of cuprate high temperature superconductors. The d-p model is more realistic model for cuprates. The superconducting condensation energy obtained by adopting the Gutzwiller ansatz is in reasonable agreement with the condensation energy estimated for YBa2Cu3O7. We show the phase diagram of the ground state using this method. We have further investigated the stability of striped and checkerboard states in the under-doped region. Holes doped in a half-filled square lattice lead to an incommensurate spin and charge density wave. The relationship of the hole density x and incommensurability δ, δ~x, is satisfied in the lower doping region, as indicated by the variationalMonte Carlocalculations for the two-dimensional Hubbard model. A checkerboard-like charge-density modulation with a roughly period has also been observed by scanning tunneling microscopy experiments in Bi2212 and Na-CCOC compounds. We have performed a variational Monte Carlo simulation on a two-dimensional t-t′-t″- U Hubbard model with a Bi-2212 type band structure and found that the period checkerboard spin modulation, that is characterized by multi Q vectors, is indeed stabilized. We have further performed an investigation by using a quantumMonte Carlomethod, which is a numerical method that can be used to simulate the behavior of correlated electron systems. We present a new algorithm of the quantum Monte Carlo diagonalization that is a method for the evaluation of expectation value without the negative sign problem. We compute pair correlation functions and show that pair correlation is indeed enhanced with hole doping.
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.