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Communications in Theoretical Physics

Calculation of the Binding Energy in the One-Dimensional Hubbard Model by the VMC Method

Yongjun Liu, Jinming Dong, Chang-De Gong

Published Jun 15, 1994• 0 citations
Publisher

Abstract

Using variational Monte-Carlo method and choosing an antiferromagnetic trial wave function, we have calculated the binding energies just off half-filling in one-dimensional Hubbard chains. We found the binding for the lattice length N being less than 40 in the degenerate case, but no binding at all in the non-degenerate case. However, with lattice length increasing gradually, difference between the two cases becomes smaller and smaller. Finally, there is no binding at all no matter whether or not the chains have a degeneracy at the Fermi level in the non-interacting limit. It may be an indication of the independence of the final calculation results on the boundary conditions for the large enough system or in thermodynamic limit.

DOI
10.1088/0253-6102/21/4/403

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Using variational Monte-Carlo method and choosing an antiferromagnetic trial wave function, we have calculated the binding energies just off half-filling in one-dimensional Hubbard chains. We found the binding for the lattice length N being less than 40 in the degenerate case, but no binding at all in the non-degenerate case. However, with lattice length increasing gradually, difference between the two cases becomes smaller and smaller. Finally, there is no binding at all no matter whether or not the chains have a degeneracy at the Fermi level in the non-interacting limit. It may be an indication of the independence of the final calculation results on the boundary conditions for the large enough system or in thermodynamic limit.
Using the Bethe ansatz, we calculate the binding energies of two holes just off half-filling in one-dimensional Hubbard chains. Somewhat surprisingly, we find binding comparable to that seen in two-dimensional systems in chains that have a degeneracy at the Fermi level in the noninteracting (U=0) limit. The binding initially IincreasesR with lattice size before beginning a slow decay to zero in the thermodynamic limit. We discuss the general implications of our results for binding energy and other numerical calculations of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ models.
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.
We use the density-matrix renormalization group to calculate the phase diagram of the one-dimensional extended Hubbard chain at half filling. We find that for one specific U,U,U, the critical value of Vc{V}_{c}Vc​ for the charge-density-wave versus spin-density-wave phase transition is smaller than that predicted from the Monte Carlo simulation. Our results are consistent with the strong- and weak-coupling limit results, whereas the previous results from the Monte Carlo simulations deviate from the weak-coupling limit result. As UUU increases, we find that the first-order phase transition develops gradually. The tricritical point is estimated to occur at Uc=4−−6,{U}_{c}=4--6,Uc​=4−−6, not Uc=3{U}_{c}=3Uc​=3 as previously predicted from the Monte Carlo simulation. Due to the topological difference, we observe that with the periodic boundary condition, the phase transition is more abrupt for finite chains of length N=4MN=4MN=4M than for chains of N=4M+2,N=4M+2,N=4M+2, where MMM is an integer, for the same UUU and V.V.V. The difference will diminish in the thermodynamic limit.
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 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.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.