The one-dimensional Hubbard model with nearest-neighbor hopping, first- and second-nearest-neighbor particle interactions, and one particle per site is reexamined by means of a small-crystal approach---an exact solution of a four-site cluster with periodic boundary conditions. Lines of symmetry crossing as well as singular lines and points are found. Analysis of correlation functions indicates that although there are smooth, strong changes as the parameters approach the values of the charge-to-spin-density-wave transition (a suggestion of an incipient second-order transition), the actual change involves a crossover between unrelated symmetries (a clear indication of a first-order transition). The relatively simple calculation and its results agree well with previous Monte Carlo numerical simulations and settles the unresolved question of the order of the density-wave transition.
Abstract This study extended the fixed-node QMC method to a lattice model for hopping fermions using Green’s function techniques. In general, the modifications of the procedures for continuous systems to adapt them to a lattice system were minor, and there appeared to be no surprises. Applications were made to the two-dimensional Hubbard model with different on-site interactions, lattice dimensions, boundary conditions, node specifications, and trial wavefunctions for importance sampling. Results were compared with those of Hartree-Fock and variational QMC calculations. For the HF calculations with zero interaction, the exact solutions are known, and the Green’s function QMC results agreed with those within statistical error. Trial wavefunctions specifying the nodes were obtained directly as those of HF determinantal solutions for zero interaction, or as variationally optimized determinantal functions with correlation terms. The procedure for fixed nodes was to eliminate walkers crossing the boundaries of a nodal region, a procedure not quite correct, but probably a reasonable approximation. The results in all cases were consistent with the HF and variational QMC calculations, giving lower energies than either for nonzero interaction energies. The existence of a domain-wall phase suggested by the HF and variational QMC results was confirmed by the fixed-node Green’s function results. It was noted that the fixed-node method would be applicable to systems other than the Hubbard model.
We have modified an existing real space renormalization group scheme to remove some physically inconsistent features and used the rectified method to study the ground state of the one-dimensional Hubbard model with nearest-neighbour repulsion and the bond-charge interaction for half filling. For the nearest-neighbour interaction we find a phase diagram which agrees nicely with the Monte Carlo results. For the case of bond-charge interaction we have reproduced some available exact results and found some interesting information in the parameter regime where no exact solution is available.
The Hubbard model is one of the most widely studied many-body models for interacting electrons in solids. Here we present a study of the Extended Hubbard model, which provides a more realistic description of real materials by including some longer-range Coulomb interactions. We discuss the derivation of the model, numerical methods for many-body systems, the ground state of the one-dimensional model, and an application of the model to organic charge-transfer solids. Our discussion of numerical methods includes details of using the Constrained Path Monte Carlo procedure to treat extended interactions and possible extension of the method for imaginary times. The chapter on the one-dimensional model focuses on the region V>U, where previous work had suggested the possibility of superconductivity. Our results show that instead phase separation is prevalent in that parameter regime. The second application is to model the organic charge-transfer salts, which are intermediate between one and two dimensions. Here we find a coexistence of charge and spin ordering that helps to explain the results of recent experiments on these materials.
We consider a generalized Hubbard model with on-site interaction U, nearest-neighbor repulsion V, and nearest-neighbor hopping for spin \ensuremath{\sigma}, which depends on the sum of particles ${\mathrm{m}}_{\mathrm{\ensuremath{\sigma}}\ensuremath{-}|\mathrm{A}\mathrm{m}}$ with opposite spin in the two sites involved. The hopping matrix elements are denoted by
tAA,
tAB,
tBB for ${\mathrm{m}}_{\mathrm{\ensuremath{\sigma}}\ensuremath{-}|\mathrm{A}\mathrm{m}}$=0,1,2, respectively. For 0
tABtAA=
tBB, we have determined the regions of parameters for which the ground-state (GS) of the one-dimensional system is a charge-density wave (CDW), a spin-density wave (SDW), or a metal (M), using Hartree-Fock, exact diagonalization of finite chains and quantum Monte Carlo. The results agree qualitatively with the exactly solvable limit of
tAB=0. For 0
tABtAA=
tBB, the GS is a M for sufficiently low values of U and V. In contrast, when
tAA+
tBB-
2tAB=0, our results suggest that the GS is either a CDW or a SDW, with the boundary between them lying near the line U=2V.
We present results from a constrained path Monte Carlo (CPMC) study of a modified two-dimensional Hubbard model. We include more general forms of the band structure and electron interaction in order to examine their effects on ground-state properties, such as electron pairing correlations. Both next-nearest neighbor hopping, t', and third-nearest neighbor hopping, t″, are introduced in the Hamiltonian. A nearest neighbor interaction of strength V is also included. We carry out CPMC calculations on system sizes up to 16 x 16, at various electron fillings, to investigate the ground state of the model for different values of these parameters. For realistic systems, these calculations indicate that Hubbard-like models are not capable of showing enhanced pairing. The modified Hamiltonian also presents an opportunity to more closely examine the accuracy and robustness of the CPMC algorithm. Results of further benchmark calculations involving CPMC are presented. These benchmarks confirm that CPMC is able to show enhanced pairing in systems where such behavior is known to exist.