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Physical review. B, Condensed matter

Ground-state properties of the two-dimensional Hubbard model

H. Q. Lin

Published Oct 1, 1991• 28 citations
Publisher

Abstract

Using all appropriate symmetries for a two-dimensional square system, we have been able to diagonalize exactly the Hubbard Hamiltonian on the 4\ifmmode\times\else\texttimes\fi{}4 and 18-site lattices. We calculate the ground-state energies, hole-hole interactions, and various correlation functions as functions of the Coulomb repulsion U and band filling, and compare the Hubbard-model results with those obtained from the t-J and strong-coupling models.

DOI
10.1103/physrevb.44.7151

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Using all appropriate symmetries for a two-dimensional square system, we have been able to diagonalize exactly the Hubbard Hamiltonian on the 4\ifmmode\times\else\texttimes\fi{}4 and 18-site lattices. We calculate the ground-state energies, hole-hole interactions, and various correlation functions as functions of the Coulomb repulsion U and band filling, and compare the Hubbard-model results with those obtained from the t-J and strong-coupling models.
We investigate the Hubbard model on a two-dimensional cluster of 18 sites using Lanczos algorithm for different strengths of Coulomb repulsion in the low doping region. Energies and correlation functions are given for half-filling and two holes. In the strong coupling regime we compare our results with the $t\ensuremath{-}J$ model. We confirm the tendency of holes to repel, although a significant value of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductive order parameter is measured. These data represent a benchmark for future quantum simulations.
The ground-state wave function and energy are calculated for the one-band Hubbard Hamiltonian on a 4\ifmmode\times\else\texttimes\fi{}4 cluster of the square lattice up to first and second order of the on-site Coulomb repulsion U, respectively. Changes induced by a deviation of the number of electrons from half-filling (one electron per lattice site) are closely followed. Charge-charge and spin-spin correlation functions are directly obtained from the ground-state wave function and show the existence of antiferromagnetic correlations that extend over the whole cluster at half-filling. Correlations among sites separated by more than a lattice site diminish very rapidly as holes are introduced into the cluster. The analysis is extended to the negative-U case. Ground-state properties change qualitatively.
The single-band Hubbard model (both with and without first-neighbor interactions) is studied by exact diagonalization on several one- and two-dimensional clusters, ranging from four to nine sites, including rings and arrangements of squares. Occupancies include the half-filled band (one electron per site) and configurations with one and two holes in the half-filled band. Ground-state properties are determined for a wide range of parameters. We discuss the phase diagram in the (U,V) plane and consider spin, charge, and pairing correlations. The binding of a pair of holes is investigated. Circumstances in which binding and pairing are enhanced by first-neighbor interactions are described.
The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling JJc{J}_{c}Jc​\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}Jc{J}_{c}Jc​ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.
We exactly diagonalize the effective Hamiltonian obtained from the Hubbard model in the strong-coupling limit for a two-dimensional $\sqrt{10}\ifmmode\times\else\texttimes\fi{}\sqrt{10}$ size square lattice. The effective Hamiltonian operates on a restricted Hilbert space containing only states with singly occupied sites, which makes the diagonalization possible within reasonable computational time for the above lattice. The ground-state energy and wave function are obtained for several values of the coupling ratio tU\frac{t}{U}Ut​ and the doping fraction xxx. We find three different phases in the (x,tUx,\frac{t}{U}x,Ut​) phase diagram. The first is characterized by antiferromagnetic order which extends to the longest possible distance in the 10-site system. The second and third phases are characterized by antiferromagnetic and ferromagnetic short-range correlations, respectively. We comment on the possible relevance of the results to the recently discovered high-temperature copper-oxide superconductors.
We use the C4vC_{4v}C4v​ symmetry group of the 4-site Hubbard model to construct a ground state variational wave function of two- and four interacting electrons. In the limit U→0U\rightarrow 0U→0, ground state energies of the two- and four interacting electrons system is of the order −4t-4t−4t. The variational wave function of the four interacting electrons obtained using the B1B_1B1​ irreducible representation is valid for on-site Coulomb repulsion, while the one obtained using the A1A_1A1​ representation is valid for negative values of the Coulomb interaction. The system exhibits antiferromagnetic correlations.