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International Journal of Modern Physics B

HOLE-HOLE CORRELATIONS IN THE 2D HUBBARD MODEL

Sandro Sorella, Alberto Parola, Erio Tosatti

Published Jan 1, 1991• 3 citations
Publisher

Abstract

The hole-hole correlation function in the 2D Hubbard model is investigated in the low doping regime both via Monte Carlo simulations for U=4 and Lanczos diagonalization on a 16 site cluster for very large U. The results are compared with available theoretical predictions for the strong coupling limit. There is a clear short range repulsion, but no signature of hole-hole attraction is found in either case. At U=4 correlations are still remarkably weak-coupling-like. For large U, they become rather similar to those of hard core bosons.

DOI
10.1142/s0217979291000109

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The hole-hole correlation function in the 2D Hubbard model is investigated in the low doping regime both via Monte Carlo simulations for U=4 and Lanczos diagonalization on a 16 site cluster for very large U. The results are compared with available theoretical predictions for the strong coupling limit. There is a clear short range repulsion, but no signature of hole-hole attraction is found in either case. At U=4 correlations are still remarkably weak-coupling-like. For large U, they become rather similar to those of hard core bosons.
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.
We use exact diagonalization in order to study the infinite-U limit of the two-dimensional Hubbard model. As well as looking at single-particle correlations, such as ${\mathit{n}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}$=〈${\mathit{c}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}$〉, we also study N-particle correlation functions, which compare the relative positions of all the particles in different models. In particular we study 16- and 18-site clusters and compare the charge correlations in the Hubbard model with those of spinless fermions and hard-core bosons. We find that although low densities of holes favor a ``locally ferromagnetic'' fermionic description, the correlations at larger densities resemble those of pure hard-core bosons surprisingly well.
The ground state of the two-dimensional (2D) Hubbard model is investigated by adopting improved wave functions that take into account intersite electron correlation beyond the Gutzwiller ansatz. The ground-state energy is lowered considerably, giving the best estimate of the ground-state energy for the 2D Hubbard model. There is a crossover from weakly to strongly correlated regions as the on-site Coulomb interaction UUU increases. The antiferromagnetic correlation induced by UUU is reduced for hole doping when UUU is large, being greater than the bandwidth, thus increasing the kinetic energy gain. The spin and charge fluctuations are induced in the strongly correlated region. These antiferromagnetic and kinetic charge fluctuations induce electron pairings, which would result in high-temperature superconductivity.
We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order U2{U}^{2}U2 contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.
The interactions between holes in the Hubbard model, in the low density, intermediate- to strong-coupling limit, are investigated by systematically improving mean field calculations. The configuration-interaction basis set is constructed by applying to local unrestricted Hartree-Fock configurations all lattice translations and rotations. It is shown that this technique reproduces, correctly, the properties of the Heisenberg model, in the limit of large U. Upon doping, dressed spin polarons in neighboring sites have an increased kinetic energy and an enhanced hopping rate. Both effects are of the order of the hopping integral and lead to an effective attraction at intermediate couplings. The numerical results also show that when more than two holes are added to the system, they do not tend to cluster, but rather hole pairs remain far apart. Hole-hole correlations are also calculated and shown to be in qualitative agreement with exact calculations for $4\ifmmode\times\else\texttimes\fi{}4$ clusters. In particular our results indicate that for intermediate coupling the hole-hole correlation attains a maximum when the holes are in the same sublattice at a distance of 2\sqrt{2}2​ times the lattice spacing, in agreement with exact results and the $t\ensuremath{-}J$ model. The method is also used to derive known properties of the quasiparticle band structure of isolated spin polarons.
In this study we have computed the pair correlation functions in the two-dimensional Hubbard model using a quantum Monte Carlo method. We employ a new diagonalization algorithm in the quantum Monte Carlo method which is free from the negative sign problem. We show that the d-wave pairing correlation function is indeed enhanced slightly for the positive on-site Coulomb interaction U when doping away from the half-filling. When the system size becomes large, the pair correlation function P d increases for U > 0 compared to the non-interacting case, while P d is suppressed for U > 0 when the system size is small. The enhancement ratio P d [ U ]/ P d [ U = 0] will give a criterion on the existence of superconductivity. The ratio P d [ U ]/ P d [ U = 0] increases almost linearly ∝ L when the system size L × L is increased. This increase is a good indication of the existence of a superconducting phase in the two-dimensional Hubbard model. There is, however, no enhancement of pair correlation functions in the half-filled case, which indicates the absence of superconductivity without hole doping.