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Physical review. B, Condensed matter

Correlations in the two-dimensional Hubbard model

Adriana Moreo, Douglas J. Scalapino

Published Apr 1, 1991• 54 citations
Publisher

Abstract

Using Monte Carlo simulations, we have studied the question of phase separation and pairing in the doped two-dimensional Hubbard model. For a range of U/t values greater than as well as less than the bandwidth, we find no evidence for phase separation when the system is doped away from half filling. In agreement with previous work, we find that the d-wave and extended s-wave pairing correlations are enhanced by the interaction vertex but that none of the eight different pairing correlations that we studied increases as a function of lattice size. We have also examined the dependence of the kinetic energy, which determines the total optical spectral weight, on U/t and filling 〈n〉.

DOI
10.1103/physrevb.43.8211

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Using Monte Carlo simulations, we have studied the question of phase separation and pairing in the doped two-dimensional Hubbard model. For a range of U/t values greater than as well as less than the bandwidth, we find no evidence for phase separation when the system is doped away from half filling. In agreement with previous work, we find that the d-wave and extended s-wave pairing correlations are enhanced by the interaction vertex but that none of the eight different pairing correlations that we studied increases as a function of lattice size. We have also examined the dependence of the kinetic energy, which determines the total optical spectral weight, on U/t and filling 〈n〉.
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.
Quantum Monte Carlo simulations are used to explore the phase diagram of the negative-U Hubbard model. As the system is doped away from half filling, 〈n〉=1, the s-wave pair-field correlations are enhanced and the charge-density-wave correlations suppressed. There is no indication of phase separation. Away from half filling, the pair-field correlations are consistent with Kosterlitz-Thouless scaling and a finite Tc{\mathit{T}}_{\mathit{c}}Tc​ which peaks near 〈n〉=1 but vanishes at exactly half filling.
Superconducting correlation of the Hubbard model on a square lattice is reinvestigated by quantum Monte Carlo method. Results from more than one order of magnitude larger Monte Carlo steps than before are presented to discuss the convergence of the data to the ground state. By examining the dependence on the choice of initial trial states as well as on the Trotter number in detail, the superconducting correlation in the ground state is reliably obtained up to 12×12 lattice away from the half-filling and up to 16×16 lattice at the half-filling, which substantially improve former results near the half-filling. Various channels of pairing in the ground state show no system size dependence and remain quite short ranged at any filling including the region close to the half-filling. Numerical data show further evidence that neither extended s - nor d -wave superconducting state is possible in the two-dimensional Hubbard model.
We present the results of a quantum Monte Carlo study of the extended sss and the ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing correlation functions for the two-dimensional Hubbard model, computed with the constrained-path Monte Carlo method. For small lattice sizes and weak interactions, we find that the ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing correlations are stronger than the extended sss pairing correlations and are positive when the pair separation exceeds several lattice constants. As the system size or the interaction strength increases, the magnitude of the long-range part of both correlation functions vanishes.
To clarify the effects of next-nearest-neighbor electron hopping ${t}^{\ensuremath{'}}$ on the spin, charge, and pairing correlations, we studied the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ Hubbard model using the constrained-path Monte Carlo technique. We found that a negative ${t}^{\ensuremath{'}}$ suppresses the spin fluctuations near the Brillouin zone point $M=(\ensuremath{\pi},\ensuremath{\pi})$ and the charge fluctuations near the point $X=(\ensuremath{\pi},0),$ while a positive ${t}^{\ensuremath{'}}$ has the opposite effect. For a negative ${t}^{\ensuremath{'}},$ we also found that the long-range vertex contributions to the pairing correlations are heavily suppressed, but the local contributions are enhanced. Over the entire hole doping range studied, the d-wave long-range vertex contributions to pairing dominated those of the extended s-wave vertex contributions. In general, we did not find any tendency for these correlations to grow with increasing system size. We also did a limited number of simulations with an attractive nearest neighbor density-density interaction added to the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ model. It enhanced the long-range vertex contributions to the pairing correlations only for weak values of U.
We have studied the behavior of pairing correlations and susceptibilities in the repulsive Hubbard model on a two-dimensional square lattice by Monte Carlo simulations. Lattice sizes up to 8\ifmmode\times\else\texttimes\fi{}8, temperatures down to T=0.083 (in units where the bandwidth W=8), interaction strengths up to U=16, and several band fillings ranging from (1/2-filled to (1/4-filled were considered. Our results suggest that the model does not exhibit superconductivity. The implications with respect to possible mechanisms for high-Tc{T}_{c}Tc​ superconductivity in oxides are discussed.