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arXiv:cond-mat/9703017 · Physical Review Letters

Pairing Correlations in the Two-Dimensional Hubbard Model

Shiwei Zhang, Joseph Carlson, James E. Gubernatis

Published Jun 9, 1997• 164 citations
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Abstract

We present the results of a quantum Monte Carlo study of the extended $s$ 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 $s$ 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.

DOI
10.1103/physrevlett.78.4486

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We present the results of a quantum Monte Carlo study of the extended sss and the dx2\ensuremath−y2{d}_{{x}^{2}{\ensuremath{-}y}^{2}}dx2\ensuremath−y2​ 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 dx2\ensuremath−y2{d}_{{x}^{2}{\ensuremath{-}y}^{2}}dx2\ensuremath−y2​ 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.
A linked cluster expansion of the Gutzwiller variational wave function is used to study the one- and two-dimensional Hubbard model; in parallel the variational results are compared with quantum Monte Carlo simulations on finite-size lattices. Detailed results are presented for the magnetic and pairing correlations as a function of wave vector, electron density, and interaction strength U. There is no sign for superconducting long-range order although both the d-wave and the extended s-wave pairing fluctuations are substantially enhanced as U increases.
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.
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.
The d-wave pair-pair correlation of the two-dimensional Hubbard model is investigated by the power-Lanczos method. A Jastrow-BCS type wave function is used in our variational study. By assuming a monotonic behavior of the long-range pairing correlation under Lanczos iteration, we found there is no long-range pairing order in the intermediate interaction region (U ≃ 10), which is consistent with the former result by the constrained-path Monte Carlo method. We also present numerical results obtained by the exact diagonalization of the 4×4 lattice in supporting our conclusion. PACS numbers: 74.20.-z, 71.18.+y I.
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〉.
Quantum Monte Carlo is used to investigate the possibility of ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductivity in the two-dimensional repulsive Hubbard model. A small energy scale relevant to possible pairing requires a care [i.e., sufficiently small level separation between the kkk points $(\ensuremath{\delta}k,\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}})$ and $(\ensuremath{\pi}\ensuremath{-}\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}},\ensuremath{\delta}{k}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})$ with small $\ensuremath{\delta}k$'s] to detect enhanced correlations in finite-size studies, as motivated from a previous study on Hubbard ladders. Our calculation indeed detects long-tailed enhancements in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation when the system is near, but not exactly at, half-filling.