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arXiv:cond-mat/9711125 · Physical review. B, Condensed matter

Quantum Monte Carlo study of hole binding and pairing correlations in the three-band Hubbard model

M. Guerrero, James E. Gubernatis, Shiwei Zhang

Published May 15, 1998• 55 citations
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Abstract

We simulated the three-band Hubbard model using the constrained-path Monte Carlo (CPMC) method in search for a possible superconducting ground state. The CPMC is a ground-state method that is free of the exponential scaling of computing time with system size. We calculated the binding energy of a pair of holes for systems up to $6\ifmmode\times\else\texttimes\fi{}4$ unit cells. We also studied the pairing correlation functions versus distance for both the $d$-wave and extended $s$-wave channels in systems up to $6\ifmmode\times\else\texttimes\fi{}6$. We found that holes bind for a wide range of parameters and that the binding increased as the system size is increased. However, the pairing correlation functions decay quickly with distance. For the extended $s$ channel, we found that as the Coulomb interaction ${U}_{d}$ on the Cu sites is increased, the long-range part of the correlation functions is suppressed and fluctuates around zero. For the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ channel, we found that the correlations decay rapidly with distance towards a small positive value. However, this value becomes smaller as the interaction ${U}_{d}$ or the system size is increased.

DOI
10.1103/physrevb.57.11980

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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.
Using the constrained path Monte Carlo method, we simulated the two-dimensional, three-band Hubbard model to study pairing, charge, and spin correlations as a function of electron and hole doping and the Coulomb repulsion Vpd{V}_{\mathrm{pd}}Vpd​ between charges on neighboring Cu and O lattice sites. As a function of distance, both the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave and extended s-wave pairing correlations decayed quickly. In the charge-transfer regime, increasing Vpd{V}_{\mathrm{pd}}Vpd​ decreased the long-range part of the correlation functions in both channels, while in the mixed-valent regime, it increased the long-range part of the s-wave behavior but decreased that of the d-wave behavior. Still the d-wave behavior dominated. At a given doping, increasing Vpd{V}_{\mathrm{pd}}Vpd​ increased the spin-spin correlations in the charge-transfer regime but decreased them in the mixed-valent regime. Also, increasing Vpd{V}_{\mathrm{pd}}Vpd​ suppressed the charge-charge correlations between neighboring Cu and O sites. Electron and hole doping away from half-filling was accompanied by a rapid suppression of antiferromagnetic correlations. Our results suggest that adding a repulsive Vpd{V}_{\mathrm{pd}}Vpd​ to the model does not enhance its tendency toward superconductivity even though the behavior of the spin structure factor is consistent with the properties of some high-temperature superconducting materials.
To gain a deeper understanding of the high-temperature superconductivity in doped cuprates, the constrained-path quantum Monte Carlo method was used to explore the superconducting pairing correlation function and effective pairing interaction in the three-band Hubbard model. Our main observation was that in the low doping region, compared to other symmetries, [Formula: see text]-wave superconducting pairing dominates. The Coulomb interaction [Formula: see text] suppresses [Formula: see text]-wave pairing correlation, while the effective pairing interaction is both enhanced by the increasing [Formula: see text] and the nearest attraction. These results are used to understand the superconducting state in copper oxide high-temperature superconductors.
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.
We benchmark constrained-path Monte Carlo (CPMC) on the triangular-lattice Hubbard model for several fillings and UUU values and show that symmetry-adapted trial wave functions substantially improve quantitative accuracy. Away from half-filling, simple free-electron-based trials that preserve the ground state symmetry yield energy deviations ≲1%\lesssim 1\%≲1% from exact diagonalization and density matrix renormalization group results. At half-filling, strong frustration in the intermediate to large UUU regimes necessitates symmetry-projected trials to reach comparable accuracy, where both free-electron and symmetry-broken Hartree-Fock trials incur substantial constraint bias. Since the computational cost of CPMC with symmetry projection scales polynomially with system size, our results motivate its use as a practical route for studying competing ground states in strongly correlated, frustrated systems.
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.