The irreducible particle-particle interaction ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{I}}$ of the two-dimensional Hubbard model is calculated by quantum Monte Carlo simulations. The Bethe-Salpeter equation for the pair-wave function is then solved using the Monte Carlo data for ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{I}}$ and the single-particle Green's function. The dominant pairing correlations in the intermediate-coupling regime and at small dopings are discussed.
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.
We study the three-band Hubbard model for the cuprate superconductors using Quantum Monte Carlo simulations and the fluctuation exchange approximation including a self-consistent many-body renormalization of the single-particle propagator. The energy dispersion of the low-lying one-particle excitations is in very good agreement with spectroscopic measurements for a hole doping concentration δ = 0.25 of the CuO2 planes. At the same doping, we examine the spin susceptibility yielding a non-ordered phase with short-range incommensurate fluctuations near the antiferromagnetic wave vector. In the superconducting regime, the interaction vertex for the pairing correlation function in the extended s-wave channel shows a maximum for δ = 0.20, resembling the dependence of transition temperature Tc on doping.
A recent paper by Matuttis and Ito questions the numerical accuracy of a widely-used fermion Monte Carlo algorithm. They also claim that the increase in the d-wave pairfield susceptibility χd(T) of a doped 4×4 Hubbard model at low temperature, previously found using this algorithm, is an artifact due to numerical errors. Here, we provide tests which show that this algorithm is numerically accurate and show that the simulation of χd for a 2×2 lattice agrees with exact diagonalization results. We also provide more complete data for χd on a 4×4 lattice that is consistent with our previous results.
Monte Carlo results for a two-dimensional Hubbard model in the intermediate-coupling regime U=4t are compared with a diagrammatic spin-fluctuation approximation. Simulations on an 8\ifmmode\times\else\texttimes\fi{}8 lattice doped away from half filling were carried out down to temperatures of order 140 of the bandwidth. Results for the spin susceptibility \ensuremath{\chi}(q,i\ensuremath{\omega}
m), the electron self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}
n), various pair-field susceptibilities, and the irreducible particle-particle scattering vertex \ensuremath{\Gamma}(p\ensuremath{'},i\ensuremath{\omega}${}_{\mathrm{n}\ensuremath{'}}$|p,i\ensuremath{\omega}
n) were obtained. A random-phase approximation for \ensuremath{\chi}(q,i\ensuremath{\omega}
m) with a renormalized Coulomb coupling U\ifmmode\bar\else\textasciimacron\fi{} is shown to provide a fit to the Monte Carlo data. A similar approximation for the Berk-Schrieffer spin-fluctuation interaction also provides a reasonable fit to the self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}
n) in the region explored by the Monte Carlo data. However, a similar approximation for the irreducible particle-particle interaction failed to reproduce the Monte Carlo results. Higher-order vertex corrections were calculated, but significant discrepancies with Monte Carlo results for \ensuremath{\Gamma} remain.