(1) The wave number dependence of the static spin susceptibility χ( q ) and (2) the effective interaction between quasiparticles mediated by spin fluctuations in the dilute two dimensional Hubbard model are investigated. The quantum Monte Carlo method (QMC) and the fluctuation exchange approximation (FLEX) are applied to this model. It is found that a maximum at wave number q =0 emerges and develops in χ( q ) as the interaction U gets strong. It is also found that the most attractive component of the effective interaction mediated by spin fluctuations is the p -wave component in the strong coupling region where a maximum at q =0 devolops in χ( q ).
Using numerical dynamic cluster quantum Monte Carlo results, we study a simple approximation for the pairing interaction of a two-dimensional Hubbard model with an on-site Coulomb interaction
U equal to the bandwidth. We find that with an effective temperature-dependent coupling
U(T) and the numerically calculated spin susceptibility $\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$, the
d-wave pairing interaction is well approximated by $\frac{3}{2}{\overline{U}}^{2}\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$.
A dynamic cluster quantum Monte Carlo algorithm is used to study a spin-susceptibility representation of the pairing interaction for the two-dimensional Hubbard model with an on-site Coulomb interaction equal to the bandwidth for various doping levels. We find that the pairing interaction is well approximated by $\frac{3}{2}\overline{U}{(T)}^{2}\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$, with an effective temperature and doping dependent coupling
U(T) and the numerically calculated spin susceptibility $\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$. We show that at low temperatures,
U may be accurately determined from a corresponding spin-susceptibility based calculation of the single-particle self-energy. We conclude that the strength of the
d-wave pairing interaction, characterized by the mean-field transition temperature, can be determined from a knowledge of the dressed spin susceptibility and the nodal quasiparticle spectral weight. This has important implications with respect to the questions of whether spin fluctuations are responsible for pairing in the high-
Tc cuprates.
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.
Susceptibilities in the static limit ω→0 are evaluated to show that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional Hubbard model for optimal band parameters. By using quantum Monte Carlo methods, we have examined the size dependence of the spin susceptibility χ( Q ) at half filling and the pair susceptibilities with d - and s -wave symmetries for the repulsive and attractive interactions, respectively. The numerical results for systems of up to 14×14 sites show that the s -wave pair susceptibility scales as χ pair ∼ L 2 for the attractive case, in agreement with previous quantum Monte Carlo studies. The scaling χ pair ∼ L 2 also holds for the d -wave pair susceptibility for the repulsive Hubbard model if we adjust the band parameter t '. Therefore, it is likely that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional repulsive Hubbard model.