Systematic analysis of a spin-susceptibility representation of the pairing interaction in the two-dimensional Hubbard model

Abstract

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 $\frac{3}{2}\overline{U}{\left(T\right)}^2\chi (K-K^{\prime })$, with an effective temperature and doping dependent coupling $\overline{U}\left(T\right)$ and the numerically calculated spin susceptibility $\chi (K-K^{\prime })$. We show that at low temperatures, $\overline{U}$ 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-$T_c$ cuprates.