Two-Dimensional Hubbard Model at Low Electron Density

The two dimensional Hubbard model with repulsive interaction is studied in the t-matrix approximation, which is the valid approximation in low electron density. The particle-particle correlation function and the t-matrix are obtained analytically. By use of these analytical results it is shown that the t-matrix and hence the self-energy have no singularities in the upper half plane of the complex energy variable. The phase angle δ(2kF + q, ω), with kF being the Fermi wave vector, in the limit of q=0 and ω = 0 is found to depend strongly on the ratio ω/q and is not always zero. The case of the attractive interaction is also briefly studied.