Phase diagram of the t-U² Hamiltonian of the weak coupling Hubbard model

We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction $U$, for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small $U$ is mapped onto an effective Hamiltonian with the attractive interaction using the canonical transformation: $H_{eff}=e^S He^{-S}$. The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling $n_e$ and the next nearest-neighbor transfer $t'$. The d-wave pairing is dominant for the square lattice in a wide range of $n_e$ and $t'$. The d-wave pairing is also stable for the square lattice with anisotropic $t'$. The three-band $d$-$p$ model is also investigated, for which the d-wave pairing is stable in a wide range of $n_e$ and $t_{pp}$ (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter $-t'$ could not be so large so that the optimum doping rate is in the range of $0.8 <n_e <0.9$.