Phase transition between d-wave and anisotropic s-wave gaps in high-temperature oxides superconductors

We numerically study models for superconductivity with two interactions: V-> due to antiferromagnetic (AF) fluctuations and V-< due to phonons, in a weak coupling approach to the high-temperature superconductivity. The nature of the two interactions is considerably different: V-> is positive and sharply peaked at (+/-pi,+/-pi) while V-< is negative and peaked at (0, 0) due to weak phonon screening. The superconductivity is mainly induced LS V-<. The positive interaction V-> (AF) is not effective in superconductivity, but important to give d-wave superconductivity. The gap order parameter Delta(k) is constant (s-wave) at an extremely overdoped region and it becomes anisotropic as doping is reduced. Then there exists a first-order phase transition between anisotropic 8-wave and d-wave gaps as doping is reduced further. These results are qualitatively in agreement with preceding works; they should be modified in the strongly underdoped region by the presence of antiferromagnetic pseudogap due to the fluctuations.