Renormalization group approach to anisotropic superconductors at finite temperature

A renormalization group (RG) analysis of the superconductive instability of an anisotropic fermionic system is developed at a finite temperature. The method appears a natural generalization of Shankar's approach to interacting fermions and of Weinberg's discussion about anisotropic superconductors at T=0. The need of such an extension is fully justified by the effectiveness of the RG at the critical point. Moreover the relationship between the RG and a mean-field approach is clarified, and a scale-invariant gap equation is discussed at a renormalization level in terms of the eigenfunctions of the interaction potential, regarded as the kernel of an integral operator on the Fermi surface. At the critical point, the gap function is expressed by a single eigenfunction and no symmetry mixing is allowed. As an illustration of the method we discuss an anisotropic tight-binding model for some classes of high $T_c$ cuprate superconductors, exhibiting a layered structure. Some indications on the nature of the pairing interaction emerge from a comparison of the model predictions with the experimental data.