Effects of an electronic topological transition for anisotropic low-dimensional superconductors

We study the superconducting properties of a two-dimensional superconductor in the proximity to an electronic topological transition (ETT). In contrast to the three-dimensional (3D) case, we find that the superconducting gap at $T=0,$ the critical temperature $T_c,$ and the impurity scattering rate are characterized by a nonmonotonic behavior, with maxima occurring close to the ETT. We derive analytical expressions for the value of such maxima both in the s-wave and in the d-wave cases. Such expressions are in good qualitative agreement with the phenomenological trend recently observed for $T_c^{\max }$ as a function of the hopping ratio $t^{\prime }/t$ across several cuprate compounds. We further analyze the effect of an ETT on the Ginzburg-Landau stiffness $\eta .$ Instead of vanishing at the ETT, as could be expected, thus giving rise to an increase of the fluctuation effects, in the case of momentum-independent electron-electron interaction, we find $\eta \ne 0,$ as a result of an integration over the whole Fermi surface.