d-wave pairing in tetragonal superconductors

Using group-theoretical methods, we classify into 11 symmetry classes all translationally invariant singlet-paired superconducting-gap functions whose quasiparticle energy spectrum preserves the symmetry of a tetragonal crystal. We show that for the gap function of symmetry ${\mathrm{\Delta }}^{\mathit{d}+}$(cos${\mathit{k}}_{\mathit{x}}$-cos${\mathit{k}}_{\mathit{y}}$) the two logarithmic singularities in the density of states as a function of energy are a robust consequence of changes in topology of the surfaces of constant quasiparticle energy. Removing the node in this gap function by adding an imaginary component of symmetry i${\mathrm{\Delta }}^{\mathit{d}\mathrm{-}}$ sin${\mathit{k}}_{\mathit{x}}$ sin${\mathit{k}}_{\mathit{y}}$ results in the only possible nodeless gap function that does not involve s-wave pairing, and hence is the only nodeless singlet superconductor that is not suppressed by a Hubbard-type repulsion. We discuss the possible relation to ab-plane tunneling experiments in the high-${\mathit{T}}_{\mathit{c}}$ compound ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_8$.