Derivation of the superconducting gap equation for the noncentrosymmetric superconductor Li2Pt3B

We present here the mathematical background of our approach, presented in Phys. Rev. B 86, 134526 (2012) regarding the gap function and symmetry for the noncentrosymmetric (NCS) superconductor $Li_2Pt_3B$. As revealed by the experiment, this NCS superconductor gives rise to line nodes in the superconducting order parameter, which is responsible for many of its experimental behaviors. Owing to the enhanced d-character of the relevant bands that cross the Fermi level,the system gets weakly correlated. The nature and symmetry of this nodal behavior is explained from a microscopic viewpoint. In this article starting with an Hubbard model relevant for this NCS system by considering the effect of the onsite Coulomb repulsion on the pairing potential perturbatively, we extract the superconducting gap equation. Further analysis of this equation predicts a $s_\pm$ wave gap function with line nodes as the most promising candidate in the superconducting state.