Fate of superconductivity in three-dimensional disordered Luttinger semimetals

Superconducting instability can occur in three-dimensional quadratic band crossing semimetals only at a finite coupling strength due to the vanishing of density of states at the quadratic band touching point. Since realistic materials are always disordered to some extent, we study the effect of short-ranged-correlated disorder on this superconducting quantum critical point using a controlled loop-expansion applying dimensional regularization. The renormalization group (RG) scheme allows us to determine the RG flows of the various interaction strengths and shows that disorder destroys the superconducting quantum critical point. In fact, the system exhibits a runaway flow to strong disorder.