Exotic Superconductivity with Enhanced Energy Scales in Three Band Crossing Materials

Three band crossings can arise in three dimensional quantum materials with certain space group symmetries. The low energy Hamiltonian supports spin $\textit{one}$ fermions and a flat band. We study the pairing problem in this setting. We write down a minimal BCS Hamiltonian and decompose it into spin-orbit coupled irreducible pairing channels. We then solve the resulting gap equations in channels with zero total angular momentum. We find that in the spin singlet channel (and also in an unusual `spin quintet' channel), superconductivity is enormously enhanced, with a possibility for the critical temperature to be $\textit{linear}$ in interaction strength. Meanwhile, in the spin triplet channel, the superconductivity exhibits features of conventional BCS theory due to the absence of flat band pairing. Three band crossings thus represent an exciting new platform for realizing exotic superconducting states with enhanced energy scales. We also discuss the effects of doping, nonzero temperature, and of retaining additional terms in the $\mathbf{k} \cdot \mathbf{p}$ expansion of the Hamiltonian.