Unconventional Superconductivity in spin-3/2 fermions

We study unconventional superconductivity in spin-orbit-coupled three-dimensional electronic systems without inversion symmetry, featuring gapless fermions with the Hamiltonian given by H=p\cdot J, where J are the generators of total angular momentum of 3/2. These systems have a Fermi points with a fixed ratio of the Fermi velocities. A favorable short-range interaction can lead to a d-wave superconductivity which is described by a complex tensor order parameter. We investigate the structure of the corresponding Ginzburg-Landau free energy and demonstrate that already at the quartic level the superconducting state of the system is uniquely determined. For a chemical potential right at the Fermi point, the ground state of the system is given by an uniaxial nematic state. In the case of a finite chemical potential, we find that the cyclic state is favored as a ground state. The cyclic state breaks time-reversal symmetry maximally, has no average magnetization, and exhibits robust small Bogoliubov Fermi surfaces in the excitation spectrum of the quasiparticles.