Topological p-ip superconductor and Bogoliubov Fermi surface in Rhombohedral tetra-layer graphene

We performed a random phase approximation (RPA) calculation for a spin-valley polarized model of the rhombohedral tetra-layer graphene to study the possibility of chiral superconductor from the Kohn-Luttinger mechanism. We included the realistic band structure and form factor in our calculation and solved the self-consistent equation numerically sampling 20,000 points in momentum space. Besides the metalic phase, we find the phase diagram consists of a superconducting phase with $C=-1$, a superconducting phase with $C=0$ as well as a superconducting phase which hosts Bogoliubov Fermi surfaces. Suprisingly we find that FFLO superconductor with a non-zero momentum $\mathbf Q$ is favored compared to $\mathbf Q=0$.