Topological incommensurate Fulde-Ferrell-Larkin-Ovchinnikov 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 by sampling 20,000 points in the momentum space at a given temperature. Around the Van-Hove singularity (VHS), we find p-ip pairing with Chern number switching from $C=-1$ to $C=0$ through a gap closing at $\mathbf k=(0,0)$ (defined relative to $\mathbf K$). Although the superconductor is generically fully gapped at low temperature, we find Bogoliubov Fermi surface at temperature just below mean field $T_c$. Besides, through calculation of the free energy, we conclude that the optimal Cooper pair momentum $\mathbf Q$ is generically finite and can be as large as $0.1 k_F$. We dub the $\mathbf Q\neq 0$ phase as an incommensurate Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) superconductor to distinguish it from the $\mathbf Q=0$ phase. Compared to the $\mathbf Q=0$ phase, our incommensurate $\mathbf Q$ phase is a nematic superconductor if it is in the Fulde-Ferrell(FF) phase or exhibts charge density wave (CDW) if it is in the Larkin-Ovchinnikov (LO) phase. Our work demonstrates the rhombohedral tetra-layer graphene as a wonderful platform to explore Majorana zero-mode, FFLO physics and Bogoliubov fermi surface within one single platform.