We performed a random-phase approximation calculation for a spin-valley-polarized model of the rhombohedral tetralayer graphene to study the possibility of a 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, we find $p\text{−}ip$ pairing with the 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 the Bogoliubov Fermi surface at a 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 $0.1k_F$. We dub the $\mathbf{Q}\ne 0$ 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 phase or exhibits a charge density wave if it is in the Larkin-Ovchinnikov phase. Our work demonstrates the rhombohedral tetralayer graphene as a wonderful platform to explore the Majorana zero-mode, FFLO physics, and the Bogoliubov Fermi surface within one single platform.

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• Fermi surface
• Topological order
• Graphene
• Strongly correlated systems
• Superconductors