Ordered states near ordinary and higher-order Van Hove points in graphene systems.

We study the crossover from ordinary van Hove singularity (oVHP) to higher order van Hove singularity (hoVHP) in a model used for Bernal bilayer graphene and rhombohedral trilayer graphene. Low energy states in these systems are located near K and K' points on the hexagonal lattice. Under density variation, the electronic structure undergoes a topological van Hove transition from three disconnected Fermi surfaces to a single Fermi surface via either three separate oVHP or a single hoVHP. We model this behavior by introducing a parameter, which allows us to interpolate between oVHP and hoVHP. Our goal is to analyze how the set of potential ordered states changes between oVHP and hoVHP. We apply a parquet remornalization group (RG) analysis to detect the leading instabilities in the particle-hole and particle-particle channels. We find that there is a substantial evolution of the ordered state between oVHP and hoVHP. The evolution includes disappearance/reemergence and pair production/annihilation of the fixed points of RG trajectories.