Weak Coupling Instabilities in Chern Bands with Multiple Van Hove Singularities

With the advent of fabrication of systems with nanometer scale superlattices, such as nanopatterned superlattices and moiré systems, large magnetic fluxes Φ per unit cell (on the order of the flux quantum Φ₀ = h/e) have become accessible. Such systems exhibit the Hofstadter single particle spectrum composed of Chern bands that generalize the Landau levels, raising the question of potential instabilities due to interactions. We consider systems with square and hexagonal lattices with Φ= (p/q) Φ₀ for various coprime integers p and q that have 2q and 3q Van Hover singularities in each Chern band respectively at which the density of states diverges. We further assume that the chemical potential is tuned to the Van Hove singularities, which corresponds to half-filled Landau levels in the limit of infinite q. We study the resulting instabilities using a parquet renormalization group approach and analyze the competition between superconductivity and charge and spin density wave instabilities.