Parquet renormalization group analysis of weak-coupling instabilities with multiple high-order Van Hove points inside the Brillouin zone

We analyze the weak-coupling instabilities that may arise when multiple high-order Van Hove points are present inside the Brillouin zone. The model we consider is inspired by twisted bilayer graphene, although the analysis should be more generally applicable. We employ a parquet renormalization group analysis to identify the leading weak-coupling instabilities, supplemented with a Ginzburg-Landau treatment to resolve any degeneracies. Hence we identify the leading instabilities that can occur from weak repulsion with the powerlaw divergent density of states. Five correlated phases are uncovered along distinct stable fixed trajectories, including s-wave ferromagnetism, p-wave chiral/helical superconductivity, d-wave chiral superconductivity, f -wave valley-polarized order, and p-wave polar valley-polarized order. The phase diagram is stable against band deformations which preserve the high-order Van Hove singularity.