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 power-law 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.