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We analyze the weak-coupling instabilities that may arise when multiplehigh-order Van Hove points are present inside the Brillouin zone. The model weconsider is inspired by twisted bilayer graphene, although the analysis shouldbe more generally applicable. We employ a parquet renormalization groupanalysis to identify the leading weak-coupling instabilities, supplemented witha Ginzburg-Landau treatment to resolve any degeneracies. Hence we identify theleading instabilities that can occur from weak repulsion with the power-lawdivergent density of states. Five correlated phases are uncovered alongdistinct stable fixed trajectories, including $s$-wave ferromagnetism, $p$-wavechiral/helical superconductivity, $d$-wave chiral superconductivity, $f$-wavevalley-polarized order, and $p$-wave polar valley-polarized order. The phasediagram is stable against band deformations which preserve the high-order VanHove singularity.

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