A chiral twist on the high-$T_c$ phase diagram in Moir\'e heterostructures

We show that the large orbital degeneracy inherent in Moir\'e heterostructures naturally gives rise to a `high-$T_c$' like phase diagram with a chiral twist - wherein an exotic $\textit{quantum anomalous Hall}$ insulator phase is flanked by chiral $d+id$ superconducting domes. Specifically, we analyze repulsively interacting fermions on hexagonal (triangular or honeycomb) lattices near Van Hove filling, with an ${\rm SU}(N_f)$ flavor degeneracy. This model is inspired by recent experiments on graphene Moir\'e heterostructures. At this point, a nested Fermi surface and divergent density of states give rise to strong ($\ln^2$) instabilities to correlated phases, the competition between which can be controllably addressed through a combination of weak coupling parquet renormalization group and Landau-Ginzburg analysis. For $N_f=2$ (i.e. spin degeneracy only) it is known that chiral $d+id$ superconductivity is the unambiguously leading weak coupling instability. Here we show that $N_f\geq4$ leads to a richer (but still unambiguous and fully controllable) behavior, wherein at weak coupling the leading instability is to a fully gapped and chiral $\textit{Chern insulator}$, characterized by a spontaneous breaking of time reversal symmetry and a quantized Hall response. Upon doping this phase gives way to a chiral $d+id$ superconductor. We further consider deforming this minimal model by introducing an orbital splitting of the Van Hove singularities, and discuss the resulting RG flow and phase diagram. Our analysis thus bridges the minimal model and the practical Moir\'e band structures, thereby providing a transparent picture of how the correlated phases arise under various circumstances. Meanwhile, a similar analysis on the square lattice predicts a phase diagram where (for $N_f>2$) a nodal staggered flux phase with `loop current' order gives way upon doping to a nodal $d$-wave superconductor.