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Using the continuum model for low energy non-interacting electronic structureof moir\'e van der Waals heterostructures developed by Bistritzer and MacDonald[1], we study the competition between spin, charge, and superconducting orderin twisted bilayer graphene. Surprisingly, we find that for a range of smallangles inclusive of the so-called magic angle, this model features robust Fermipockets that preclude any Mott insulating phase at weak coupling. However, aFermi surface reconstruction at $\theta \gtrsim 1.2^{\circ}$ gives emergent vanHove singularities without any Fermi pockets. Using a hot-spot model for Fermisurface patches around these emergent saddle points, we develop a random-phaseapproximation from which we obtain a phase diagram very similar to thatobtained recently by Isobe, Yuan, and Fu using the parquet renormalizationgroup [2] but with additional insights. For example, our model shows strongnesting around time-reversal symmetric points at a moderate doping of $\sim 2\times 10^{11} \text{cm}^{-2}$ away from the van Hove singularity. When thisnesting dominates, we predict that charge-order enhances singletsuperconductivity, while spin-order suppresses superconductivity. Our theoryalso provides additional possibilities for the case of unnested Fermi surfaces.

Singlet superconductivity enhanced by charge order in nested twisted bilayer graphene Fermi surfaces