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Correlated insulators and superconductivity have been observed in"magic-angle" twisted bilayer graphene, when the nearly flat bands close toneutrality are partially filled. While a momentum-space continuum modelaccurately describes these flat bands, interaction effects are moreconveniently incorporated in tight-binding models. We have previously shownthat no fully symmetric tight-binding model can be minimal, in the sense ofcapturing just the flat bands, so extended models are unavoidable. Here, weintroduce a family of tight-binding models that capture the flat bands whilesimultaneously retaining all symmetries. In particular, we construct threeconcrete models with five, six, or ten bands per valley and per spin. Thesemodels are also faithful, in that the additional degrees of freedom representenergy bands further away from neutrality, and they serve as optimal startingpoints for a controlled study of interaction effects. Furthermore, ourconstruction demonstrates the "fragile topology" of the nearly flat bands;i.e., the obstruction to constructing exponentially localized Wannier functionscan be resolved when a particular set of trivial bands is added to the model.

Faithful Tight-binding Models and Fragile Topology of Magic-angle Bilayer Graphene