A remarkable recent experiment has observed Mott insulator and proximate superconductor phases in twisted bilayer graphene when electrons partly fill a nearly flat mini-band that arises a 'magic' twist angle. However, the nature of the Mott insulator, origin of superconductivity and an effective low energy model remain to be determined. We propose a Mott insulator with intervalley coherence that spontaneously breaks U(1) valley symmetry, and describe a mechanism that selects this order over the competing magnetically ordered states favored by the Hunds coupling. We also identify symmetry related features of the nearly flat band that are key to understanding the strong correlation physics and constrain any tight binding description. First, although the charge density is concentrated on the triangular lattice sites of the moiré pattern, the Wannier states of the tight-binding model must be centered on different sites which form a honeycomb lattice. Next, spatially localizing electrons derived from the nearly flat band necessarily breaks valley and other symmetries within any mean-field treatment, which is suggestive of a valley-ordered Mott state, and also dictates that additional symmetry breaking is present to remove symmetry-enforced band contacts. Tight-binding models describing the nearly flat mini-band are derived, which highlight the importance of further neighbor hopping and interactions. We discuss consequences of this picture for superconducting states obtained on doping the valley ordered Mott insulator. We show how important features of the experimental phenomenology may be explained and suggest a number of further experiments for the future. We also describe a model for correlated states in trilayer graphene heterostructures and contrast it with the bilayer case. Contents 14 X. Proposed future experiments 15 XI. Conclusion 15 References 16 A. Lattice and symmetries 18 B. Valley Symmetry and Wannier Obstruction 19 C. Wannier functions 19 D. Tight-binding model 21 E. Hartree-Fock theory for selection of IVC ordering 23 F. Spin-orbital model for Mott insulators in trilayer graphene 26 arXiv:1803.09742v2 [cond-mat.str-el]

A remarkable feature of the band structure of bilayer graphene at small twist angle is the appearance of isolated bands near neutrality, whose bandwidth can be reduced at certain magic angles (e.g., θ ∼ 1.05• ). In this regime, correlated insulating states and superconductivity have been experimentally observed. A microscopic description of these phenomena requires an understanding of universal aspects of the band structure, which we discuss here. First, we point out the importance of emergent symmetries, such as valley conservation, which are excellent symmetries in the limit of small twist angles and dictate qualitative features of the band structure. These have sometimes been overlooked when discussing commensurate approximants to the band structure, which we also review here, and solidify their connection with the continuum theory which incorporates all emergent symmetries. Finally, we discuss obstructions to writing tight-binding models of just the isolated bands, and in particular a symmetry-based diagnostic of these obstructions, as well as relations to band topology and strategies for resolving the obstruction. Especially, we construct a four-band model where the two lower isolated bands realize all identified Wannier obstructions of the single-valley nearly flat bands of twisted bilayer graphene.

Correlated insulators and superconductivity have been observed in "magic-angle" twisted bilayer graphene, when the nearly flat bands close to neutrality are partially filled. While a momentum-space continuum model accurately describes these flat bands, interaction effects are more conveniently incorporated in tight-binding models. We have previously shown that no fully symmetric tight-binding model can be minimal, in the sense of capturing just the flat bands, so extended models are unavoidable. Here, we introduce a family of tightbinding models that capture the flat bands while simultaneously retaining all symmetries. In particular, we construct three concrete models with five, six, or ten bands per valley and per spin. These models are also faithful, in that the additional degrees of freedom represent energy bands further away from neutrality, and they serve as optimal starting points for a controlled study of interaction effects. Furthermore, our construction demonstrates the "fragile topology" of the nearly flat bands; i.e., the obstruction to constructing exponentially localized Wannier functions can be resolved when a particular set of trivial bands is added to the model. arXiv:1808.02482v3 [cond-mat.str-el]