I will present an overview of the current theoretical understanding of the remarkable phenomenology of the twisted bilayer graphene (TBG) near the magic angle. This includes the early insights obtained using the 2D exponentially localized Wannier basis which revealed a qualitative difference between the effect of the interactions in the TBG narrow bands and the much studied narrow band, whose width is small due to the exponentially vanishing overlap of the well separated localized orbitals i.e. an atomic limit. The non-trivial band topology thus leads to a new form of the interaction Hamiltonian projected onto the narrow band basis, showing why the usual Hubbard model anti-ferromagnetic exchange fails and why the system turns ferromagnetic with an approximate spin-valley SU(4) symmetry. I will also present results of the DMRG calculations showing that the strong coupling limit phase diagram of the lattice model introduced in Ref. hosts a period-2 stripe insulator and a Chern insulator at an odd integer filling. Such phases, together with a nematic, are also found in a more detailed DMRG calculation using hybrid Wannier basis.
I will then describe a novel renormalization group procedure that we developed in order to integrate out the remote bands, allowing us to systematically project onto the narrow bands, to quantify the strength of the residual interactions relative to the Coulomb renormalized kinetic energy, and to justify the strong coupling approach. Interestingly, the progressive elimination of remote bands leads to an increase in the Fermi velocity -- well known for the monolayer graphene -- as well as a compensating increase of the AB interlayer tunneling, in precisely such a way as to make the magic angle condition essentially independent of the Coulomb interaction. At the same time, RG reveals a Coulomb induced decrease of the AA relative to the AB tunneling, thus approaching the so-called chiral limit with an enhanced U(4)×U(4) symmetry.
Finally, I will present results of the exact calculation of the single particle, and of charge neutral collective modes, energy spectra from the residual Coloumb interactions within the renormalized narrow bands in the strong coupling limit[5-7]. The softening of a branch of collective modes indeed marks the enlarged U(4)×U(4) symmetry as the chiral limit is approached, while the Coulomb induced dispersion of the single particle bands will be used to explain the experimentally observed cascades of transitions in TBG as cascades between light and heavy fermions[6-7].
 Jian Kang and Oskar Vafek, Phys. Rev. X 8, 031088 (2018)  Jian Kang and Oskar Vafek, Phys. Rev. Lett. 122, 246401 (2019)  Bin-Bin Chen, Yuan Da Liao, Ziyu Chen, Oskar Vafek, Jian Kang, Wei Li, Zi Yang Meng arXiv:2011.07602 (to appear in Nature Communications)  Jian Kang and Oskar Vafek, Phys. Rev. B 102, 035161 (2020)  Oskar Vafek and Jian Kang Phys. Rev. Lett. 125, 257602 (2020)  Jian Kang, B. Andrei Bernevig and Oskar Vafek arXiv:2104:01145  Oskar Vafek and Jian Kang, Phys. Rev. B 104, 075143 (2021)
Recording of this seminar can be found here.