Density wave and topological superconductivity in the magic-angle-twisted bilayer-graphene

The model dependence in the study of the magic-angle twisted bilayer-graphene (MA-TBG) is an important issue in the research area. It has been argued previously that the two-band tight-binding (TB) model (per spin and valley) cannot serve as a start point for succeeding studies as it cannot correctly describe the topological aspect of the continuum-theory model near the Dirac nodes in the mini Brillouin zone (MBZ). For this purpose, we adopt the faithful TB model [Phys. Rev. B 99 195455 (2019)] with five bands (per spin and valley) as our start point, which is further equipped with extended Hubbard interactions. Then after systematic random-phase-approximation (RPA) based calculations, we study the electron instabilities of this model, including the density wave (DW) and superconductivity (SC), near the van Hove singularity (VHS). Our results are as follows. In the case neglecting the tiny inter-valley exchange interaction, the exact SU(2)K × SU(2)K′ symmetry leads to the degeneracy between the inter-valley charge DW (CDW) and the spin DW (SDW) (which would be mixed then), and that between the singlet d + id-wave and triplet p + ip-wave topological SCs. When a realistic tiny inter-valley exchange interaction is turned on with nonzero coefficient (J H ≠ 0), the SDW or CDW is favored respectively at the critical point, determined by J H → 0- or J H → 0+. In the mean time, the degeneracy between the singlet d + id-wave and triplet p + ip-wave topological SCs is also lifted up by the tiny J H. These results are highly similar to the results of our previous study [arXiv:2003.09513] adopting the two-band TB model, with the reason lying in that both models share the same symmetry and Fermi-surface (FS) nesting character near the VHS. Such a similarity suggests that the low-energy physics of the doped MA-TBG is mainly determined by the symmetry and the shape of the FS of the doped system, and is insensitive to other details of the band structure, including the topological aspects near the Dirac nodes in the MBZ.