Mirror symmetry decomposition in double-twisted multilayer graphene systems

Due to the observed superconductivity, the alternating twisted trilayer graphene (ATTLG) has drawn great research interest very recently, in which three single-layered graphene (SLG) are stacked in an alternating twist way. If one or several of the SLG in ATTLG are replaced by multilayer graphene, we get a double-twisted multilayer graphene (DTMLG). In this work, we theoretically illustrate that, if the DTMLG has a mirror symmetry along the z direction like the ATTLG, there exists a mirror symmetry decomposition (MSD), by which the DTMLG can be exactly decoupled into two subsystems with opposite parity. The two subsystems are either a twisted multilayer graphene (single twist) or a multilayer graphene, depending on the stacking configuration. Such MSD can give a clear interpretation of all the novel features of the moire band structures of DTMLG, e.g., the fourfold degenerate flat bands, and the enlarged magic angle. Meanwhile, in such DTMLG, the parity becomes a new degree of freedom of the electrons, so that we can define a parity-resolved Chern number for the moire flat bands. More importantly, the MSD implies that all the novel correlated phases in the twisted multilayer graphene should also exist in the corresponding DTMLGs since they have the same Hamiltonian in form. Specifically, according to the MSD, we predict that the superconductivity should exist in the (1 + 3 + 1 )-DTMLG.