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Atomistic Hartree theory of twisted double bilayer graphene near the magic angle


 Twisted double bilayer graphene (tDBLG) is a moiré material that has recently generated signiﬁcant interest because of the observation of correlated phases near the magic angle. We carry out atomistic Hartree theory calculations to study the role of electron-electron interactions in the normal state. In contrast to twisted bilayer graphene (tBLG), we ﬁnd that such interactions do not result in signiﬁcant doping-dependent deformations of the electronic band structure. However, interactions play an important role for the electronic structure in the presence of a perpendicular electric ﬁeld as they screen the external ﬁeld. Finally, we analyze the contribution of the Hartree potential to the crystal ﬁeld, i.e. the onsite energy diﬀerence between the inner and outer layers. We ﬁnd that the onsite energy obtained from Hartree theory has the same sign, but a smaller magnitude compared to previous studies who determined the onsite energy by ﬁtting tight-binding results to ab initio density-functional theory (DFT) band structures. To understand this quantitative diﬀerence, we analyze the ab initio Kohn-Sham potential obtained from DFT and ﬁnd that a subtle interplay of electron-electron and electron-ion interactions determines the magnitude of the onsite potential.